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+arXiv:2301.08418v1  [math.CT]  20 Jan 2023
+Measurings of Hopf algebroids and morphisms in cyclic (co)homology
+theories
+Abhishek Banerjee *
+Surjeet Kour †
+Abstract
+In this paper, we consider measurings between Hopf algebroids and show that they induce morphisms on cyclic homology
+and cyclic cohomology. We also consider comodule measurings between SAYD modules over Hopf algebroids. These measur-
+ings induce morphisms on cyclic (co)homology of Hopf algebroids with SAYD coefficients. Finally, we obtain morphisms on
+cyclic homology induced by measurings of cyclic comp modules over operads with multiplication.
+MSC(2020) Subject Classification: 16T15, 16E40, 18D50
+Keywords: Hopf algebroids, cyclic (co)homology, SAYD modules, comp modules
+1
+Introduction
+Let k be a field, C be a k-coalgebra and A, B be k-algebras. In [20], Sweedler introduced the notion of a coalgebra measuring
+as a kind of generalized morphism between algebras. More precisely, a C-measuring from A to B consists of a k-linear map
+φ : C −→ Vectk(A, B) satisfying
+φ(c)(aa′) =
+�
+φ(c(1)(a)φ(c(2))(a′)
+φ(c)(1) = ǫ(c)1
+(1.1)
+for any a, a′ ∈ A. Here, ∆(c) = � c(1) ⊗ c(2) denotes the coproduct on C and ǫ : C −→ k denotes the counit. Since then, the
+notion of a measuring has been widely studied in the literature by several authors (see, for instance, [1], [3], [4], [9], [10], [11],
+[12],[22], [23], [24]).
+In [2], we studied how coalgebra measurings induce morphisms between Hochschild homology groups of algebras. The purpose
+of this paper is to take this idea one step further. Our aim is to consider cocommutative coalgebra morphisms between Hopf
+algebroids and show that they induce morphisms in cyclic homology and cyclic cohomology. We begin by showing that there are
+universal measurings which give an enrichment of the category HAlgk of Hopf algebroids over the category of cocommutative
+coalgebras. If U is a Hopf algebroid, the cyclic module C•(U) defining its cyclic homology groups as well as the cocyclic
+module C•(U) defining its cyclic cohomology groups are defined in [13]. We show that a cocommutative coalgebra measuring
+between two Hopf algebroids induces morphisms on their corresponding cyclic homology and cyclic cohomology groups. If a
+Hopf algebroid is commutative, we know from [15] that there is a shuffle product on its Hochschild homology groups. We show
+that a measuring between commutative Hopf algebroids induces an algebra measuring between the corresponding Hochschild
+homology rings with respct to this shuffle product. This also gives us an enrichment of commutative Hopf algebroids over
+cocommutative coalgebras.
+Thereafter, we consider comodule measurings of SAYD modules over Hopf algebroids. Accordingly, we obtain an enrichment
+of the “global category” of SAYD modules (see Theorem 5.9) over the “global category” of comodules. The cyclic homology
+*Department of Mathematics, Indian Institute of Science, Bangalore. Email: abhishekbanerjee1313@gmail.com
+†Department of Mathematics, Indian Institute of Technology, Delhi. Email: koursurjeet@gmail.com
+1
+
+and the cyclic cohomology of a Hopf algebroid with coefficients in an SAYD module was defined in [14]. We show that a
+comodule measuring induces morphisms between cyclic (co)homology with SAYD coefficients. In the final part of the paper,
+we work with pairs of the form (O, M), where M is a cyclic unital comp module over a non-Σ operad O with multiplication
+in the sense of [16]. The cyclic homology groups of such a comp module were also defined in [16]. We consider comodule
+measurings between such pairs and show that they induce morphisms in cyclic homology.
+2
+Measurings of Hopf algebroids
+Throughout, k is a field and let Vectk be the category of k-vector spaces. Let A be a unital k-algebra. In order to define left
+and right bialgebroids, as well as Hopf algebroids in later sections, we will frequently need both the algebra A and its opposite
+algebra Aop. For this, we will often write the algebra A as AL, while Aop will often be written as AR.
+An (s, t)-ring over A consists of a unital k-algebra U along with two k-algebra morphisms s : A −→ U and t : Aop −→ U whose
+images commute in U, i.e., s(a1)t(a2) = t(a2)s(a1) for any a1, a2 ∈ U. The morphisms s and t are often referred to as source
+and target maps respectively. These morphisms introduce an (A, A)-bimodule structure on U given by left multiplication
+a1 · h · a2 := s(a1)t(a2)h
+a1, a2 ∈ A, h ∈ H
+(2.1)
+The left and right A-module structures on U in (2.1) allow us to consider the tensor product U ⊗A U. The following subspace
+of U ⊗A U is known as the Takeuchi product
+U ×A U := {� ui ⊗A u′
+i ∈ U ⊗A U | � uit(a) ⊗A u′
+i = � ui ⊗A u′
+is(a), ∀ a ∈ A}
+(2.2)
+It is well known (see, for instance, [13, § 2]) that the Takeuchi product U ×A U is a unital subalgebra of U ⊗A U.
+From now onwards, we also fix a unital k-algebra U. The multiplication on U will be denoted by µU. Since the category of
+(A, A)-bimodules is monoidal, we can consider coalgebra objects in this category. We now recall the notion of a left Hopf
+algebroid (see, for instance, [5], [13], [21]). For several closely related notions, see [18], [19].
+Definition 2.1. A left bialgebroid UL := (U, AL, sL, tL, ∆L, ǫL) over k consists of the following data:
+(1) A unital k-algebra AL
+(2) A unital k-algebra U which carries the structure of an (sL, tL) ring over AL.
+(3) A coalgebra object (U, ∆L : U −→ U ⊗AL U, ǫL : U −→ AL) in the category of (AL, AL)-bimodules satisfying the following
+conditions:
+(i) ∆L : U −→ U ⊗AL U factors through U ×A U ⊆ U ⊗AL U.
+(ii) ǫL(usL(ǫL(u′))) = ǫL(uu′) = ǫL(utL(ǫL(u′))) for all u, u′ ∈ H.
+A morphism (F, f) : (U, AL, sL, tL, ∆L, ǫL) = UL −→ U′
+L = (U′, A′
+L, s′
+L, t′
+L, ∆′
+L, ǫ′
+L) of left bialgebroids consists of a pair of
+k-algebra morphisms F : U −→ U′ and f : AL −→ A′
+L such that
+F ◦ sL = s′
+L ◦ f
+F ◦ tL = t′
+L ◦ f
+∆′
+L ◦ F = (F ⊗φ F) ◦ ∆L
+f ◦ ǫL = ǫ′
+L ◦ F
+(2.3)
+We will denote the category of left bialgebroids over k by LBialgk.
+If UL = (H, AL, sL, tL, ∆L, ǫL) is a left bialgebroid, we employ standard Sweedler notation to write ∆L(u) = � u(1) ⊗ u(2) for any
+u ∈ H and suppress the summation sign throughout. We now recall the notion of Hopf algebroid from [5, Definition 4.1].
+Definition 2.2. A Hopf algebroid U = (UL, S ) over k consists of the following data:
+(1) A left bialgebroid UL = (U, AL, sL, tL, ∆L, ǫL) over k.
+(2) An involutive anti-automorphism S : U −→ U of the k-algebra U which satisfies S ◦ tL = sL as well as
+S (u(1))(1)u(2) ⊗ S (u(1))(2) = 1H ⊗ S (u)
+S (u2)1 ⊗ S (u(2))(2)u(1) = S (u) ⊗ 1U
+(2.4)
+2
+
+as elements of U ⊗AL U, for all u ∈ U.
+A morphism (F, f) : U = (UL, S ) −→ (U′
+L, S ′) = U′ of Hopf algebroids is a morphism in LBialgk that also satisfies S ′◦F = f ◦S .
+We will denote the category of Hopf bialgebroids over k by HAlgk.
+We remark here that in this paper we will always assume the antipode on a Hopf algebroid U = (UL, S ) is involutive, i.e.,
+S 2 = id. However, this condition is not part of the original definition due to B¨ohm and Szlach´anyi in [5]. Further, it is shown in
+[5, Proposition 4.2] that a Hopf algebroid (UL, S ) is equivalent to a datum consisting of a left bialgebroid and a right bialgebroid
+connected by an antipode.
+We now recall the classical notion of a coalgebra measuring due to Sweedler [20]. Let R, R′ be k-algebras and C be a k-
+coalgebra. Then, a C-measuring from R to R′ consists of a morphism ψ : C −→ Vectk(R, R′) such that
+ψ(x)(ab) =
+�
+ψ(x(1))(a)ψ(x(2))
+ψ(x)(1R) = ǫC(x)1R′
+∀ a, b ∈ R
+(2.5)
+where the coproduct ∆C : C −→ C ⊗ C is given by ∆C(x) = � x(1) ⊗ x(2) for any x ∈ C and ǫC : C −→ k is the counit. The
+measuring as in (2.5) is said to be cocommutative if the coalgebra C is cocommutative. In this paper, we will only consider
+cocommutative measurings. By abuse of notation, if ψ : C −→ Vectk(R, R′) is a coalgebra measuring, we will often write the
+morphism ψ(x) ∈ Vectk(R, R′) simply as c : R −→ R′ for any x ∈ C.
+We are now ready to introduce the notion of measuring between Hopf algebroids.
+Definition 2.3. Let U = (U, S ) = (U, AL, sL, tL, ∆L, ǫL, S ) and U′ = (U′, S ′) = (U′, AL, s′
+L, t′
+L, ∆′
+L, ǫ′
+L, S ′) be Hopf algebroids
+over k. Let C be a cocommutative k-coalgebra. A C-measuring (Ψ, ψ) from U to U′ consists of a pair of measurings
+Ψ : C −→ Vectk(U, U′)
+ψ : C −→ Vectk(AL, A′
+L)
+(2.6)
+such that the following diagrams commute for any x ∈ C
+AL
+sL
+−−−−−−→ U
+c
+�
+�c
+A′
+L
+s′
+L
+−−−−−−→ U′
+AL
+tL
+−−−−−−→ U
+c
+�
+�c
+A′
+L
+t′
+L
+−−−−−−→ U′
+U
+S
+−−−−−−→ U
+c
+�
+�c
+U′
+S ′
+−−−−−−→ U′
+(2.7)
+U
+ǫL
+−−−−−−→ AL
+c
+�
+�c
+U′
+ǫ′
+L
+−−−−−−→ A′
+L
+U
+∆L
+−−−−−−→ U ⊗AL U
+c
+�
+�c
+U′
+∆′
+L
+−−−−−−→ U′ ⊗A′
+L U′
+(2.8)
+where the arrow c : U ⊗AL U −→ U′ ⊗A′
+L U′ is defined by setting c(u1 ⊗ u2) := x(1)(h) ⊗ x(2)(u2) for u1 ⊗ u2 ∈ U ⊗AL U.
+Before proceeding further, we need to verify the following fact.
+Lemma 2.4. For any x ∈ C, the morphism c : H ⊗AL U −→ U′ ⊗A′
+L U′ defined by setting c(u1 ⊗ u2) := x(1)(h) ⊗ x(2)(u2) for
+u1 ⊗ u2 ∈ U ⊗AL U is well-defined.
+Proof. We consider u1, u2 ∈ uL and a ∈ AL. Using the fact that Ψ : C −→ Vectk(U, U′) is a measuring and applying the
+conditions in (2.7) and (2.8), we see that
+c((u1 · a) ⊗ u2) = c(tL(a)u1 ⊗ u2)
+= x(1)(tL(a)u1) ⊗ x(2)(u2) = x(1)(tL(a))x(2)(u1) ⊗ x(3)(u2)
+= t′
+L(x(1)(a))x(2)(u1) ⊗ x(3)(u2)
+= t′
+L(x(2)(a))x(1)(u1) ⊗ x(3)(u2)
+(because C is cocommutative)
+= x(1)(u1) · x(2)(a) ⊗ x(3)(u2) = x(1)(u1) ⊗ x(2)(a) · x(3)(u2)
+= x(1)(u1) ⊗ s′
+L(x(2)(a))x(3)(u2) = x(1)(u1) ⊗ x(2)(sL(a))x(3)(u2)
+= x(1)(u1) ⊗ x(2)(sL(a)u2) = x(1)(u1) ⊗ x(2)(a · u2)
+= c(u1 ⊗ (a · u2))
+(2.9)
+□
+3
+
+If U = (UL, S ) and U′ = (U′
+L, S ′) are Hopf algebroids over k, we now consider the subspace
+V(U, U′) ⊆ Vectk(U, U′) × Vectk(AL, A′
+L)
+(2.10)
+given by setting
+V(U, U′) := {(F, f) | FsL = s′
+L f, FtL = t′
+L f, FS = S ′F and fǫL = ǫ′
+LF }
+(2.11)
+We note that a measuring from U to U′ by means of a cocommutative coalgebra C has an underlying morphism (Ψ, ψ) : C −→
+V(U, U′).
+Let Coalgk denote the category of k-coalgebras. We know that the forgetful functor Coalgk −→ Vectk has a right adjoint
+C : Vectk −→ Coalgk. In other words, we have natural isomorphisms
+Vectk(C, V) � Coalgk(C, C(V))
+(2.12)
+for any k-coalgebra C and any k-vector space V.
+Proposition 2.5. Let U = (UL, S ) and U′ = (U′
+L, S ′) be Hopf algebroids over k. Then, there exists a cocommutative coalgebra
+Mc(U, U′) and a measuring (Φ, φ) from U to U′ satisfying the following universal property: given any measuring (Ψ, ψ) : C −→
+V(U, U′) with a cocommutative coalgebra C, there exists a unique morphism ξ : C −→ Mc(U, U′) of coalgebras making the
+following diagram commutative
+Mc(U, U′)
+(Φ,φ)
+� V(U, U′)
+C
+ξ
+�■■■■■■■■■■
+(Ψ,ψ)
+�✇
+✇
+✇
+✇
+✇
+✇
+✇
+✇
+✇
+(2.13)
+Proof. We set V := V(U, U′) and consider the canonical morphism π(V) : C(V) −→ V ⊆ Vectk(uL, u′
+L) × Vectk(AL, A′
+L) from
+the cofree coalgebra C(V) induced by the adjunction in (2.12). We now set Mc(U, U′) := � D, where the sum is taken over all
+cocommutative subcoalgebras of C(V) such that the restriction π(V)|D : D −→ V = V(U, U′) is a measuring. It is clear that this
+sum is still a cocommutative coalgebra, and that the restriction (Φ, φ) := π(V)|Mc(U,U′) gives a measuring from U to U′.
+In general, if (Ψ, ψ) : C −→ V = V(U, U′) is a cocommutative measuring, the adjunction in (2.12) shows that it factors through
+ξ : C −→ C(V). Then, ξ(C) ⊆ C(V) is a cocommutative coalgebra such that the restriction π(V)|ξ(C) is a measuring. By
+definition, it follows that ξ(C) ⊆ Mc(U, U′). This proves the result.
+□
+From (2.10) and (2.11) it is clear that given Hopf algebroids U = (UL, S ), U′ = (U′
+L, S ′) and U′′ = (U′′
+L, S ′′), the composition
+of morphisms induces a canonical map
+V(U, U′) ⊗ V(U′, U′′)
+◦
+−→ V(U, U′′)
+(2.14)
+We denote by CoCoalgk the category of cocommutative coalgebras over k. We know that this category is symmetric monoidal
+and our objective is to show that the category HAlgk of Hopf algebroids is enriched over CoCoalgk. For this we need the
+following result.
+Proposition 2.6. Let U = (UL, S ), U′ = (U′
+L, S ′) and U′′ = (U′′
+L, S ′′) be Hopf algebroids over k. Suppose that we have a
+measuring (Ψ, ψ) : C −→ V(U, U′) from U to U′ and a measuring (Ψ′, ψ′) : C′ −→ V(U′, U′′) from U′ to U′′. Then, the
+following
+(Ψ′, ψ′) ◦ (Ψ, ψ) : C ⊗ C′ (Ψ,ψ)⊗(Ψ′,ψ′)
+−−−−−−−−−−→ V(U, U′) ⊗ V(U′, U′′)
+◦
+−→ V(U, U′′)
+(2.15)
+determines a measuring from U to U′′.
+Proof. It is easy to verify that the compositions
+C ⊗ C′ Ψ⊗Ψ′
+−−−−→ Vectk(U, U′) ⊗ Vectk(U′, U′′)
+◦
+−−−−−−→ Vectk(U, U′′)
+C ⊗ C′ ψ⊗ψ′
+−−−−→ Vectk(AL, A′
+L) ⊗ Vectk(A′
+L, A′′
+L)
+◦
+−−−−−−→ Vectk(AL, A′′
+L)
+(2.16)
+4
+
+give coalgebra measurings from U to U′′ and from AL to A′′
+L respectively. For c ⊗ c′ ∈ C ⊗ C′ and u ∈ U, we also see that
+∆′′
+L((c ⊗ c′)(u)) = ∆′′
+L(c′(c(u)))
+= c′
+(1)(c(u)(1)) ⊗ c′
+(2)(c(u)(2))
+= c′
+(1)(x(1)(u(1))) ⊗ c′
+(2)(x(2)(u(2)))
+= (c′ ⊗ c)(1)(u(1)) ⊗ (c′ ⊗ c)(2)(u(2))
+(2.17)
+It is also clear that the morphism in (2.15) satisfies all the other conditions in Definition 2.3. This proves the result.
+□
+Theorem 2.7. The category HAlgk of Hopf algebroids is enriched over the category CoCoalgk of cocommutative k-coalgebras.
+Proof. Given Hopf algebroids U = (UL, S ) and U′ = (U′
+L, S ′), we consider the “hom object” Mc(U, U′) which lies in CoCoalgk.
+The composition of these hom objects is obtained as follows: if U, U′ and U′′ are Hopf algebroids, we obtain as in Proposition
+2.6 a measuring
+Mc(U, U′) ⊗ Mc(U′, U′′) −→ V(U, U′′)
+(2.18)
+Applying the universal property in Proposition 2.5, we now have a morphism of coalgebras Mc(U, U′) ⊗ Mc(U′, U′′) −→
+Mc(U, U′′). The unit object in CoCoalgk is k treated as a coalgebra over itself. Then, we have a unit map
+k −→ V(U, U) ⊆ Vectk(U, U) × Vectk(AL, AL)
+t �→ (t · iduL, t · idAL)
+(2.19)
+which induces a morphism k −→ Mc(U, U) of cocommutative coalgebras. Together with the composition of hom objects in
+(2.18), we see that HAlgk is enriched over CoCoalgk.
+□
+From now onwards, we will denote by HALGk the category of Hopf algebroids enriched over the symmetric monoidal category
+CoCoalgk of cocommutative k-algebras.
+3
+Morphisms on cyclic (co)homology and Hopf-Galois maps
+Let U = (UL, S ) = (U, AL, sL, tL, ∆L, ǫL) be a Hopf algebroid over k. We now recall from [13, § 2] the cocyclic module C•(U)
+that computes the cyclic cohomology of the Hopf algebroid U. For n ≥ 1, we put
+Cn(U) := U ⊗AL ⊗ · · · ⊗AL U
+��������������������������������������
+n-times
+(3.1)
+and set C0(U) := AL. For n ≥ 1, the face maps δi : Cn(U) −→ Cn+1(U) are defined by
+δi(u1 ⊗ ... ⊗ un) :=
+
+1 ⊗ u1 ⊗ ... ⊗ un
+if i = 0
+u1 ⊗ .... ⊗ ∆Lui ⊗ ... ⊗ un
+if 1 ≤ i ≤ n
+u1 ⊗ .... ⊗ un ⊗ 1
+if i = n + 1
+(3.2)
+For n = 0, there are two maps δ0 := tL : C0(U) = AL −→ C1(U) = U and δ1 := sL : C0(U) = AL −→ C1(U) = U. The
+degeneracy maps σi : Cn(U) −→ Cn−1(U) are given by
+σi(u1 ⊗ ... ⊗ un) := u1 ⊗ ... ⊗ ǫL(ui+1) · ui+2 ⊗ ... ⊗ un
+0 ≤ i ≤ n − 1
+(3.3)
+The cyclic operator τn : Cn(U) −→ Cn(U) is defined by setting
+τn(u1 ⊗ ... ⊗ un) := (S (u1)(1) · u2) ⊗ .... ⊗ (S (u1)(n−1) · un) ⊗ S (u1)(n)
+(3.4)
+Since we have assumed that the antipode S is involutive, it follows from [13, Theorem 2.1] that C•(U) is indeed a cocyclic
+module. We will denote by HC•(U) the cyclic cohomology groups of the Hopf algebroid U by U. The Hochschild cohomology
+groups of the Hopf algebroid U will then be denoted by HH•(U).
+5
+
+Let U, U′ be Hopf algebroids and let (Ψ, ψ) : C −→ V(U, U′) be a measuring from U to U′. For each x ∈ C, we now define a
+family of morphisms
+Ψ
+n(x) : Cn(U) −→ Cn(U′)
+Ψ
+n(x)(u1 ⊗ ... ⊗ un) := x(u1 ⊗ ... ⊗ un) = x(1)(u1) ⊗ ... ⊗ x(n)(un)
+∀ n ≥ 0
+(3.5)
+We now prove the first main result of this section.
+Proposition 3.1. Let C be a cocommutative coalgebra and let (Ψ, ψ) : C −→ V(U, U′) be a measuring of Hopf algebroids.
+For each x ∈ C, the family {Ψ
+n(x) : Cn(U) −→ Cn(U′)}n≥0 gives a morphism of cyclic modules. In particular, we have induced
+morphisms
+Ψ
+•
+hoc(x) : HH•(U) −→ HH•(U)
+Ψ
+•
+cy(x) : HC•(U) −→ HC•(U)
+(3.6)
+on Hochschild and cyclic cohomologies for each x ∈ C.
+Proof. For each x ∈ C, we start by showing that Ψ
+n+1(x) ◦ δi = δ′
+i ◦ Ψ
+n(x) : Cn(U) −→ Cn+1(H ′), where δi and δ′
+i are the face
+maps on the respective cocyclic modules C•(U) and C•(U′). If i = 0 or i = n + 1, this is immediately clear from the definition
+in (3.2) and the action in (3.5). For 1 ≤ i ≤ n, we see that
+Ψ
+n+1(x) ◦ δi(u1 ⊗ ... ⊗ un)
+= Ψ
+n+1(x)(u1 ⊗ .... ⊗ ∆Lui ⊗ ... ⊗ un)
+= x(1)(u1) ⊗ ... ⊗ x(i)(ui
+(1)) ⊗ x(i+1)(ui
+(2)) ⊗ .... ⊗ x(n+1)(un)
+= x(1)(u1) ⊗ ∆L(x(i)(ui)) ⊗ ... ⊗ x(n)(un) = δ′
+i ◦ Ψ
+n(x)(u1 ⊗ ... ⊗ un)
+Next, we verify that Ψ
+n−1(x) ◦ σi = σ′
+i ◦ Ψ
+n(x), where σi and σ′
+i are the degeneracies on the respective cocyclic modules C•(U)
+and C•(U′).
+Ψ
+n−1(x) ◦ σi(u1 ⊗ ... ⊗ un)
+= Ψ
+n−1(x)(u1 ⊗ ... ⊗ ǫL(ui+1) · ui+2 ⊗ ... ⊗ un)
+= x(1)(u1) ⊗ ... ⊗ x(i+1)(ǫL(ui+1) · ui+2) ⊗ ... ⊗ x(n−1)(un)
+= x(1)(u1) ⊗ ... ⊗ x(i+1)(ǫL(ui+1)) · x(i+2)(ui+2) ⊗ ... ⊗ x(n)(un)
+= x(1)(u1) ⊗ ... ⊗ ǫL(x(i+1)(ui+1)) · x(i+2)(ui+2) ⊗ ... ⊗ x(n)(un)
+= σ′
+i ◦ Ψ
+n(x)(u1 ⊗ ... ⊗ un)
+Finally, we show that Ψ
+n(x)◦τn = τ′
+n ◦Ψ
+n(x), where τn and τ′
+n are the cyclic operators on the respective cocyclic modules C•(U)
+and C•(U′).
+Ψ
+n(x) ◦ τn(u1 ⊗ ... ⊗ un)
+= Ψ
+n(x)((S (u1)(1) · u2) ⊗ .... ⊗ (S (u1)(n−1) · un) ⊗ S (u1)(n))
+= x(1)((S (u1)(1) · u2)) ⊗ .... ⊗ x(n−1)((S (u1)(n−1) · un) ⊗ x(n)(S (u1)(n))
+= x(1)(S (u1)(1)) · x(2)(u2) ⊗ .... ⊗ x(2n−3)(S (u1)(n−1)) · x(2n−2)(un) ⊗ x(2n−1)(S (u1)(n))
+= x(1)(S (u1)(1)) · x(n+1)(u2) ⊗ .... ⊗ x(n−1)(S (u1)(n−1)) · x(2n−1)(un) ⊗ x(n)(S (u1)(n))
+= (x(1)(S (u1)))(1) · x(2)(u2) ⊗ .... ⊗ (x(1)(S (u1)))(n−1) · x(n)(un) ⊗ (x(1)(S (u1)))(n)
+= (S (x(1)(u1)))(1) · x(2)(u2) ⊗ .... ⊗ (S (x(1)(u1)))(n−1) · x(n)(un) ⊗ (S (x(1)(u1)))(n)
+= τ′
+n ◦ Ψ
+n(x)(u1 ⊗ ... ⊗ un)
+□
+We continue with a Hopf algebroid U = (UL, S ) = (U, AL, sL, tL, ∆L, ǫL). As mentioned in Section 2, we set AR := Aop
+L = Aop.
+Following [5, § 4], we also set
+sR := tL
+tR := S ◦ tL = sL
+(3.7)
+Then, U becomes an (AR, AR)-bimodule by right multiplication as follows
+a1 · h · a2 := hsR(a2)tR(a1) = htL(a2)sL(a1)
+h ∈ H, a1, a2 ∈ AR
+(3.8)
+We now consider
+S rl : H ⊗AR H −→ H ⊗AL H
+u1 ⊗ u2 �→ S (u2) ⊗ S (u1)
+S lr := S −1
+rl : H ⊗AL H −→ H ⊗AR H
+u1 ⊗ u2 �→ S (u2) ⊗ S (u1)
+(3.9)
+6
+
+as well as
+∆R := S lr ◦ ∆L ◦ S : U −→ U ⊗AR U
+ǫR := ǫL ◦ S : U −→ AR
+(3.10)
+We know from [5, § 4] that the datum UR := (U, AR, sR, tR, ∆R, ǫR) defines a right bialgebroid over k. We now adopt the Sweedler
+notation ∆R(u) := u[1] ⊗ u[2] for any u ∈ U in order to distinguish it from the left coproduct ∆L(u) = u(1) ⊗ u(2). More explicitly,
+we have
+∆R(u) = u[1] ⊗ u[2] = S (S (u)(2)) ⊗ S (S (u)(1))
+∀ u ∈ U
+(3.11)
+Now let U, U′ be Hopf algebroids and consider (F, f) ∈ V(U, U′). From the conditions in (2.11) and the definitions in (3.7) and
+(3.10), we already have
+FsR = s′
+R f
+FtR = t′
+R f
+FS = S ′F
+fǫR = ǫ′
+RF
+(3.12)
+We now need the following result.
+Lemma 3.2. Let C be a cocommutative coalgebra and (Ψ, ψ) : C −→ V(U, U′) a measuring of Hopf algebroids. Then, for
+each x ∈ C, there is a well defined morphism
+x : U −→ U ⊗AR U
+u1 ⊗ u2 �→ x(1)(u1) ⊗ x(2)(u2)
+(3.13)
+which fits into the following commutative diagram
+U
+∆R
+−−−−−−→ U ⊗AR U
+x
+�
+�x
+U′
+∆′
+R
+−−−−−−→ U′ ⊗A′
+R U′
+(3.14)
+Proof. We consider u1, u2 ∈ uL and a ∈ AR. Using the fact that Ψ : C −→ Vectk(U, U′) is a measuring and applying the
+conditions in (3.12), we see that
+c((u1 · a) ⊗ u2) = c(u1sR(a) ⊗ u2)
+= x(1)(u1sR(a)) ⊗ x(2)(u2) = x(1)(u1)x(2)(sR(a)) ⊗ x(3)(u2)
+= x(1)(u1)s′
+R(x(2)(a)) ⊗ x(3)(u2)
+= x(1)(u1) · x(2)(a) ⊗ x(3)(u2)
+= x(1)(u1) ⊗ x(2)(a) · x(3)(u2) = x(1)(u1) ⊗ x(3)(u2)t′
+R(x(2)(a))
+= x(1)(u1) ⊗ x(3)(u2)x(2)(tR(a))
+= x(1)(u1) ⊗ x(2)(u2)x(3)(tR(a))
+(as C is cocommutative)
+= x(1)(u1) ⊗ x(2)(u2tR(a)) = x(1)(u1) ⊗ x(2)(a · u2) = c(u1 ⊗ (a · u2))
+It follows that the morphism in (3.13) is well defined. It remains to verify the condition in (3.14). For u ∈ U and x ∈ C, we
+have
+c(∆R(u)) = c(S (S (u)(2)) ⊗ S (S (u)(1)))
+= x(1)(S (S (u)(2))) ⊗ x(2)(S (S (u)(1)))
+= S ′(x(1)(S (u)(2))) ⊗ S ′(x(2)(S (u)(1)))
+= S ′(x(2)(S (u)(2))) ⊗ S ′(x(1)(S (u)(1)))
+(as C is cocommutative)
+= S ′(c(S (u))(2)) ⊗ S ′(c(S (u))(1))
+= S ′(S ′(c(u))(2)) ⊗ S ′(S ′(c(u))(1)) = ∆′
+R(c(u))
+□
+We now recall from [13, § 2.3.1] the cyclic module C•(U) defining the cyclic homology of a Hopf algebroid U. For n ≥ 0, we
+set
+Cn(U) := U ⊗AR ⊗ · · · ⊗AR U
+��������������������������������������
+n-times
+(3.15)
+7
+
+and C0(U) := AR. The face maps di : Cn(U) −→ Cn−1(U) are defined by setting
+di(u1 ⊗ ... ⊗ un) :=
+
+ǫR(u1)u2 ⊗ ... ⊗ un
+if i = 0
+u1 ⊗ ... ⊗ uiui+1 ⊗ ... ⊗ un
+if i ≤ i ≤ n − 1
+u1 ⊗ ... ⊗ un−1ǫR(S (un))
+if i = n
+(3.16)
+The degeneracies si : Cn(U) −→ Cn+1(U) are defined as
+si(u1 ⊗ ... ⊗ un) :=
+� 1 ⊗ u1 ⊗ ... ⊗ un
+if i = 0
+u1 ⊗ ... ⊗ ui ⊗ 1 ⊗ ui+1 ⊗ .... ⊗ un
+if 1 ≤ i ≤ n
+(3.17)
+The cyclic operators tn : Cn(U) −→ Cn(U) are given by
+tn(u1 ⊗ ... ⊗ un) := S (u1
+(2)...un−1
+(2) un) ⊗ u1
+(1) ⊗ u2
+(1) ⊗ ... ⊗ un−1
+(1)
+(3.18)
+The Hochschild homology groups of the Hopf algebroid U will then be denoted by HH•(U) and the cyclic homology groups
+by HC•(U). We will now prove the homological counterpart for Proposition 3.1.
+Proposition 3.3. Let C be a cocommutative coalgebra and let (Ψ, ψ) : C −→ V(U, U′) be a measuring of Hopf algebroids. For
+each x ∈ C, the family
+Ψn(x) : Cn(U) −→ Cn(U′)
+u1 ⊗ ... ⊗ un �→ x(u1 ⊗ ... ⊗ un) = x(1)(u1) ⊗ ... ⊗ x(n)(un)
+(3.19)
+for n ≥ 0 gives a morphism of cyclic modules. In particular, we have induced morphisms
+Ψhoc
+• (x) : HH•(U) −→ HH•(U′)
+Ψcy
+• (x) : HC•(U) −→ HC•(U′)
+(3.20)
+on Hochschild and cyclic homologies for each x ∈ C.
+Proof. Using the properties in (3.12) and the fact that Ψ : C −→ Vectk(U, U′) is a measuring, it may easily be verified that the
+maps Ψ•(x) commute with the respective face maps and degeneracy maps on the cyclic modules C•(U) and C•(U′). Moreover,
+if tn and t′
+n are the respective cyclic operators on C•(U) and C•(U′), we have for each x ∈ C
+c(tn(u1 ⊗ ... ⊗ un))
+= c(S (u1
+(2)...un−1
+(2) un) ⊗ u1
+(1) ⊗ u2
+(1) ⊗ ... ⊗ un−1
+(1) )
+= x(1)(S (u1
+(2)...un−1
+(2) un)) ⊗ x(2)(u1
+(1)) ⊗ x(3)(u2
+(1)) ⊗ ... ⊗ x(n)(un−1
+(1) )
+= S ′(x(1)(u1
+(2))...x(n−1)(un−1
+(2) )x(n)(un)) ⊗ x(n+1)(u1
+(1)) ⊗ ... ⊗ x(2n−1)(un−1
+(1) )
+= S ′(x(2)(u1
+(2))...x(2n−2)(un−1
+(2) )x(2n−1)(un)) ⊗ x(1)(u1
+(1)) ⊗ ... ⊗ x(2n−3)(un−1
+(1) )
+= S ′(x(1)(u1)(2)...x(n−1)(un−1)(2)x(n)(un)) ⊗ x(1)(u1)(1) ⊗ ... ⊗ x(n−1)(un−1)(1)
+= t′
+n(x(1)(u1) ⊗ ... ⊗ x(n)(un))
+□
+Our final aim in this section is to show that the morphisms induced by a measuring of Hopf algebroids are well behaved with
+respect to cyclic duality. More precisely, we know from [13, § 2.3.3] that there are Hopf-Galois maps
+ξn(U) : Cn(U)
+�
+−→ Cn(U)
+u1 ⊗ ... ⊗ un �→ u1
+(1) ⊗ u1
+(2)u2
+(1) ⊗ u1
+(3)u2
+(2)u3
+(1) ⊗ ... ⊗ u1
+(n)u2
+(n−1)....un−1
+(2) un
+(3.21)
+inducing isomorphisms between C•(U) and C•(U). We now have the following result.
+Proposition 3.4. Let C be a cocommutative coalgebra and let (Ψ, ψ) : C −→ V(U, U′) be a measuring of Hopf algebroids.
+Then for each x ∈ C, the following diagram commutes
+Cn(U)
+ξn(U)
+−−−−−−→ Cn(U)
+Ψn(x)
+�
+�Ψ
+n(x)
+Cn(U)
+ξn(U)
+−−−−−−→ Cn(U)
+(3.22)
+8
+
+Proof. We put N := n(n + 1)/2. Using the fact that Ψ : C −→ Vectk(U, U′) is a measuring and that C is cocommutative we
+have
+c(ξn(U)(u1 ⊗ ... ⊗ un))
+= c(u1
+(1) ⊗ u1
+(2)u2
+(1) ⊗ u1
+(3)u2
+(2)u3
+(1) ⊗ ... ⊗ u1
+(n)u2
+(n−1)....un−1
+(2) un)
+= x(1)(u1
+(1)) ⊗ x(2)(u1
+(2))x(3)(u2
+(1)) ⊗ ... ⊗ x(N+1−n)(u1
+(n))....x(N−1)(un−1
+(2) )x(N)(un)
+= x(1)(u1
+(1)) ⊗ x(2)(u1
+(2))x(n+1)(u2
+(1)) ⊗ ... ⊗ x(n)(u1
+(n))....x(N−1)(un−1
+(2) )x(N)(un)
+= ξn(U′)(x(1)(u1) ⊗ ... ⊗ x(n)(un))
+This proves the result.
+□
+4
+Shuffle products and the enrichment of the category of commutative Hopf alge-
+broids
+We recall from Section 2 the category HALGk of Hopf algebroids over k, enriched over the symmetric monoidal category
+of CoCoalgk of cocommutative k-coalgebras.
+By a commutative Hopf algebroid, we will mean a Hopf algebroid U =
+(U, AL, sL, tL, ∆L, ǫL) such that H and AL = A = AR are commutative rings.
+Let cHALGk denote the full subcategory of HALGk consisting of commutative Hopf algebroids. Then, cHALGk is also en-
+riched over CoCoalgk. In this section, we will obtain a second enrichment of commutative Hopf algebroids in cocommutative
+coalgebras, by using the shuffle product in Hochschild homology.
+We know from [17, § 4.2] that the Hochschild homology of a commutative algebra is equipped with a shuffle product structure.
+For a commutative Hopf algebroid U = (U, AL, sL, tL, ∆L, ǫL), we now recall from [15, § 4.4.1] the (p, q)-shuffle product
+shpq(U) : Cp(U) ⊗ Cq(U) −→ Cp+q(U)
+(4.1)
+which is given by the formula (for p, q ≥ 1)
+shpq(U)((u1 ⊗ ... ⊗ up) ⊗ (up+1 ⊗ ... ⊗ up+q)) :=
+�
+σ∈S h(p,q)
+sgn(σ)(uσ−1(1) ⊗ ... ⊗ uσ−1(p+q))
+(4.2)
+Here S h(p, q) is the set of (p, q)-shuffles, i.e.,
+S h(p, q) := {σ ∈ S p+q | σ(1) < ... < σ(p); σ(p + 1) < ... < σ(p + q)}
+(4.3)
+For p = q = 0, the shuffle product is given by setting sh00(U) to be the multiplication on A. Further, one has (see [15, § 4.4.1])
+shp0(U) : Cp(U) ⊗ C0(U) −→ Cp(U)
+(u1 ⊗ ... ⊗ up) ⊗ a �→ (tL(a)u1 ⊗ ... ⊗ up)
+sh0q(U) : C0(U) ⊗ Cq(U) −→ Cq(U)
+a ⊗ (u1 ⊗ ... ⊗ up) �→ (u1 ⊗ ... ⊗ uqtL(a))
+(4.4)
+for p ≥ 1 and q ≥ 1. There is now an induced product structure shpq(U) : HHp(U) ⊗ HHq(U) −→ HHp+q(U) which makes the
+the Hochschild homology HH•(U) :=
+�
+n≥0
+HHp(U) of a commutative Hopf algebroid U into a graded algebra (see [15, § 4.4.1])
+that we denote by (HH•(U), sh(U)).
+Proposition 4.1. Let U, U′ be commutative Hopf algebroids. Let C be a cocommutative coalgebra and let (Ψ, ψ) : C −→
+V(U, U′) be a measuring of Hopf algebroids. Then, the induced K-linear map
+Ψhoc : C −→ HomK(HH•(U), HH•(U′))
+x �→ (Ψhoc
+• (x) : HH•(U) −→ HH•(U′))
+(4.5)
+gives a measuring of algebras from (HH•(U), sh(U)) to (HH•(U′), sh(U′)).
+9
+
+Proof. The unit in (HH•(U), sh(U)) is given by the class of the unit 1A ∈ A = C0(U). Since ψ : C −→ HomK(A, A′) gives in
+particular a measuring from A to A′, we have Ψhoc
+• (x)(1A) = 1A′. We now note that for any x ∈ C and p, q ≥ 1, we have
+Ψp+q(x)(shpq(U)((u1 ⊗ ... ⊗ up) ⊗ (up+1 ⊗ ... ⊗ up+q)))
+= Ψp+q(x)
+�
+�
+σ∈S h(p,q)
+sgn(σ)(uσ−1(1) ⊗ ... ⊗ uσ−1(p+q))
+�
+=
+�
+σ∈S h(p,q)
+sgn(σ)(x(1)(uσ−1(1)) ⊗ ... ⊗ x(p+q)(uσ−1(p+q)))
+=
+�
+σ∈S h(p,q)
+sgn(σ)(xσ−1(1)(uσ−1(1)) ⊗ ... ⊗ xσ−1(p+q)(uσ−1(p+q)))
+(because C is cocommutative)
+= shpq(U)((x(1)(u1) ⊗ ... ⊗ x(p)(up)) ⊗ (x(p+1)(up+1) ⊗ ... ⊗ x(p+q)(up+q)))
+(4.6)
+For p ≥ 1, we have
+Ψp(x)(shp0(U)((u1 ⊗ ... ⊗ up) ⊗ a)
+= Ψp(x)((tL(a)u1 ⊗ ... ⊗ up))
+= (x(1)(tL(a)u1) ⊗ ... ⊗ x(p)(up))
+= (x(1)(tL(a))x(2)(u1) ⊗ ... ⊗ x(p+1)(up))
+= (tL(x(p+1)(a)))x(1)(u1) ⊗ ... ⊗ x(p)(up))
+= shp0(U)((x(1)(u1) ⊗ ... ⊗ x(p)(up)) ⊗ x(p+1)(a))
+(4.7)
+We can similarly verify the case for sh0q with q ≥ 1 and for sh00. This proves the result.
+□
+Our next objective is to use Proposition 4.1 to obtain an enrichment of commutative Hopf algebroids over the category of
+cocommutative coalgebras. For that we recall the following fact: if R, R′ are k-algebras, the category of coalgebra measurings
+from R to R′ contains a final object ϕ(R, R′) : M(R, R′) −→ Vectk(R, R′) (see Sweedler [20]). Then, M(R, R′) is known as the
+universal measuring coalgebra. We let Mc(R, R′) be the cocommutative part of the coalgebra M(R, R′). Then, the restriction
+ϕc(R, R′) : Mc(R, R′) ֒→ M(R, R′) −→ Vectk(R, R′) becomes the final object in the category of cocommutative coalgebra
+measurings from R to R′ (see [9, Proposition 1.4], [10]). Further, the objects Mc(R, R′) give an enrichment of k-algebras over
+cocommutative k-coalgebras.
+We now define the enriched category
+�
+cHALGk whose objects are commutative Hopf algebroids over k and whose hom-objects
+are defined by setting
+�
+cHALGk(U, U′) := Mc((HH•(U), sh(U)), (HH•(U′), sh(U′))) ∈ CoCoalgk
+(4.8)
+for commutative Hopf algebroids U, U′. Since each (HH•(U), sh(U)) is an algebra, we also have a canonical morphism k −→
+Mc((HH•(U), sh(U)), (HH•(U), sh(U))) of cocommutative coalgebras.
+Lemma 4.2. Let U, U′ be commutative Hopf algebroids. Then, there is a canonical morphism of cocommutative coalgebras
+τ(U, U′) : Mc(U, U′) −→ Mc((HH•(U), sh(U)), (HH•(U′), sh(U′)))
+(4.9)
+Proof. By definition, (Φ, φ) : Mc(U, U′) −→ V(U, U′) is a cocommutative measuring from U to U′. By Proposition 4.1,
+this induces a measuring of algebras from (HH•(U), sh(U)) to (HH•(U′), sh(U′)). By the universal property of the universal
+cocommutative measuring coalgebra Mc((HH•(U), sh(U)), (HH•(U′), sh(U′))), we now obtain an induced morphism τ(U, U′)
+as in (4.9).
+□
+Theorem 4.3. There is a CoCoalgk enriched functor cHALGk −→
+�
+cHALGk which is identity on objects and whose mapping
+on hom-objects is given by
+τ(U, U′) : cHALGk(U, U′) = Mc(U, U′) −→ Mc((HH•(U), sh(U)), (HH•(U′), sh(U′))) =
+�
+cHALGk(U, U′)
+(4.10)
+for commutative Hopf algebroids U, U′ over k.
+10
+
+Proof. Let U, U′, U′′ be commutative Hopf algebroids. We show that the following diagram commutes
+Mc(U, U′) ⊗ Mc(U′, U′′)
+◦
+−−−−−−→
+Mc(U, U′′)
+τ(U,U′)⊗τ(U′,U′′)
+�
+�τ(U,U′′)
+Mc(HH•(U), HH•(U′)) ⊗ Mc(HH•(U′), HH•(U′′))
+◦
+−−−−−−→ Mc(HH•(U), HH•(U′′))
+(4.11)
+The top horizontal composition ◦ : Mc(U, U′) ⊗ Mc(U′, U′′) −→ Mc(U, U′′) in (4.11) is obtained from Theorem 2.7, while the
+bottom horizontal composition ◦ : Mc(HH•(U), HH•(U′)) ⊗ Mc(HH•(U′), HH•(U′′)) −→ Mc(HH•(U), HH•(U′′)) is obtained
+from the enrichment of algebras in cocommutative coalgebras.
+From Lemma 4.2 and Theorem 2.7, we note that all the maps in (4.11) are morphisms of cocommutative coalgebras. It follows
+from the property of the universal cocommutative measuring coalgebra Mc(HH•(U), HH•(U′′)) that in order to show that (4.11)
+commutes, it suffices to verify that the following two compositions are equal
+Mc(U, U′) ⊗ Mc(U′, U′′)
+τ(U,U′)⊗τ(U′,U′′)
+�
+Mc(HH•(U), HH•(U′)) ⊗ Mc(HH•(U′), HH•(U′′))
+◦
+�
+Mc(HH•(U), HH•(U′′))
+�ϕc(HH•(U),HH•(U′′))
+Vectk(HH•(U), HH•(U′′))
+Mc(U, U′) ⊗ Mc(U′, U′′)
+�◦
+Mc(U, U′′)
+�τ(U,U′′)
+Mc(HH•(U), HH•(U′′))
+�ϕc(HH•(U),HH•(U′′))
+Vectk(HH•(U), HH•(U′′))
+(4.12)
+For the sake of convenience, we denote the left vertical composition in (4.12) by ψ1 and the right vertical composition by ψ2.
+We now consider x ∈ Mc(U, U′), y ∈ Mc(U′, U′′) and (u1 ⊗ ... ⊗ up) ∈ Cp(U). We see that
+ψ2(x ⊗ y)(u1 ⊗ ... ⊗ up)
+= (y ◦ x)(u1 ⊗ ... ⊗ up)
+= (y ◦ x)(1)(u1) ⊗ ... ⊗ (y ◦ x)(p)(up)
+(4.13)
+Since ◦ : Mc(U, U′) ⊗ Mc(U′, U′′) −→ Mc(U, U′′) is a morphism of coalgebras, we note that (y ◦ x)(1) ⊗ ... ⊗ (y ◦ x)(p) =
+(y(1) ◦ x(1)) ⊗ ... ⊗ (y(p) ◦ x(p)). Combining with (4.13), we see that the right vertical composition in (4.12) may be described
+explicitly as
+ψ2(x ⊗ y)(u1 ⊗ ... ⊗ up) = (y(1) ◦ x(1))(u1) ⊗ ... ⊗ (y(p) ◦ x(p))(up) = y(1)(x(1)(u1)) ⊗ ... ⊗ y(p)(x(p)(up))
+(4.14)
+On the other hand, we note that the following diagram is commutative
+Mc(U, U′) ⊗ Mc(U′, U′′)
+◦(τ(U,U′)⊗τ(U′,U′′))
+−−−−−−−−−−−−−−−−→
+Mc(HH•(U), HH•(U′′))
+(ϕc(HH•(U),HH•(U′))◦τ(U,U′))⊗
+�(ϕc(HH•(U′),HH•(U′′))◦τ(U′,U′′))
+ϕc(HH•(U),HH•(U′′))
+�
+Vectk(HH•(U), HH•(U′)) ⊗ Vectk(HH•(U′), HH•(U′′))
+◦
+−−−−−−→
+Vectk(HH•(U), HH•(U′′))
+(4.15)
+From (4.15), it follows that the left vertical composition in (4.12) may be described explicitly as
+ψ1(x ⊗ y)(u1 ⊗ ... ⊗ up)
+= y(x(u1 ⊗ ... ⊗ up))
+= y(1)(x(1)(u1)) ⊗ ... ⊗ y(p)(x(p)(up))
+(4.16)
+From (4.14) and (4.16), we see that ψ1 = ψ2 and hence the diagram (4.11) commutes. Similarly by considering the coalgebra k
+and using the fact that the p-th iterated coproduct ∆p(1) = 1 ⊗ ... ⊗ 1(p-times), we see that the following compositions are equal
+k −→ Mc(HH•(U), HH•(U))
+ϕc(HH•(U),HH•(U))
+−−−−−−−−−−−−−−−→ Vectk(HH•(U), HH•(U))
+k −→ Mc(U, U)
+τ(U,U)
+−−−−−→ Mc(HH•(U), HH•(U))
+ϕc(HH•(U),HH•(U))
+−−−−−−−−−−−−−−−→ Vectk(HH•(U), HH•(U))
+(4.17)
+11
+
+It follows from (4.17) that the following diagram commutes
+k
+�
+�●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+Mc(HH•(U), HH•(U))
+Mc(U, U)
+τ(U,U)
+�❧
+❧
+❧
+❧
+❧
+❧
+❧
+❧
+❧
+❧
+❧
+❧
+❧
+(4.18)
+This proves the result.
+□
+5
+Comodule measurings for SAYD modules
+Let U = (U, AL, sL, tL, ∆L, ǫL, S ) be a Hopf algebroid. From now onwards, we set Ae := A ⊗k Aop and define
+ηL : Ae = A ⊗k Aop
+sL⊗tL
+−−−−→ U ⊗ U −→ U
+(5.1)
+where the second arrow in (5.1) is the multilplication on U. Following [14, § 2], we note that there are now four commuting
+actions of A on U which are denoted as follows
+a ⊲ u ⊳ b := sL(a)tL(b)u
+a ◮ u ◭ b := usL(b)tL(a)
+a, b ∈ A, u ∈ U
+(5.2)
+By Definition 2.1, we then have an A-coring
+∆L : U −→ U⊳ ⊗A ⊲U
+ǫL : U −→ A
+(5.3)
+The left action ◮ of A on U may be treated as a right action of Aop on U. Similarly, the right action ⊳ of A on U may be treated
+as a left action by Aop. Accordingly, we may consider the tensor product
+◮U ⊗Aop U⊳ := U ⊗k U/span{a ◮ u ⊗ v − u ⊗ v ⊳ a | u, v ∈ U, a ∈ A}
+(5.4)
+There is now a Hopf-Galois map (see [5], [14], [19])
+β(U) : ◮U ⊗Aop U⊳ −→ U⊳ ⊗A ⊲U
+u ⊗Aop v �→ u(1) ⊗A u(2)v
+(5.5)
+Since U is a Hopf algebroid, it follows (see [5, Proposition 4.2]) that the morphism β(U) in (5.5) is a bijection. Accordingly, in
+the notation of [14], [19], we write
+u+ ⊗Aop u− := β(U)−1(u ⊗A 1)
+u ∈ U
+(5.6)
+In this section, we will consider comodule measurings between stable anti-Yetter Drinfeld modules over Hopf algebroids. For
+this, we first recall the notion of comodule measuring between ordinary modules. Let R, R′ be rings and let P, P′ be modules
+over R and R′ respectively. Then, a comodule measuring from P to P′ consists of a pair of maps (see [4], [12])
+ψ : C −→ Vectk(R, R′)
+ω : D −→ Vectk(P, P′)
+(5.7)
+where C is a k-coalgebra, D is a right C-comodule, ψ : C −→ Vectk(R, R′) is a coalgebra measuring and
+ω(y)(pr) = y(pr) = y(0)(p)y(1)(r) = ω(y(0))(p)ψ(y(1))(r)
+(5.8)
+for y ∈ D, p ∈ P and r ∈ R. For U = (U, AL, sL, tL, ∆L, ǫL, S ), we will now recall the notions of U-modules, U-comodules and
+stable anti-Yetter Drinfeld modules.
+Definition 5.1. (see [14, § 2.4]) Let U = (U, AL, sL, tL, ∆L, ǫL, S ) be a Hopf algebroid. A right U-module P is a right module
+over the k-algebra U. Because of the ring homomorphism ηL : Ae −→ U, any right U-module P is also equipped with a right
+Ae-module structure (or (A, A)-bimodule structure) given by
+b ◮ p ◭ a = p(a ⊗ b) = pηL((a ⊗ 1)(1 ⊗ b)) = psL(a)tL(b)
+(5.9)
+for (a ⊗ b) ∈ Ae = A ⊗k Aop and p ∈ P.
+12
+
+Definition 5.2. (see [6], [8], [14], [18]) Let U = (U, AL, sL, tL, ∆L, ǫL, S ) be a Hopf algebroid. A left U-comodule P is a left
+comodule over the A-coring (U, ∆L : U −→ U ⊗AL U, ǫL : U −→ AL). In particular, a left U-comodule P is equipped with a left
+A-module structure (a, p) �→ ap as well as a left A-module map
+∆P : P −→ U⊳ ⊗A P
+p �→ p(−1) ⊗ p(0)
+(5.10)
+Following [14, § 2.5], we note that any left U-comodule P also carries a right A-module structure given by setting
+pa := ǫL(p(−1)sL(a))p(0)
+(5.11)
+for p ∈ P, a ∈ A. This makes any left U-comodule P into a right Ae = A ⊗k Aop-module by setting
+p(a ⊗ b) = bpa = bǫL(p(−1)sL(a))p(0)
+(5.12)
+for p ∈ P and (a ⊗ b) ∈ Ae.
+Definition 5.3. (see [14, Definition 2.7]) Let U = (U, AL, sL, tL, ∆L, ǫL, S ) be a Hopf algebroid. A stable anti-Yetter Drinfeld
+module (or SAYD module) P over U consists of the following
+(1) A right U-module structure on P denoted by (p, u) �→ pu for p ∈ P and u ∈ U.
+(2) A left U-comodule structure on P given by ∆P : P −→ U⊳ ⊗A P.
+(3) The right Ae-module structure on P induced by (5.9) coincides with the right Ae-module structure on P as in (5.12):
+psL(a)tL(b) = b ◮ p ◭ a = bǫL(p(−1)sL(a))p(0)
+(5.13)
+(4) For u ∈ U and p ∈ P, one has
+∆P(pu) = u−p(−1)u+(1) ⊗A p(0)u+(2)
+(5.14)
+(5) Stability condition: for any p ∈ P, one has p(0)p(−1) = p.
+Lemma 5.4. Let R, R′ be k-algebras and let Re = R ⊗k Rop, R′e = R′ ⊗k R′op be their respective enveloping algebras. Let C be
+a cocommutative k-coalgebra and let ψ : C −→ Vectk(R, R′) be a measuring. Then,
+ψe : C −→ Vectk(Re, R′e)
+ψe(c)(r ⊗ r′) = c(r1 ⊗ r2) = c(1)(r1) ⊗ c(2)(r2) = ψ(c(1))(r1) ⊗ ψ(c(2))(r2)
+(5.15)
+is a measuring of algebras.
+Proof. From (5.15), it is immediate that c(1 ⊗ 1) = ǫC(c)(1 ⊗ 1), where ǫC is the counit on C. Since C is cocommutative, we
+have for (r1 ⊗ r2), (r3 ⊗ r4) ∈ Re
+c((r1 ⊗ r2)(r3 ⊗ r4)) = c(r1r3 ⊗ r4r2)
+= c(1)(r1r3) ⊗ c(2)(r4r2)
+= c(1)(r1)c(2)(r3) ⊗ c(3)(r4)c(4)(r2)
+= c(1)(r1)c(3)(r3) ⊗ c(4)(r4)c(2)(r2)
+= (c(1)(r1) ⊗ c(2)(r2))(c(3)(r3) ⊗ c(4)(r4))
+= c(1)(r1 ⊗ r2)c(2)(r3 ⊗ r4)
+(5.16)
+□
+Lemma 5.5. Let U = (U, S ) = (U, AL, sL, tL, ∆L, ǫL, S ) and U′ = (U′, S ′) = (U′, AL, s′
+L, t′
+L, ∆′
+L, ǫ′
+L, S ′) be Hopf algebroids over
+k. Let P (resp. P′) be an SAYD-module over U (resp. U′). Let C be a cocommutative k-coalgebra and D be a right C-comodule.
+Suppose that we are given the following data
+Ψ : C −→ Vectk(U, U′)
+ψ : C −→ Vectk(A, A′)
+Ω : D −→ Vectk(P, P′)
+(5.17)
+such that
+13
+
+(1) (Ψ, ψ) is a measuring of Hopf algebroids from U to U′.
+(2) (Ψ, Ω) is a comodule measuring from the right U-module P to the right U′ module P′.
+Then, we have:
+(a) (ψe, Ω) is a comodule measuring from the right Ae-module P to the right A′e-module P′.
+(b) For each d ∈ D, the following morphism is well-defined
+d : U⊳ ⊗A P −→ U′
+⊳ ⊗A′ P′
+d(u ⊗A p) := Ψ(d(1))(u) ⊗A′ Ω(d(0))(p)
+(5.18)
+Proof. (a) Since C is cocommutative, we already know from Lemma 5.4 that ψe : C −→ Vectk(Ae, A′e) is a coalgebra measuring
+from Ae to A′e. We now consider (a ⊗ b) ∈ Ae = A ⊗k Aop. By (5.9), we know that p(a ⊗ b) = psL(a)tL(b). For any d ∈ D, we
+now have
+Ω(d)(p(a ⊗ b)) = Ω(d)(psL(a)tL(b))
+= Ω(d(0))(p)Ψ(d(1))(sL(a)tL(b))
+= Ω(d(0))(p)Ψ(d(1))(sL(a))Ψ(d(2))(tL(b))
+= Ω(d(0))(p)s′
+L(ψ(d(1))(a))t′
+L(ψ(d(2))(b))
+= Ω(d(0))(p)(ψ(d(1)(a)) ⊗ ψ(d(2)(b)))
+= Ω(d(0))(p)(ψe(d(1))(a ⊗ b))
+(5.19)
+(b) Since P and P′ are SAYD modules, it follows from the definition in (5.9) and the condition in (5.13) that
+ap = ptL(a)
+a′p′ = p′t′
+L(a′)
+a ∈ A, a′ ∈ A′, p ∈ P, p′ ∈ P′
+(5.20)
+where the left hand side of the equalities in (5.20) comes from the left A-module action on P (resp. the left A′-module action
+on P′) appearing in the structure map ∆P : P −→ U⊳ ⊗A P (resp. the structure map ∆′
+P′ : P′ −→ U′
+⊳ ⊗A′ P′). For a ∈ A, u ∈ U
+and p ∈ P, we now see that
+d(u ⊗A ap)
+= Ψ(d(1))(u) ⊗A′ Ω(d(0))(ap)
+= Ψ(d(1))(u) ⊗A′ Ω(d(0))(ptL(a))
+(using (5.20))
+= Ψ(d(2))(u) ⊗A′ Ω(d(0))(p)Ψ(d(1))(tL(a))
+= Ψ(d(2))(u) ⊗A′ Ω(d(0))(p)t′
+L(ψ(d(1))(a))
+= Ψ(d(2))(u) ⊗A′ ψ(d(1))(a)Ω(d(0))(p)
+(using (5.20))
+= Ψ(d(2))(u) ⊳ ψ(d(1))(a) ⊗A′ Ω(d(0))(p)
+= t′
+L(ψ(d(1))(a))Ψ(d(2))(u) ⊗A′ Ω(d(0))(p)
+(using (5.2))
+(5.21)
+On the other hand, we also have
+d(u ⊳ a ⊗A p)
+= d(tL(a)u ⊗A p)
+(using (5.2))
+= Ψ(d(1))(tL(a)u) ⊗A′ Ω(d(0))(p)
+= Ψ(d(1))(tL(a))Ψ(d(2))(u) ⊗A′ Ω(d(0))(p)
+= t′
+L(ψ(d(1))(a))Ψ(d(2))(u) ⊗A′ Ω(d(0))(p)
+(5.22)
+This proves the result.
+□
+We are now ready to introduce the notion of a comodule measuring between SAYD modules.
+Definition 5.6. Let U = (U, S ) = (U, AL, sL, tL, ∆L, ǫL, S ) and U′ = (U′, S ′) = (U′, AL, s′
+L, t′
+L, ∆′
+L, ǫ′
+L, S ′) be Hopf algebroids
+over k. Let P (resp. P′) be an SAYD-module over U (resp. U′). Let C be a cocommutative coalgebra. Then, a (right) measuring
+comodule over (C, Ψ, ψ) from P to P′ consists of the following data
+Ψ : C −→ Vectk(U, U′)
+ψ : C −→ Vectk(A, A′)
+Ω : D −→ Vectk(P, P′)
+(5.23)
+such that
+(1) (Ψ, ψ) is a measuring of Hopf algebroids from U to U′.
+14
+
+(2) (Ψ, Ω) is a comodule measuring from the right U-module P to the right U′ module P′.
+(3) For any d ∈ D, the following diagram commutes
+P
+∆P
+−−−−−−→ U⊳ ⊗A P
+d:=Ω(d)
+�
+d
+�
+P′
+∆′
+P′
+−−−−−−→ U′
+⊳ ⊗A′ P′
+(5.24)
+where the right vertical morphism is as defined in (5.18)
+We will now construct universal measuring comodules. By definition, the right comodules over a k-coalgebra C are coalgebras
+over the comonad
+⊗k C : Vectk −→ Vectk. Accordingly, the forgetful functor Comod − C −→ Vectk from the category of
+right C-comodules has a right adjoint (see, for instance, [7, § 2.4]) that we denote by RC, i.e., we have natural isomorphisms
+Vectk(D, V) � Comod − C(D, RC(V))
+(5.25)
+for any D ∈ Comod − C and V ∈ Vectk.
+Theorem 5.7. Let U = (U, AL, sL, tL, ∆L, ǫL, S ) and U′ = (U′, AL, s′
+L, t′
+L, ∆′
+L, ǫ′
+L, S ′) be Hopf algebroids over k. Let P (resp. P′)
+be an SAYD-module over U (resp. U′). Let C be a cocommutative coalgebra and (Ψ, ψ) : C −→ V(U, U′) be a measuring of
+Hopf algebroids.
+Then, there exists a measuring (C, Ψ, ψ)-comodule (QC(P, P′), Θ : QC(P, P′) −→ Vectk(P, P′)) satisfying the following property:
+given any measuring (C, Ψ, ψ)-comodule (D, Ω : D −→ Vectk(P, P′)) from P to P′, there exists a morphism χ : D −→ QC(P, P′)
+of right C-comodules such that the following diagram is commutative
+QC(P, P′)
+Θ
+� Vectk(P, P′)
+D
+χ
+�❍❍❍❍❍❍❍❍❍
+Ω
+�t
+t
+t
+t
+t
+t
+t
+t
+t
+t
+(5.26)
+Proof. We put V := Vectk(P, P′). By the adjunction in (5.25), there is a canonical morphism ρ(V) : RC(V) −→ V of vector
+spaces. We set QC(P, P′) := � Q, where the sum is taken over all right C-subcomodules over RC(V) such that the restriction
+ρ(V)|Q : Q −→ V = Vectk(P, P′) is a (C, Ψ, ψ)-comodule measuring from P to P′ in the sense of Definition 5.6. It is clear that
+Θ : ρ(V)|QC(P, P′) : QC(P, P′) −→ V = Vectk(P, P′) is a (C, Ψ, ψ)-measuring comodule.
+Additionally, given a measuring (C, Ψ, ψ)-comodule (D, Ω : D −→ Vectk(P, P′)) from P to P′, the adjunction in (5.25) gives a
+morphism χ : D −→ RC(V). But then we notice that ρ(V)|χ(D) : χ(D) −→ V is a measuring (C, Ψ, ψ)-comodule, whence it
+follows that the image χ(D) ⊆ QC(P, P′). The result is now clear.
+□
+Lemma 5.8. Let U = (U, AL, sL, tL, ∆L, ǫL, S ), U′ = (U′, AL, s′
+L, t′
+L, ∆′
+L, ǫ′
+L, S ′) and U′′ = (U′′, A′′
+L, s′′
+L, t′′
+L , ∆′′
+L, ǫ′′
+L , S ′′) be Hopf
+algebroids over k. Let P, P′ and P′′ be SAYD modules over U, U′ and U′′ respectively. Suppose that we have:
+(1) Ψ : C −→ Vectk(U, U′), ψ : C −→ Vectk(A, A′) and Ω : D −→ Vectk(P, P′) giving the data of a measuring comodule from
+P to P′.
+(2) Ψ′ : C′ −→ Vectk(U′, U′′), ψ′ : C′ −→ Vectk(A′, A′′) and Ω : D′ −→ Vectk(P′, P′′) giving the data of a measuring
+comodule from P′ to P′′.
+Then, the following
+(Ψ′, ψ′) ◦ (Ψ, ψ) : C ⊗ C′ (Ψ,ψ)⊗(Ψ′,ψ′)
+−−−−−−−−−−→ V(U, U′) ⊗ V(U′, U′′)
+◦
+−→ V(U, U′′)
+Ω′ ◦ Ω : D ⊗ D′ Ω⊗Ω′
+−−−−→ Vectk(P, P′) ⊗ Vectk(P′, P′′)
+◦−→ Vectk(P, P′′)
+(5.27)
+gives the data of a measuring comodule from P to P′′. There is also a canonical morphism of right (C ⊗ C′)-comodules
+QC(P, P′) ⊗ QC′(P′, P′′) −→ QC⊗C′(P, P′′)
+(5.28)
+15
+
+Proof. We know from Proposition 2.6 that (Ψ′, ψ′) ◦ (Ψ, ψ) : C ⊗ C′ −→ V(U, U′′) is a measuring of Hopf algebroids. It
+may also be directly verified that ((Ψ′, ψ′) ◦ (Ψ, ψ), Ω′ ◦ Ω) is a comodule measuring from the right U-module P to the right
+U′′-module P′′. To check the condition (5.24) in Definition 5.6, we observe that for any d ⊗ d′ ∈ D ⊗ D′, u ∈ U and p ∈ P:
+(d ⊗ d′)(u ⊗A p) = (d ⊗ d′)(1)(u) ⊗A′′ (d ⊗ d′)(0)(p) = d′
+(1)(d(1)(u)) ⊗A′′ d′
+(0)(d(0)(p)) = d′(d(u ⊗A p)))
+(5.29)
+Since the measurings (Ψ, ψ, Ω) and (Ψ′, ψ′, Ω′) both satisfy the condition in (5.24), it is clear that so does (Ψ′◦Ψ, ψ′ ◦ψ, Ω′ ◦Ω).
+Hence, (5.27) gives the data of a measuring comodule from P to P′′. By definition, QC(P, P′) (resp. QC′(P′, P′′)) is a measuring
+comodule from P to P′ (resp. from P′ to P′′). From (5.27) it now follows that QC(P, P′) ⊗ QC′(P′, P′′) is a measuring comodule
+from P to P′′. The morphism in (5.28) is now obtained by the universal property of QC⊗C′(P, P′′).
+□
+We now consider the “global category of comodules” Comodk whose objects are pairs (C, D), where C is a cocommutative k-
+coalgebra and D is a right C-comodule. A morphism (f, g) : (C, D) −→ (C′, D′) in Comodk consists of a k-coalgebra morphism
+f : C −→ C′ and a morphism g : D −→ D′ of C′-comodules, where D is treated as a C′-comodule by corestriction of scalars.
+It is clear that putting (C, D) ⊗ (C′, D′) := (C ⊗ C′, D ⊗ D′) makes Comodk into a symmetric monoidal category.
+Theorem 5.9. Let S AYDk be the category given by:
+(a) Objects: pairs (U, P), where U is a Hopf-algebroid and P is an S AYD-module over U
+(b) Hom-objects: for pairs (U, P), (U′, P′) ∈ S AYDk, we set
+S AYDk((U, P), (U′, P′)) := (Mc(U, U′), QMc(U,U′)(P, P′)) ∈ Comodk
+(5.30)
+Then, S AYDk is enriched over the symmetric monoidal category Comodk.
+Proof. For any (U, P) ∈ S AYDk, the scalar multiples of the identity map give a morphism k −→ Mc(U, U) of k-coalgebras,
+and along with the universal property in Theorem 5.7 give a morphism k −→ QMc(U,U)(P, P). We now consider (U, P), (U′, P′),
+(U′′, P′′) ∈ S AYDk. Applying Lemma 5.8 with C = Mc(U, U′) and C′ = Mc(U′, U′′), we obtain a morphism QMc(U,U′)(P, P′) ⊗
+QMc(U′,U′′)(P, P′) −→ QMc(U,U′)⊗Mc(U′,U′′)(P, P′′) of (Mc(U, U′) ⊗ Mc(U′, U′′))-comodules. From the proof of Theorem 2.7, we
+already have a morphism Mc(U, U′) ⊗ Mc(U′, U′′) −→ Mc(U, U′′) of k-coalgebras. Combining, we have a morphism
+S AYDk((U, P), (U′, P′)) ⊗ S AYDk((U′, P′), (U′′, P′′)) −→ (Mc(U, U′′), QMc(U,U′)⊗Mc(U′,U′′)(P, P′′))
+(5.31)
+in Comodk. In (5.31), QMc(U,U′)⊗Mc(U′,U′′)(P, P′′) is treated as a Mc(U, U′′)-module via the morphism Mc(U, U′)⊗Mc(U′, U′′) −→
+Mc(U, U′′) of k-coalgebras. From the proof of Theorem 2.7, we also know that the morphism Mc(U, U′) ⊗ Mc(U′, U′′) −→
+Mc(U, U′′) arises from the universal property of Mc(U, U′′) applied to the measuring Mc(U, U′) ⊗ Mc(U′, U′′) −→ V(U, U′) ⊗
+V(U′, U′′)
+◦−→ V(U, U′′). Hence, the canonical map QMc(U,U′)⊗Mc(U′,U′′)(P, P′′) −→ Vectk(P, P′′) gives a measuring when treated
+as a Mc(U, U′′)-comodule. The universal property of QMc(U,U′′)(P, P′′) as in Theorem 5.7 now yields a morphism
+(Mc(U, U′′), QMc(U,U′)⊗Mc(U′,U′′)(P, P′′)) −→ (Mc(U, U′′), QMc(U,U′′)(P, P′′))
+(5.32)
+in Comodk. Composing (5.32) with (5.31), we obtain the required composition of Hom-objects S AYDk((U, P), (U′, P′)) ⊗
+S AYDk((U′, P′), (U′′, P′′)) −→ S AYDk((U, P), (U′′, P′′)). This proves the result.
+□
+6
+Comodule measurings and morphisms on cyclic (co)homology
+Throughout this section, we fix the following: let U = (U, AL, sL, tL, ∆L, ǫL, S ), and U′ = (U′, A′
+L, s′
+L, t′
+L, ∆′
+L, ǫ′
+L, S ′) be Hopf
+algebroids over k. Let P and P′ be SAYD modules over U and U′ respectively. Let (Ψ, ψ) : C −→ V(U, U′) be a cocommutative
+measuring and let Ω : D −→ Vectk(P, P′) be a (C, Ψ, ψ)-measuring comodule from P to P′.
+Since U, U′ are Hopf algebroids, we have recalled in Section 5 that the morphisms β(U) : ◮U ⊗Aop U⊳ −→ U⊳ ⊗A ⊲U and
+β(U′) : ◮U′ ⊗A′op U′
+⊳ −→ U′
+⊳ ⊗A′ ⊲U′ in the notation of (5.5) are bijections. We now need the following result.
+16
+
+Lemma 6.1. For each x ∈ C, the following diagram commutes:
+U⊳ ⊗A ⊲U
+β(U)−1
+−−−−−−→
+◮U ⊗Aop U⊳
+x
+�
+x
+�
+U′
+⊳ ⊗A′ ⊲U′
+β(U′)−1
+−−−−−−→ ◮U′ ⊗A′op U′
+⊳
+(6.1)
+Here, the left vertical map is given by u1 ⊗A u2 �→ x(1)(u1) ⊗A′ x(2)(u2) and the right vertical map by u1 ⊗Aop u2 �→ x(1)(u1) ⊗A′op
+x(2)(u2).
+Proof. It is easy to see that the vertical morphisms in (6.1) are well-defined. Further, since β(U) and β(U′) are invertible, it
+suffices to check that the following diagram commutes
+◮U ⊗Aop U⊳
+β(U)
+−−−−−−→ U⊳ ⊗A ⊲U
+x
+�
+�x
+◮U′ ⊗A′op U′
+⊳
+β(U′)
+−−−−−−→ U′
+⊳ ⊗A′ ⊲U′
+(6.2)
+We now see that for u ⊗Aop v ∈ ◮U ⊗Aop U⊳ and x ∈ C, we have
+x(β(U)(u ⊗Aop v)) = x(u(1) ⊗A u(2)v) = x(1)(u(1)) ⊗A x(2)(u(2))x(3)(v) = (x(1)(u))(1) ⊗A (x(1)(u))(2)x(2)(v) = β(U′)(x(1)(u) ⊗ x(2)(v))
+This proves the result.
+□
+From Lemma 6.1, it follows in the notation of (5.6) that we have
+x(1)(u+) ⊗A′op x(2)(u−) = x(u+ ⊗Aop u−) = β(U′)−1(x(u ⊗A 1)) = x(u)+ ⊗A′op x(u)−
+(6.3)
+for each u ∈ U. We now recall from [14, Theorem 4.1] that the Hochschild homology groups HH•(U; P) (resp. the cyclic
+homology groups HC•(U; P)) of U with coefficients in the SAYD module P are obtained from the cyclic module C•(U; P) :=
+P ⊗Aop (◮U⊳)⊗Aop• with operators as follows (where ¯u := u1 ⊗Aop ⊗... ⊗Aop un, p ∈ P)
+di(p ⊗Aop ¯u) :=
+
+p ⊗Aop u1 ⊗Aop · · · ⊗Aop un−1tL(ǫL(un))
+if i = 0
+p ⊗Aop u1 ⊗Aop · · · ⊗Aop un−iun−i+1 ⊗Aop . . .
+if 1 ≤ i ≤ n − 1
+pu1 ⊗Aop u2 ⊗Aop · · · ⊗Aop un
+if i = n
+si(p ⊗Aop ¯u) :=
+
+p ⊗Aop u1 ⊗Aop · · · ⊗Aop un ⊗Aop 1
+if i = 0
+p ⊗Aop · · · ⊗Aop un−i ⊗Aop 1 ⊗Aop un−i+1 ⊗Aop . . .
+if 1 ≤ i ≤ n − 1
+p ⊗Aop 1 ⊗Aop u1 ⊗Aop · · · ⊗Aop un
+if i = n
+tn(p ⊗Aop ¯u) := p(0)u1
++ ⊗Aop u2
++ ⊗Aop · · · ⊗Aop un
++ ⊗Aop un
+− . . .u1
+−p(−1)
+(6.4)
+We now have the following result.
+Proposition 6.2. For each y ∈ D, the family
+Ωn(y) : Cn(U; P) −→ Cn(U′; P′)
+p ⊗ u1 ⊗ ... ⊗ un �→ y(p ⊗ u1 ⊗ ... ⊗ un) = y(0)(p) ⊗ y(1)(u1) ⊗ ... ⊗ y(n)(un)
+(6.5)
+for n ≥ 0 gives a morphism of cyclic modules. In particular, we have induced morphisms
+Ωhoc
+• (y) : HH•(U; P) −→ HH•(U′; P′)
+Ωcy
+• (y) : HC•(U; P) −→ HC•(U′; P′)
+(6.6)
+on Hochschild and cyclic homologies for each y ∈ D.
+17
+
+Proof. From the fact that C is cocommutative and the conditions in Definition 5.6, it is clear that the morphisms Ωn(y) are well
+defined, as well as the fact that they commute with the face maps and degeneracies appearing in the cyclic modules C•(U; P)
+and C•(U′; P′) as in (6.4). To verify that the morphisms in (6.5) also commute with the cyclic operators, we note that for
+p ⊗Aop u1 ⊗Aop ⊗... ⊗Aop un ∈ Cn(U; P)
+y(tn(p ⊗ u1 ⊗ ... ⊗ un)) = y(p(0)u1
++ ⊗ u2
++ ⊗ · · · ⊗ un
++ ⊗ un
+− . . . u1
+−p(−1))
+= y(0)(p(0))y(1)(u1
++) ⊗ y(2)(u2
++) ⊗ · · · ⊗ y(n)(un
++) ⊗ y(n+1)(un
+−) . . .y(2n)(u1
+−)y(2n+1)(p(−1))
+= y(0)(p(0))y(2)(u1
++) ⊗ y(4)(u2
++) ⊗ · · · ⊗ y(2n)(un
++) ⊗ y(2n+1)(un
+−) . . . y(3)(u1
+−)y(1)(p(−1))
+(since C is cocommutative)
+= y(0)(p)(0)y(1)(u1
++) ⊗ y(3)(u2
++) ⊗ · · · ⊗ y(2n−1)(un
++) ⊗ y(2n)(un
+−) . . . y(2)(u1
+−)y(0)(p)(−1)
+(using (5.24))
+= y(0)(p)(0)y(1)(u1)+ ⊗ y(2)(u2)+ ⊗ · · · ⊗ y(n)(un)+ ⊗ y(n)(un)− . . . y(1)(u1)−y(0)(p)(−1)
+(using (6.3))
+This proves the result.
+□
+We now come to cyclic cohomology. For this, we recall that from [14, Theorem 1.1] that the Hochschild cohomology groups
+HH•(U; P) (resp. the cyclic cohomology groups HC•(U; P)) of U with coefficients in the SAYD module P are obtained from
+the cocyclic module C•(U; P) := (⊲U⊳)⊗A• ⊗A P with operators as follows (where ¯u := u1 ⊗A ⊗... ⊗A un, p ∈ P)
+δi(¯u ⊗A p)
+=
+
+1 ⊗A u1 ⊗A · · · ⊗A un ⊗A p
+if i = 0
+u1 ⊗A · · · ⊗A ∆L(ui) ⊗A · · · ⊗A un ⊗A p
+if 1 ≤ i ≤ n
+u1 ⊗A · · · ⊗A un ⊗A p(−1) ⊗A p(0)
+if i = n + 1
+δi(p)
+=
+� 1 ⊗A p
+if j = 0
+p(−1) ⊗A p(0)
+if j = 1
+σi(¯u ⊗A p)
+= u1 ⊗A · · · ⊗A ǫL(ui+1) ⊗A · · · ⊗A un ⊗A p
+0 ≤ i ≤ n − 1
+τn(¯u ⊗A p)
+= u1
+−(1)u2 ⊗A · · · ⊗A u1
+−(n−1)un ⊗A u1
+−(n)p(−1) ⊗A p(0)u1
++
+(6.7)
+We now have the following result.
+Proposition 6.3. For each y ∈ D, the family
+Ω
+n(y) : Cn(U; P) −→ Cn(U′; P′)
+u1 ⊗ ... ⊗ un ⊗ p �→ y(u1 ⊗ ... ⊗ un ⊗ p) = y(1)(u1) ⊗ ... ⊗ y(n)(un) ⊗ y(0)(p)
+(6.8)
+for n ≥ 0 gives a morphism of cocyclic modules. In particular, we have induced morphisms
+Ω
+•
+hoc(y) : HH•(U; P) −→ HH•(U′; P′)
+Ω
+•
+cy(y) : HC•(U; P) −→ HC•(U′; P′)
+(6.9)
+on Hochschild and cyclic cohomologies for each y ∈ D.
+Proof. It is clear that the morphisms in (6.8) are well-defined. For y ∈ D and i = n + 1 in (6.7), we note that
+y(δn+1(u1 ⊗ · · · ⊗ un ⊗ p))
+= y(1)(u1) ⊗ . . . y(n)(un) ⊗ y(n+1)(p(−1)) ⊗ y(0)(p(0))
+= y(2)(u1) ⊗ . . . y(n+1)(un) ⊗ y(1)(p(−1)) ⊗ y(0)(p(0))
+(since C is cocommutative)
+= y(1)(u1) ⊗ . . . y(n)(un) ⊗ (y(0)(p))(−1) ⊗ y(0)(p)(0)
+(using (5.24))
+(6.10)
+Similarly, we may verify that the morphisms in (6.8) commute with the face and degeneracy maps appearing in (6.7). To show
+that they also commute with the cyclic operators appearing in (6.7), we note that for u1 ⊗ ... ⊗ un ⊗ p ∈ Cn(U; P) and y ∈ D, we
+have
+y(τn(u1 ⊗ ... ⊗ un ⊗ p)) = y(u1
+−(1)u2 ⊗A · · · ⊗A u1
+−(n−1)un ⊗A u1
+−(n)p(−1) ⊗A p(0)u1
++)
+= y(1)(u1
+−(1))y(2)(u2) ⊗A · · · ⊗A y(2n−3)(u1
+−(n−1))y(2n−2)(un) ⊗A y(2n−1)(u1
+−(n))y(2n)(p(−1)) ⊗A y(0)(p(0)u1
++)
+= y(1)(u1
+−(1))y(n+1)(u2) ⊗A · · · ⊗A y(n−1)(u1
+−(n−1))y(2n−1)(un) ⊗A y(n)(u1
+−(n))y(2n)(p(−1)) ⊗A y(0)(p(0)u1
++)
+= y(1)(u1
+−)(1)y(2)(u2) ⊗A · · · ⊗A y(1)(u1
+−)(n−1)y(n)(un) ⊗A y(1)(u1
+−)(n)y(n+1)(p(−1)) ⊗A y(0)(p(0)u1
++)
+= y(2)(u1
+−)(1)y(3)(u2) ⊗A · · · ⊗A y(2)(u1
+−)(n−1)y(n+1)(un) ⊗A y(2)(u1
+−)(n)y(n+2)(p(−1)) ⊗A y(0)(p(0))y(1)(u1
++)
+= y(1)(u1)−(1)y(2)(u2) ⊗A · · · ⊗A y(1)(u1)−(n−1)y(n)(un) ⊗A y(1)(u1)−(n)y(n+1)(p(−1)) ⊗A y(0)(p(0))y(1)(u1)+
+(using (6.3))
+= y(1)(u1)−(1)y(2)(u2) ⊗A · · · ⊗A y(1)(u1)−(n−1)y(n)(un) ⊗A y(1)(u1)−(n)y(0)(p)(−1) ⊗A y(0)(p)(0)y(1)(u1)+
+(using (5.24))
+This proves the result.
+□
+18
+
+Finally, we recall from [14, § 4.3] that there are Hopf-Galois isomorphisms relating the modules C•(U; P) and C•(U; P)
+ξn(U; P) : Cn(U; P)
+�
+−→ Cn(U; P)
+p ⊗ u1 ⊗ · · · ⊗ un �→ u1
+(1) ⊗ u1
+(2)u2
+(1) ⊗ · · · ⊗ u1
+(n)u2
+(n−1) . . .un−1
+(2) un ⊗ p
+(6.11)
+We will conclude this section by showing that the morphisms induced by comodule measurings of SAYD modules are compat-
+ible with the Hopf-Galois isomorphisms in (6.11).
+Theorem 6.4. Let U = (U, AL, sL, tL, ∆L, ǫL, S ), and U′ = (U′, A′
+L, s′
+L, t′
+L, ∆′
+L, ǫ′
+L, S ′) be Hopf algebroids over k. Let P and P′
+be SAYD modules over U and U′ respectively. Let (Ψ, ψ) : C −→ V(U, U′) be a cocommutative measuring and let Ω : D −→
+Vectk(P, P′) be a (C, Ψ, ψ)-measuring comodule from P to P′. Then, for each y ∈ D, the following diagram commutes
+Cn(U; P)
+ξn(U;P)
+−−−−−−→ Cn(U; P)
+Ωn(y)
+�
+�Ω
+n(y)
+Cn(U; P)
+ξn(U;P)
+−−−−−−→ Cn(U; P)
+(6.12)
+Proof. We set N := n(n − 1)/2. For y ∈ D and p ⊗ u1 ⊗ · · · ⊗ un ∈ Cn(U; P), we see that
+Ω
+n(y)(ξn(U; P)(p ⊗ u1 ⊗ · · · ⊗ un))
+= Ω
+n(y)(u1
+(1) ⊗ u1
+(2)u2
+(1) ⊗ · · · ⊗ u1
+(n)u2
+(n−1) . . . un−1
+(2) un ⊗ p)
+= y(1)(u1
+(1)) ⊗ y(2)(u1
+(2))y(3)(u2
+(1)) ⊗ · · · ⊗ y(N+1)(u1
+(n))y(N+2)(u2
+(n−1)) . . . y(N+n−1)(un−1
+(2) )y(N+n)(un) ⊗ y(0)(p)
+= y(1)(u1
+(1)) ⊗ y(2)(u1
+(2))y(n+1)(u2
+(1)) ⊗ · · · ⊗ y(n)(u1
+(n))y(2n−1)(u2
+(n−1)) . . .y(N+n−1)(un−1
+(2) )y(N+n)(un) ⊗ y(0)(p)
+= y(1)(u1)(1) ⊗ y(1)(u1)(2)y(2)(u2)(1) ⊗ · · · ⊗ y(1)(u1)(n)y(2)(u2)(n−1) . . .y(n−1)(un−1)(2)y(n)(un) ⊗ y(0)(p)
+= ξn(U; P)(Ωn(y)(p ⊗ u1 ⊗ · · · ⊗ un))
+(6.13)
+□
+7
+Operads with multiplication, comp modules and morphisms on cyclic homology
+We start the final section by recalling from Kowalzig [16] the following two notions.
+Definition 7.1. (see [16, Definition 2.2]) A non-Σ operad O over k consists of the following:
+(a) A collection of vector spaces O = {O(n)}n≥0.
+(b) A family of k-linear operations ◦i : O(p)⊗O(q) −→ O(p+q−1) and an identity 1 ∈ O(1) satisfying the following conditions
+(for φ ∈ O(p), ψ ∈ O(q), χ ∈ O(r))
+φ ◦i ψ
+= 0
+if p < i or p = 0
+(φ ◦i ψ) ◦ j χ
+=
+
+(φ ◦ j χ) ◦i+r−1 ψ
+if j < i
+φ ◦i (ψ ◦ j−i+1 χ)
+if i ≤ j < q + i
+(φ ◦ j−q+1 χ) ◦i ψ
+if j ≥ q + i
+φ ◦i 1
+= 1 ◦1 φ = φ
+for i ≤ p
+(c) An operad multiplication µ ∈ O(2) and a unit e ∈ O(0) such that
+µ ◦1 µ = µ ◦2 µ
+µ ◦1 e = µ ◦2 e = 1
+(7.1)
+Definition 7.2. (see [16, Definition 3.1]) A cyclic unital comp module M over an operad O with multiplication consists of the
+following data:
+(a) A collection of vector spaces M = {M(n)}n≥0.
+19
+
+(b) A family of k-bilinear operations •i : O(p) ⊗ M(n) −→ M(n − p + 1), 0 ≤ i ≤ n + 1 − p satisfying the following conditions
+for φ ∈ O(p), ψ ∈ O(q), x ∈ M(n)
+φ •i (ψ • j x) =
+
+ψ • j (φ •i+q−1 x)
+j < i
+(φ • j−i+1 ψ) •i x
+if j − p < i ≤ j
+ψ • j−p+1 (φ •i x)
+if 1 ≤ i ≤ j − p
+as well as 1 •i x = x for i = 1, 2, ..., n.
+(c) A cyclic operator t : M(n) −→ M(n) for n ≥ 1 satisfying
+t(φ •i x) = φ •i t(x)
+(7.2)
+for φ ∈ O(p), x ∈ M(n) and 0 ≤ i ≤ n − p as well as tn+1 = id.
+We take pairs (O, M) consisting of a non-linear Σ operad O and a cyclic unital comp module M over O. We now consider
+comodule measurings between such pairs
+Definition 7.3. A comodule measuring from (O, M) to (O′, M′) consists of the following:
+(a) A cocommutative coalgebra C and a family of morphisms {Φn : C −→ Vectk(O(n), O′(n))}n≥0 satisfying
+Φp+q−1(x)(φ ◦i ψ) = Φp(x(1))(φ) ◦′
+i Φq(x(2))(φ)
+Φ2(x)(µ) = ǫ(x)µ′
+Φ0(x)(e) = ǫ(x)e′
+(7.3)
+for φ ∈ O(p), ψ ∈ O(q) and any x ∈ C.
+(b) A comodule D over C and a family of morphisms {Ψn : D −→ Vectk(M(n), M′(n))}n≥0 satisfying
+Ψn−p+1(φ •i x) = Ψp(y(0))(φ) •i Ψn(y(1))(x)
+(7.4)
+for y ∈ D, φ ∈ O(p), x ∈ M(n), 0 ≤ i ≤ n + 1 − p and also
+Ψn(y)(t(x)) = t′(Ψn(y)(x))
+(7.5)
+for y ∈ D, x ∈ M(n), where t and t′ are respectively the cyclic operators on M and M′.
+We now recall from [16, Proposition 3.5] that the cyclic homology of (O, M) is obtained from the cyclic module C•(O, M) :=
+M(•) whose cyclic operators are t : M(n) −→ M(n) and whose face maps and degeneracies are given as follows:
+di(x) := µ •i x, (0 ≤ i < n)
+dn(x) := µ •0 t(x)
+sj(x) := e • j+1 x, 0 ≤ j ≤ n
+(7.6)
+The cyclic homologies of this cyclic module will be denoted by HC•(O, M). We conclude with the following result.
+Proposition 7.4. If D is a C-measuring comodule from (O, M) to (O′, M′), then each y ∈ D induces a morphism Ψcy
+• (y) :
+HC•(O, M) −→ HC•(O′, M′) on Hochschild homologies.
+Proof. We know from (7.5) that the action of any y ∈ D commutes with the cyclic operators. From the definitions in (7.6)
+and the conditions in (7.3), it is clear that the action also commutes with the degeneracies and face maps. The result is now
+clear.
+□
+References
+[1] M. Anel and A. Joyal, Sweedler theory for (co)algebras and the bar-cobar constructions, arXiv 1309.6952 (2013).
+[2] A. Banerjee and S. Kour, On measurings of algebras over operads and homology theories, Algebr. Geom. Topol. 22 (2022), no. 3, 1113–1158.
+[3] M. Batchelor, Difference operators, measuring coalgebras, and quantum group-like objects, Adv. Math. 105 (1994), no. 2, 190–218.
+[4]
+, Measuring comodules—their applications, J. Geom. Phys. 36 (2000), no. 3-4, 251–269.
+20
+
+[5] G. B¨ohm and K. Szlach´anyi, Hopf algebroids with bijective antipodes: axioms, integrals, and duals, J. Algebra 274 (2004), no. 2, 708–750.
+[6] G. B¨ohm, Galois theory for Hopf algebroids, Ann. Univ. Ferrara Sez. VII (N.S.) 51 (2005), 233–262.
+[7] G. B¨ohm, T. Brzezi´nski, and R. Wisbauer, Monads and comonads on module categories, J. Algebra 322 (2009), no. 5, 1719–1747.
+[8] T. Brzezinski and R. Wisbauer, Corings and comodules, London Mathematical Society Lecture Note Series, vol. 309, Cambridge University Press,
+Cambridge, 2003.
+[9] L. Grunenfelder and M. Mastnak, On bimeasurings, J. Pure Appl. Algebra 204 (2006), no. 2, 258–269.
+[10]
+, On bimeasurings. II, J. Pure Appl. Algebra 209 (2007), no. 3, 823–832.
+[11] M. Hyland, I. L´opez Franco, and C. Vasilakopoulou, Hopf measuring comonoids and enrichment, Proc. Lond. Math. Soc. (3) 115 (2017), no. 5, 1118–
+1148.
+[12]
+, Measuring comodules and enrichment, arXiv 1703.10137 (2017).
+[13] N. Kowalzig and H. Posthuma, The cyclic theory of Hopf algebroids, J. Noncommut. Geom. 5 (2011), no. 3, 423–476.
+[14] N. Kowalzig and U. Kr¨ahmer, Cyclic structures in algebraic (co)homology theories, Homology Homotopy Appl. 13 (2011), no. 1, 297–318.
+[15] N. Kowalzig, Batalin-Vilkovisky algebra structures on (Co)Tor and Poisson bialgebroids, J. Pure Appl. Algebra 219 (2015), no. 9, 3781–3822.
+[16]
+, Gerstenhaber and Batalin-Vilkovisky structures on modules over operads, Int. Math. Res. Not. IMRN 22 (2015), 11694–11744.
+[17] J.-L Loday, Cyclic homology, 2nd ed., Grundlehren der mathematischen Wissenschaften, vol. 301, Springer-Verlag, Berlin, 1998. Appendix E by M. O.
+Ronco; Chapter 13 by the author in collaboration with T. Pirashvili.
+[18] P. Schauenburg, Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules, Appl. Categ. Structures 6 (1998), no. 2, 193–222.
+[19]
+, Duals and doubles of quantum groupoids (×R-Hopf algebras), New trends in Hopf algebra theory (La Falda, 1999), Contemp. Math., vol. 267,
+Amer. Math. Soc., Providence, RI, 2000, pp. 273–299.
+[20] M. E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969.
+[21] M. Takeuchi, Groups of algebras over A ⊗ A, J. Math. Soc. Japan 29 (1977), no. 3, 459–492.
+[22] C. Vasilakopoulou, Enrichment of categories of algebras and modules, arXiv 1205.6450 (2012).
+[23]
+, On enriched fibrations, Cah. Topol. G´eom. Diff´er. Cat´eg. 59 (2018), no. 4, 354–387.
+[24]
+, Enriched duality in double categories: V-categories and V-cocategories, J. Pure Appl. Algebra 223 (2019), no. 7, 2889–2947.
+21
+
diff --git a/29FAT4oBgHgl3EQfDxyZ/content/tmp_files/load_file.txt b/29FAT4oBgHgl3EQfDxyZ/content/tmp_files/load_file.txt
new file mode 100644
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--- /dev/null
+++ b/29FAT4oBgHgl3EQfDxyZ/content/tmp_files/load_file.txt
@@ -0,0 +1,1130 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf,len=1129
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='08418v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='CT]  20 Jan 2023  Measurings of Hopf algebroids and morphisms in cyclic (co)homology  theories  Abhishek Banerjee *  Surjeet Kour †  Abstract  In this paper, we consider measurings between Hopf algebroids and show that they induce morphisms on cyclic homology  and cyclic cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We also consider comodule measurings between SAYD modules over Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' These measur-  ings induce morphisms on cyclic (co)homology of Hopf algebroids with SAYD coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Finally, we obtain morphisms on  cyclic homology induced by measurings of cyclic comp modules over operads with multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  MSC(2020) Subject Classification: 16T15, 16E40, 18D50  Keywords: Hopf algebroids, cyclic (co)homology, SAYD modules, comp modules  1  Introduction  Let k be a field, C be a k-coalgebra and A, B be k-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' In [20], Sweedler introduced the notion of a coalgebra measuring  as a kind of generalized morphism between algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' More precisely, a C-measuring from A to B consists of a k-linear map  φ : C −→ Vectk(A, B) satisfying  φ(c)(aa′) =  �  φ(c(1)(a)φ(c(2))(a′)  φ(c)(1) = ǫ(c)1  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1)  for any a, a′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Here, ∆(c) = � c(1) ⊗ c(2) denotes the coproduct on C and ǫ : C −→ k denotes the counit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Since then, the  notion of a measuring has been widely studied in the literature by several authors (see, for instance, [1], [3], [4], [9], [10], [11],  [12],[22], [23], [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  In [2], we studied how coalgebra measurings induce morphisms between Hochschild homology groups of algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' The purpose  of this paper is to take this idea one step further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Our aim is to consider cocommutative coalgebra morphisms between Hopf  algebroids and show that they induce morphisms in cyclic homology and cyclic cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We begin by showing that there are  universal measurings which give an enrichment of the category HAlgk of Hopf algebroids over the category of cocommutative  coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' If U is a Hopf algebroid, the cyclic module C•(U) defining its cyclic homology groups as well as the cocyclic  module C•(U) defining its cyclic cohomology groups are defined in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We show that a cocommutative coalgebra measuring  between two Hopf algebroids induces morphisms on their corresponding cyclic homology and cyclic cohomology groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' If a  Hopf algebroid is commutative, we know from [15] that there is a shuffle product on its Hochschild homology groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We show  that a measuring between commutative Hopf algebroids induces an algebra measuring between the corresponding Hochschild  homology rings with respct to this shuffle product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' This also gives us an enrichment of commutative Hopf algebroids over  cocommutative coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Thereafter, we consider comodule measurings of SAYD modules over Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Accordingly, we obtain an enrichment  of the “global category” of SAYD modules (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='9) over the “global category” of comodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' The cyclic homology  Department of Mathematics, Indian Institute of Science, Bangalore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Email: abhishekbanerjee1313@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='com  †Department of Mathematics, Indian Institute of Technology, Delhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Email: koursurjeet@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='com  1  and the cyclic cohomology of a Hopf algebroid with coefficients in an SAYD module was defined in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We show that a  comodule measuring induces morphisms between cyclic (co)homology with SAYD coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' In the final part of the paper,  we work with pairs of the form (O, M), where M is a cyclic unital comp module over a non-Σ operad O with multiplication  in the sense of [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' The cyclic homology groups of such a comp module were also defined in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We consider comodule  measurings between such pairs and show that they induce morphisms in cyclic homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  2  Measurings of Hopf algebroids  Throughout, k is a field and let Vectk be the category of k-vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let A be a unital k-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' In order to define left  and right bialgebroids, as well as Hopf algebroids in later sections, we will frequently need both the algebra A and its opposite  algebra Aop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For this, we will often write the algebra A as AL, while Aop will often be written as AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  An (s, t)-ring over A consists of a unital k-algebra U along with two k-algebra morphisms s : A −→ U and t : Aop −→ U whose  images commute in U, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=', s(a1)t(a2) = t(a2)s(a1) for any a1, a2 ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' The morphisms s and t are often referred to as source  and target maps respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' These morphisms introduce an (A, A)-bimodule structure on U given by left multiplication  a1 · h · a2 := s(a1)t(a2)h  a1, a2 ∈ A, h ∈ H  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1)  The left and right A-module structures on U in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1) allow us to consider the tensor product U ⊗A U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' The following subspace  of U ⊗A U is known as the Takeuchi product  U ×A U := {� ui ⊗A u′  i ∈ U ⊗A U | � uit(a) ⊗A u′  i = � ui ⊗A u′  is(a), ∀ a ∈ A}  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2)  It is well known (see, for instance, [13, § 2]) that the Takeuchi product U ×A U is a unital subalgebra of U ⊗A U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  From now onwards, we also fix a unital k-algebra U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' The multiplication on U will be denoted by µU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Since the category of  (A, A)-bimodules is monoidal, we can consider coalgebra objects in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We now recall the notion of a left Hopf  algebroid (see, for instance, [5], [13], [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For several closely related notions, see [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' A left bialgebroid UL := (U, AL, sL, tL, ∆L, ǫL) over k consists of the following data:  (1) A unital k-algebra AL  (2) A unital k-algebra U which carries the structure of an (sL, tL) ring over AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  (3) A coalgebra object (U, ∆L : U −→ U ⊗AL U, ǫL : U −→ AL) in the category of (AL, AL)-bimodules satisfying the following  conditions:  (i) ∆L : U −→ U ⊗AL U factors through U ×A U ⊆ U ⊗AL U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  (ii) ǫL(usL(ǫL(u′))) = ǫL(uu′) = ǫL(utL(ǫL(u′))) for all u, u′ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  A morphism (F, f) : (U, AL, sL, tL, ∆L, ǫL) = UL −→ U′  L = (U′, A′  L, s′  L, t′  L, ∆′  L, ǫ′  L) of left bialgebroids consists of a pair of  k-algebra morphisms F : U −→ U′ and f : AL −→ A′  L such that  F ◦ sL = s′  L ◦ f  F ◦ tL = t′  L ◦ f  ∆′  L ◦ F = (F ⊗φ F) ◦ ∆L  f ◦ ǫL = ǫ′  L ◦ F  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3)  We will denote the category of left bialgebroids over k by LBialgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  If UL = (H, AL, sL, tL, ∆L, ǫL) is a left bialgebroid, we employ standard Sweedler notation to write ∆L(u) = � u(1) ⊗ u(2) for any  u ∈ H and suppress the summation sign throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We now recall the notion of Hopf algebroid from [5, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' A Hopf algebroid U = (UL, S ) over k consists of the following data:  (1) A left bialgebroid UL = (U, AL, sL, tL, ∆L, ǫL) over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  (2) An involutive anti-automorphism S : U −→ U of the k-algebra U which satisfies S ◦ tL = sL as well as  S (u(1))(1)u(2) ⊗ S (u(1))(2) = 1H ⊗ S (u)  S (u2)1 ⊗ S (u(2))(2)u(1) = S (u) ⊗ 1U  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4)  2  as elements of U ⊗AL U, for all u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  A morphism (F, f) : U = (UL, S ) −→ (U′  L, S ′) = U′ of Hopf algebroids is a morphism in LBialgk that also satisfies S ′◦F = f ◦S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  We will denote the category of Hopf bialgebroids over k by HAlgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  We remark here that in this paper we will always assume the antipode on a Hopf algebroid U = (UL, S ) is involutive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=',  S 2 = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' However, this condition is not part of the original definition due to B¨ohm and Szlach´anyi in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Further, it is shown in  [5, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2] that a Hopf algebroid (UL, S ) is equivalent to a datum consisting of a left bialgebroid and a right bialgebroid  connected by an antipode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  We now recall the classical notion of a coalgebra measuring due to Sweedler [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let R, R′ be k-algebras and C be a k-  coalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Then, a C-measuring from R to R′ consists of a morphism ψ : C −→ Vectk(R, R′) such that  ψ(x)(ab) =  �  ψ(x(1))(a)ψ(x(2))  ψ(x)(1R) = ǫC(x)1R′  ∀ a, b ∈ R  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='5)  where the coproduct ∆C : C −→ C ⊗ C is given by ∆C(x) = � x(1) ⊗ x(2) for any x ∈ C and ǫC : C −→ k is the counit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' The  measuring as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='5) is said to be cocommutative if the coalgebra C is cocommutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' In this paper, we will only consider  cocommutative measurings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' By abuse of notation, if ψ : C −→ Vectk(R, R′) is a coalgebra measuring, we will often write the  morphism ψ(x) ∈ Vectk(R, R′) simply as c : R −→ R′ for any x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  We are now ready to introduce the notion of measuring between Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let U = (U, S ) = (U, AL, sL, tL, ∆L, ǫL, S ) and U′ = (U′, S ′) = (U′, AL, s′  L, t′  L, ∆′  L, ǫ′  L, S ′) be Hopf algebroids  over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let C be a cocommutative k-coalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' A C-measuring (Ψ, ψ) from U to U′ consists of a pair of measurings  Ψ : C −→ Vectk(U, U′)  ψ : C −→ Vectk(AL, A′  L)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='6)  such that the following diagrams commute for any x ∈ C  AL  sL  −−−−−−→ U  c  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�c  A′  L  s′  L  −−−−−−→ U′  AL  tL  −−−−−−→ U  c  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�c  A′  L  t′  L  −−−−−−→ U′  U  S  −−−−−−→ U  c  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�c  U′  S ′  −−−−−−→ U′  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7)  U  ǫL  −−−−−−→ AL  c  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�c  U′  ǫ′  L  −−−−−−→ A′  L  U  ∆L  −−−−−−→ U ⊗AL U  c  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�c  U′  ∆′  L  −−−−−−→ U′ ⊗A′  L U′  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='8)  where the arrow c : U ⊗AL U −→ U′ ⊗A′  L U′ is defined by setting c(u1 ⊗ u2) := x(1)(h) ⊗ x(2)(u2) for u1 ⊗ u2 ∈ U ⊗AL U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Before proceeding further, we need to verify the following fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For any x ∈ C, the morphism c : H ⊗AL U −→ U′ ⊗A′  L U′ defined by setting c(u1 ⊗ u2) := x(1)(h) ⊗ x(2)(u2) for  u1 ⊗ u2 ∈ U ⊗AL U is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We consider u1, u2 ∈ uL and a ∈ AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Using the fact that Ψ : C −→ Vectk(U, U′) is a measuring and applying the  conditions in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='8), we see that  c((u1 · a) ⊗ u2) = c(tL(a)u1 ⊗ u2)  = x(1)(tL(a)u1) ⊗ x(2)(u2) = x(1)(tL(a))x(2)(u1) ⊗ x(3)(u2)  = t′  L(x(1)(a))x(2)(u1) ⊗ x(3)(u2)  = t′  L(x(2)(a))x(1)(u1) ⊗ x(3)(u2)  (because C is cocommutative)  = x(1)(u1) · x(2)(a) ⊗ x(3)(u2) = x(1)(u1) ⊗ x(2)(a) · x(3)(u2)  = x(1)(u1) ⊗ s′  L(x(2)(a))x(3)(u2) = x(1)(u1) ⊗ x(2)(sL(a))x(3)(u2)  = x(1)(u1) ⊗ x(2)(sL(a)u2) = x(1)(u1) ⊗ x(2)(a · u2)  = c(u1 ⊗ (a · u2))  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='9)  □  3  If U = (UL, S ) and U′ = (U′  L, S ′) are Hopf algebroids over k, we now consider the subspace  V(U, U′) ⊆ Vectk(U, U′) × Vectk(AL, A′  L)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='10)  given by setting  V(U, U′) := {(F, f) | FsL = s′  L f, FtL = t′  L f, FS = S ′F and fǫL = ǫ′  LF }  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='11)  We note that a measuring from U to U′ by means of a cocommutative coalgebra C has an underlying morphism (Ψ, ψ) : C −→  V(U, U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Let Coalgk denote the category of k-coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We know that the forgetful functor Coalgk −→ Vectk has a right adjoint  C : Vectk −→ Coalgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' In other words, we have natural isomorphisms  Vectk(C, V) � Coalgk(C, C(V))  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='12)  for any k-coalgebra C and any k-vector space V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let U = (UL, S ) and U′ = (U′  L, S ′) be Hopf algebroids over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Then, there exists a cocommutative coalgebra  Mc(U, U′) and a measuring (Φ, φ) from U to U′ satisfying the following universal property: given any measuring (Ψ, ψ) : C −→  V(U, U′) with a cocommutative coalgebra C, there exists a unique morphism ξ : C −→ Mc(U, U′) of coalgebras making the  following diagram commutative  Mc(U, U′)  (Φ,φ)  � V(U, U′)  C  ξ  �■■■■■■■■■■  (Ψ,ψ)  �✇  ✇  ✇  ✇  ✇  ✇  ✇  ✇  ✇  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='13)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We set V := V(U, U′) and consider the canonical morphism π(V) : C(V) −→ V ⊆ Vectk(uL, u′  L) × Vectk(AL, A′  L) from  the cofree coalgebra C(V) induced by the adjunction in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We now set Mc(U, U′) := � D, where the sum is taken over all  cocommutative subcoalgebras of C(V) such that the restriction π(V)|D : D −→ V = V(U, U′) is a measuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' It is clear that this  sum is still a cocommutative coalgebra, and that the restriction (Φ, φ) := π(V)|Mc(U,U′) gives a measuring from U to U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  In general, if (Ψ, ψ) : C −→ V = V(U, U′) is a cocommutative measuring, the adjunction in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='12) shows that it factors through  ξ : C −→ C(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Then, ξ(C) ⊆ C(V) is a cocommutative coalgebra such that the restriction π(V)|ξ(C) is a measuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' By  definition, it follows that ξ(C) ⊆ Mc(U, U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  □  From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='11) it is clear that given Hopf algebroids U = (UL, S ), U′ = (U′  L, S ′) and U′′ = (U′′  L, S ′′), the composition  of morphisms induces a canonical map  V(U, U′) ⊗ V(U′, U′′) −→ V(U, U′′)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='14)  We denote by CoCoalgk the category of cocommutative coalgebras over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We know that this category is symmetric monoidal  and our objective is to show that the category HAlgk of Hopf algebroids is enriched over CoCoalgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For this we need the  following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let U = (UL, S ), U′ = (U′  L, S ′) and U′′ = (U′′  L, S ′′) be Hopf algebroids over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Suppose that we have a  measuring (Ψ, ψ) : C −→ V(U, U′) from U to U′ and a measuring (Ψ′, ψ′) : C′ −→ V(U′, U′′) from U′ to U′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Then, the  following  (Ψ′, ψ′) ◦ (Ψ, ψ) : C ⊗ C′ (Ψ,ψ)⊗(Ψ′,ψ′)  −−−−−−−−−−→ V(U, U′) ⊗ V(U′, U′′) −→ V(U, U′′)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='15)  determines a measuring from U to U′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' It is easy to verify that the compositions  C ⊗ C′ Ψ⊗Ψ′  −−−−→ Vectk(U, U′) ⊗ Vectk(U′, U′′) −−−−−−→ Vectk(U, U′′)  C ⊗ C′ ψ⊗ψ′  −−−−→ Vectk(AL, A′  L) ⊗ Vectk(A′  L, A′′  L) −−−−−−→ Vectk(AL, A′′  L)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='16)  4  give coalgebra measurings from U to U′′ and from AL to A′′  L respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For c ⊗ c′ ∈ C ⊗ C′ and u ∈ U, we also see that  ∆′′  L((c ⊗ c′)(u)) = ∆′′  L(c′(c(u)))  = c′  (1)(c(u)(1)) ⊗ c′  (2)(c(u)(2))  = c′  (1)(x(1)(u(1))) ⊗ c′  (2)(x(2)(u(2)))  = (c′ ⊗ c)(1)(u(1)) ⊗ (c′ ⊗ c)(2)(u(2))  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='17)  It is also clear that the morphism in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='15) satisfies all the other conditions in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  □  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' The category HAlgk of Hopf algebroids is enriched over the category CoCoalgk of cocommutative k-coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Given Hopf algebroids U = (UL, S ) and U′ = (U′  L, S ′), we consider the “hom object” Mc(U, U′) which lies in CoCoalgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  The composition of these hom objects is obtained as follows: if U, U′ and U′′ are Hopf algebroids, we obtain as in Proposition  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='6 a measuring  Mc(U, U′) ⊗ Mc(U′, U′′) −→ V(U, U′′)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='18)  Applying the universal property in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='5, we now have a morphism of coalgebras Mc(U, U′) ⊗ Mc(U′, U′′) −→  Mc(U, U′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' The unit object in CoCoalgk is k treated as a coalgebra over itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Then, we have a unit map  k −→ V(U, U) ⊆ Vectk(U, U) × Vectk(AL, AL)  t �→ (t · iduL, t · idAL)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='19)  which induces a morphism k −→ Mc(U, U) of cocommutative coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Together with the composition of hom objects in  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='18), we see that HAlgk is enriched over CoCoalgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  □  From now onwards, we will denote by HALGk the category of Hopf algebroids enriched over the symmetric monoidal category  CoCoalgk of cocommutative k-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  3  Morphisms on cyclic (co)homology and Hopf-Galois maps  Let U = (UL, S ) = (U, AL, sL, tL, ∆L, ǫL) be a Hopf algebroid over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We now recall from [13, § 2] the cocyclic module C•(U)  that computes the cyclic cohomology of the Hopf algebroid U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For n ≥ 1, we put  Cn(U) := U ⊗AL ⊗ · · · ⊗AL U  ��������������������������������������  n-times  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1)  and set C0(U) := AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For n ≥ 1, the face maps δi : Cn(U) −→ Cn+1(U) are defined by  δi(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un) :=  \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3  1 ⊗ u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un  if i = 0  u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='. ⊗ ∆Lui ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un  if 1 ≤ i ≤ n  u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='. ⊗ un ⊗ 1  if i = n + 1  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2)  For n = 0, there are two maps δ0 := tL : C0(U) = AL −→ C1(U) = U and δ1 := sL : C0(U) = AL −→ C1(U) = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' The  degeneracy maps σi : Cn(U) −→ Cn−1(U) are given by  σi(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un) := u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ ǫL(ui+1) · ui+2 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un  0 ≤ i ≤ n − 1  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3)  The cyclic operator τn : Cn(U) −→ Cn(U) is defined by setting  τn(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un) := (S (u1)(1) · u2) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='. ⊗ (S (u1)(n−1) · un) ⊗ S (u1)(n)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4)  Since we have assumed that the antipode S is involutive, it follows from [13, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1] that C•(U) is indeed a cocyclic  module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We will denote by HC•(U) the cyclic cohomology groups of the Hopf algebroid U by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' The Hochschild cohomology  groups of the Hopf algebroid U will then be denoted by HH•(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  5  Let U, U′ be Hopf algebroids and let (Ψ, ψ) : C −→ V(U, U′) be a measuring from U to U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For each x ∈ C, we now define a  family of morphisms  Ψ  n(x) : Cn(U) −→ Cn(U′)  Ψ  n(x)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un) := x(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un) = x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(n)(un)  ∀ n ≥ 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='5)  We now prove the first main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let C be a cocommutative coalgebra and let (Ψ, ψ) : C −→ V(U, U′) be a measuring of Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  For each x ∈ C, the family {Ψ  n(x) : Cn(U) −→ Cn(U′)}n≥0 gives a morphism of cyclic modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' In particular, we have induced  morphisms  Ψ hoc(x) : HH•(U) −→ HH•(U)  Ψ cy(x) : HC•(U) −→ HC•(U)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='6)  on Hochschild and cyclic cohomologies for each x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For each x ∈ C, we start by showing that Ψ  n+1(x) ◦ δi = δ′  i ◦ Ψ  n(x) : Cn(U) −→ Cn+1(H ′), where δi and δ′  i are the face  maps on the respective cocyclic modules C•(U) and C•(U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' If i = 0 or i = n + 1, this is immediately clear from the definition  in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2) and the action in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For 1 ≤ i ≤ n, we see that  Ψ  n+1(x) ◦ δi(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un)  = Ψ  n+1(x)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='. ⊗ ∆Lui ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un)  = x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(i)(ui  (1)) ⊗ x(i+1)(ui  (2)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='. ⊗ x(n+1)(un)  = x(1)(u1) ⊗ ∆L(x(i)(ui)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(n)(un) = δ′  i ◦ Ψ  n(x)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un)  Next, we verify that Ψ  n−1(x) ◦ σi = σ′  i ◦ Ψ  n(x), where σi and σ′  i are the degeneracies on the respective cocyclic modules C•(U)  and C•(U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Ψ  n−1(x) ◦ σi(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un)  = Ψ  n−1(x)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ ǫL(ui+1) · ui+2 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un)  = x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(i+1)(ǫL(ui+1) · ui+2) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(n−1)(un)  = x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(i+1)(ǫL(ui+1)) · x(i+2)(ui+2) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(n)(un)  = x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ ǫL(x(i+1)(ui+1)) · x(i+2)(ui+2) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(n)(un)  = σ′  i ◦ Ψ  n(x)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un)  Finally, we show that Ψ  n(x)◦τn = τ′  n ◦Ψ  n(x), where τn and τ′  n are the cyclic operators on the respective cocyclic modules C•(U)  and C•(U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Ψ  n(x) ◦ τn(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un)  = Ψ  n(x)((S (u1)(1) · u2) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='. ⊗ (S (u1)(n−1) · un) ⊗ S (u1)(n))  = x(1)((S (u1)(1) · u2)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='. ⊗ x(n−1)((S (u1)(n−1) · un) ⊗ x(n)(S (u1)(n))  = x(1)(S (u1)(1)) · x(2)(u2) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='. ⊗ x(2n−3)(S (u1)(n−1)) · x(2n−2)(un) ⊗ x(2n−1)(S (u1)(n))  = x(1)(S (u1)(1)) · x(n+1)(u2) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='. ⊗ x(n−1)(S (u1)(n−1)) · x(2n−1)(un) ⊗ x(n)(S (u1)(n))  = (x(1)(S (u1)))(1) · x(2)(u2) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='. ⊗ (x(1)(S (u1)))(n−1) · x(n)(un) ⊗ (x(1)(S (u1)))(n)  = (S (x(1)(u1)))(1) · x(2)(u2) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='. ⊗ (S (x(1)(u1)))(n−1) · x(n)(un) ⊗ (S (x(1)(u1)))(n)  = τ′  n ◦ Ψ  n(x)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un)  □  We continue with a Hopf algebroid U = (UL, S ) = (U, AL, sL, tL, ∆L, ǫL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' As mentioned in Section 2, we set AR := Aop  L = Aop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Following [5, § 4], we also set  sR := tL  tR := S ◦ tL = sL  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7)  Then, U becomes an (AR, AR)-bimodule by right multiplication as follows  a1 · h · a2 := hsR(a2)tR(a1) = htL(a2)sL(a1)  h ∈ H, a1, a2 ∈ AR  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='8)  We now consider  S rl : H ⊗AR H −→ H ⊗AL H  u1 ⊗ u2 �→ S (u2) ⊗ S (u1)  S lr := S −1  rl : H ⊗AL H −→ H ⊗AR H  u1 ⊗ u2 �→ S (u2) ⊗ S (u1)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='9)  6  as well as  ∆R := S lr ◦ ∆L ◦ S : U −→ U ⊗AR U  ǫR := ǫL ◦ S : U −→ AR  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='10)  We know from [5, § 4] that the datum UR := (U, AR, sR, tR, ∆R, ǫR) defines a right bialgebroid over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We now adopt the Sweedler  notation ∆R(u) := u[1] ⊗ u[2] for any u ∈ U in order to distinguish it from the left coproduct ∆L(u) = u(1) ⊗ u(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' More explicitly,  we have  ∆R(u) = u[1] ⊗ u[2] = S (S (u)(2)) ⊗ S (S (u)(1))  ∀ u ∈ U  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='11)  Now let U, U′ be Hopf algebroids and consider (F, f) ∈ V(U, U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' From the conditions in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='11) and the definitions in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7) and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='10), we already have  FsR = s′  R f  FtR = t′  R f  FS = S ′F  fǫR = ǫ′  RF  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='12)  We now need the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let C be a cocommutative coalgebra and (Ψ, ψ) : C −→ V(U, U′) a measuring of Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Then, for  each x ∈ C, there is a well defined morphism  x : U −→ U ⊗AR U  u1 ⊗ u2 �→ x(1)(u1) ⊗ x(2)(u2)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='13)  which fits into the following commutative diagram  U  ∆R  −−−−−−→ U ⊗AR U  x  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�x  U′  ∆′  R  −−−−−−→ U′ ⊗A′  R U′  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='14)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We consider u1, u2 ∈ uL and a ∈ AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Using the fact that Ψ : C −→ Vectk(U, U′) is a measuring and applying the  conditions in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='12), we see that  c((u1 · a) ⊗ u2) = c(u1sR(a) ⊗ u2)  = x(1)(u1sR(a)) ⊗ x(2)(u2) = x(1)(u1)x(2)(sR(a)) ⊗ x(3)(u2)  = x(1)(u1)s′  R(x(2)(a)) ⊗ x(3)(u2)  = x(1)(u1) · x(2)(a) ⊗ x(3)(u2)  = x(1)(u1) ⊗ x(2)(a) · x(3)(u2) = x(1)(u1) ⊗ x(3)(u2)t′  R(x(2)(a))  = x(1)(u1) ⊗ x(3)(u2)x(2)(tR(a))  = x(1)(u1) ⊗ x(2)(u2)x(3)(tR(a))  (as C is cocommutative)  = x(1)(u1) ⊗ x(2)(u2tR(a)) = x(1)(u1) ⊗ x(2)(a · u2) = c(u1 ⊗ (a · u2))  It follows that the morphism in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='13) is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' It remains to verify the condition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For u ∈ U and x ∈ C, we  have  c(∆R(u)) = c(S (S (u)(2)) ⊗ S (S (u)(1)))  = x(1)(S (S (u)(2))) ⊗ x(2)(S (S (u)(1)))  = S ′(x(1)(S (u)(2))) ⊗ S ′(x(2)(S (u)(1)))  = S ′(x(2)(S (u)(2))) ⊗ S ′(x(1)(S (u)(1)))  (as C is cocommutative)  = S ′(c(S (u))(2)) ⊗ S ′(c(S (u))(1))  = S ′(S ′(c(u))(2)) ⊗ S ′(S ′(c(u))(1)) = ∆′  R(c(u))  □  We now recall from [13, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1] the cyclic module C•(U) defining the cyclic homology of a Hopf algebroid U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For n ≥ 0, we  set  Cn(U) := U ⊗AR ⊗ · · · ⊗AR U  ��������������������������������������  n-times  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='15)  7  and C0(U) := AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' The face maps di : Cn(U) −→ Cn−1(U) are defined by setting  di(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un) :=  \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3  ǫR(u1)u2 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un  if i = 0  u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ uiui+1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un  if i ≤ i ≤ n − 1  u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un−1ǫR(S (un))  if i = n  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='16)  The degeneracies si : Cn(U) −→ Cn+1(U) are defined as  si(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un) :=  � 1 ⊗ u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un  if i = 0  u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ ui ⊗ 1 ⊗ ui+1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='. ⊗ un  if 1 ≤ i ≤ n  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='17)  The cyclic operators tn : Cn(U) −→ Cn(U) are given by  tn(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un) := S (u1  (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='un−1  (2) un) ⊗ u1  (1) ⊗ u2  (1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un−1  (1)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='18)  The Hochschild homology groups of the Hopf algebroid U will then be denoted by HH•(U) and the cyclic homology groups  by HC•(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We will now prove the homological counterpart for Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let C be a cocommutative coalgebra and let (Ψ, ψ) : C −→ V(U, U′) be a measuring of Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For  each x ∈ C, the family  Ψn(x) : Cn(U) −→ Cn(U′)  u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un �→ x(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un) = x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(n)(un)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='19)  for n ≥ 0 gives a morphism of cyclic modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' In particular, we have induced morphisms  Ψhoc  (x) : HH•(U) −→ HH•(U′)  Ψcy  (x) : HC•(U) −→ HC•(U′)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='20)  on Hochschild and cyclic homologies for each x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Using the properties in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='12) and the fact that Ψ : C −→ Vectk(U, U′) is a measuring, it may easily be verified that the  maps Ψ•(x) commute with the respective face maps and degeneracy maps on the cyclic modules C•(U) and C•(U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Moreover,  if tn and t′  n are the respective cyclic operators on C•(U) and C•(U′), we have for each x ∈ C  c(tn(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un))  = c(S (u1  (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='un−1  (2) un) ⊗ u1  (1) ⊗ u2  (1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un−1  (1) )  = x(1)(S (u1  (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='un−1  (2) un)) ⊗ x(2)(u1  (1)) ⊗ x(3)(u2  (1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(n)(un−1  (1) )  = S ′(x(1)(u1  (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='x(n−1)(un−1  (2) )x(n)(un)) ⊗ x(n+1)(u1  (1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(2n−1)(un−1  (1) )  = S ′(x(2)(u1  (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='x(2n−2)(un−1  (2) )x(2n−1)(un)) ⊗ x(1)(u1  (1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(2n−3)(un−1  (1) )  = S ′(x(1)(u1)(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='x(n−1)(un−1)(2)x(n)(un)) ⊗ x(1)(u1)(1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(n−1)(un−1)(1)  = t′  n(x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(n)(un))  □  Our final aim in this section is to show that the morphisms induced by a measuring of Hopf algebroids are well behaved with  respect to cyclic duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' More precisely, we know from [13, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3] that there are Hopf-Galois maps  ξn(U) : Cn(U)  �  −→ Cn(U)  u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un �→ u1  (1) ⊗ u1  (2)u2  (1) ⊗ u1  (3)u2  (2)u3  (1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ u1  (n)u2  (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='.un−1  (2) un  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='21)  inducing isomorphisms between C•(U) and C•(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We now have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let C be a cocommutative coalgebra and let (Ψ, ψ) : C −→ V(U, U′) be a measuring of Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Then for each x ∈ C, the following diagram commutes  Cn(U)  ξn(U)  −−−−−−→ Cn(U)  Ψn(x)  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�Ψ  n(x)  Cn(U)  ξn(U)  −−−−−−→ Cn(U)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='22)  8  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We put N := n(n + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Using the fact that Ψ : C −→ Vectk(U, U′) is a measuring and that C is cocommutative we  have  c(ξn(U)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un))  = c(u1  (1) ⊗ u1  (2)u2  (1) ⊗ u1  (3)u2  (2)u3  (1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ u1  (n)u2  (n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='.un−1  (2) un)  = x(1)(u1  (1)) ⊗ x(2)(u1  (2))x(3)(u2  (1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(N+1−n)(u1  (n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='.x(N−1)(un−1  (2) )x(N)(un)  = x(1)(u1  (1)) ⊗ x(2)(u1  (2))x(n+1)(u2  (1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(n)(u1  (n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='.x(N−1)(un−1  (2) )x(N)(un)  = ξn(U′)(x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(n)(un))  This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  □  4  Shuffle products and the enrichment of the category of commutative Hopf alge-  broids  We recall from Section 2 the category HALGk of Hopf algebroids over k, enriched over the symmetric monoidal category  of CoCoalgk of cocommutative k-coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  By a commutative Hopf algebroid, we will mean a Hopf algebroid U =  (U, AL, sL, tL, ∆L, ǫL) such that H and AL = A = AR are commutative rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Let cHALGk denote the full subcategory of HALGk consisting of commutative Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Then, cHALGk is also en-  riched over CoCoalgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' In this section, we will obtain a second enrichment of commutative Hopf algebroids in cocommutative  coalgebras, by using the shuffle product in Hochschild homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  We know from [17, § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2] that the Hochschild homology of a commutative algebra is equipped with a shuffle product structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  For a commutative Hopf algebroid U = (U, AL, sL, tL, ∆L, ǫL), we now recall from [15, § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1] the (p, q)-shuffle product  shpq(U) : Cp(U) ⊗ Cq(U) −→ Cp+q(U)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1)  which is given by the formula (for p, q ≥ 1)  shpq(U)((u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ up) ⊗ (up+1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ up+q)) :=  �  σ∈S h(p,q)  sgn(σ)(uσ−1(1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ uσ−1(p+q))  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2)  Here S h(p, q) is the set of (p, q)-shuffles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=',  S h(p, q) := {σ ∈ S p+q | σ(1) < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' < σ(p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' σ(p + 1) < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' < σ(p + q)}  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3)  For p = q = 0, the shuffle product is given by setting sh00(U) to be the multiplication on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Further, one has (see [15, § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1])  shp0(U) : Cp(U) ⊗ C0(U) −→ Cp(U)  (u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ up) ⊗ a �→ (tL(a)u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ up)  sh0q(U) : C0(U) ⊗ Cq(U) −→ Cq(U)  a ⊗ (u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ up) �→ (u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ uqtL(a))  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4)  for p ≥ 1 and q ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' There is now an induced product structure shpq(U) : HHp(U) ⊗ HHq(U) −→ HHp+q(U) which makes the  the Hochschild homology HH•(U) :=  �  n≥0  HHp(U) of a commutative Hopf algebroid U into a graded algebra (see [15, § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1])  that we denote by (HH•(U), sh(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let U, U′ be commutative Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let C be a cocommutative coalgebra and let (Ψ, ψ) : C −→  V(U, U′) be a measuring of Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Then, the induced K-linear map  Ψhoc : C −→ HomK(HH•(U), HH•(U′))  x �→ (Ψhoc  (x) : HH•(U) −→ HH•(U′))  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='5)  gives a measuring of algebras from (HH•(U), sh(U)) to (HH•(U′), sh(U′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  9  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' The unit in (HH•(U), sh(U)) is given by the class of the unit 1A ∈ A = C0(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Since ψ : C −→ HomK(A, A′) gives in  particular a measuring from A to A′, we have Ψhoc  (x)(1A) = 1A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We now note that for any x ∈ C and p, q ≥ 1, we have  Ψp+q(x)(shpq(U)((u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ up) ⊗ (up+1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ up+q)))  = Ψp+q(x)  �  �  σ∈S h(p,q)  sgn(σ)(uσ−1(1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ uσ−1(p+q))  �  =  �  σ∈S h(p,q)  sgn(σ)(x(1)(uσ−1(1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(p+q)(uσ−1(p+q)))  =  �  σ∈S h(p,q)  sgn(σ)(xσ−1(1)(uσ−1(1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ xσ−1(p+q)(uσ−1(p+q)))  (because C is cocommutative)  = shpq(U)((x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(p)(up)) ⊗ (x(p+1)(up+1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(p+q)(up+q)))  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='6)  For p ≥ 1, we have  Ψp(x)(shp0(U)((u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ up) ⊗ a)  = Ψp(x)((tL(a)u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ up))  = (x(1)(tL(a)u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(p)(up))  = (x(1)(tL(a))x(2)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(p+1)(up))  = (tL(x(p+1)(a)))x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(p)(up))  = shp0(U)((x(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ x(p)(up)) ⊗ x(p+1)(a))  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7)  We can similarly verify the case for sh0q with q ≥ 1 and for sh00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  □  Our next objective is to use Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1 to obtain an enrichment of commutative Hopf algebroids over the category of  cocommutative coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For that we recall the following fact: if R, R′ are k-algebras, the category of coalgebra measurings  from R to R′ contains a final object ϕ(R, R′) : M(R, R′) −→ Vectk(R, R′) (see Sweedler [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Then, M(R, R′) is known as the  universal measuring coalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We let Mc(R, R′) be the cocommutative part of the coalgebra M(R, R′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Then, the restriction  ϕc(R, R′) : Mc(R, R′) ֒→ M(R, R′) −→ Vectk(R, R′) becomes the final object in the category of cocommutative coalgebra  measurings from R to R′ (see [9, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4], [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Further, the objects Mc(R, R′) give an enrichment of k-algebras over  cocommutative k-coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  We now define the enriched category  �  cHALGk whose objects are commutative Hopf algebroids over k and whose hom-objects  are defined by setting  �  cHALGk(U, U′) := Mc((HH•(U), sh(U)), (HH•(U′), sh(U′))) ∈ CoCoalgk  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='8)  for commutative Hopf algebroids U, U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Since each (HH•(U), sh(U)) is an algebra, we also have a canonical morphism k −→  Mc((HH•(U), sh(U)), (HH•(U), sh(U))) of cocommutative coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let U, U′ be commutative Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Then, there is a canonical morphism of cocommutative coalgebras  τ(U, U′) : Mc(U, U′) −→ Mc((HH•(U), sh(U)), (HH•(U′), sh(U′)))  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='9)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' By definition, (Φ, φ) : Mc(U, U′) −→ V(U, U′) is a cocommutative measuring from U to U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1,  this induces a measuring of algebras from (HH•(U), sh(U)) to (HH•(U′), sh(U′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' By the universal property of the universal  cocommutative measuring coalgebra Mc((HH•(U), sh(U)), (HH•(U′), sh(U′))), we now obtain an induced morphism τ(U, U′)  as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  □  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' There is a CoCoalgk enriched functor cHALGk −→  �  cHALGk which is identity on objects and whose mapping  on hom-objects is given by  τ(U, U′) : cHALGk(U, U′) = Mc(U, U′) −→ Mc((HH•(U), sh(U)), (HH•(U′), sh(U′))) =  �  cHALGk(U, U′)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='10)  for commutative Hopf algebroids U, U′ over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  10  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let U, U′, U′′ be commutative Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We show that the following diagram commutes  Mc(U, U′) ⊗ Mc(U′, U′′) −−−−−−→  Mc(U, U′′)  τ(U,U′)⊗τ(U′,U′′)  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�τ(U,U′′)  Mc(HH•(U), HH•(U′)) ⊗ Mc(HH•(U′), HH•(U′′)) −−−−−−→ Mc(HH•(U), HH•(U′′))  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='11)  The top horizontal composition ◦ : Mc(U, U′) ⊗ Mc(U′, U′′) −→ Mc(U, U′′) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='11) is obtained from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7, while the  bottom horizontal composition ◦ : Mc(HH•(U), HH•(U′)) ⊗ Mc(HH•(U′), HH•(U′′)) −→ Mc(HH•(U), HH•(U′′)) is obtained  from the enrichment of algebras in cocommutative coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7, we note that all the maps in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='11) are morphisms of cocommutative coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' It follows  from the property of the universal cocommutative measuring coalgebra Mc(HH•(U), HH•(U′′)) that in order to show that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='11)  commutes, it suffices to verify that the following two compositions are equal  Mc(U, U′) ⊗ Mc(U′, U′′)  τ(U,U′)⊗τ(U′,U′′)  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�  Mc(HH•(U), HH•(U′)) ⊗ Mc(HH•(U′), HH•(U′′)) \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�  Mc(HH•(U), HH•(U′′))  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�ϕc(HH•(U),HH•(U′′))  Vectk(HH•(U), HH•(U′′))  Mc(U, U′) ⊗ Mc(U′, U′′)  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�◦  Mc(U, U′′)  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�τ(U,U′′)  Mc(HH•(U), HH•(U′′))  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�ϕc(HH•(U),HH•(U′′))  Vectk(HH•(U), HH•(U′′))  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='12)  For the sake of convenience, we denote the left vertical composition in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='12) by ψ1 and the right vertical composition by ψ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  We now consider x ∈ Mc(U, U′), y ∈ Mc(U′, U′′) and (u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ up) ∈ Cp(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We see that  ψ2(x ⊗ y)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ up)  = (y ◦ x)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ up)  = (y ◦ x)(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ (y ◦ x)(p)(up)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='13)  Since ◦ : Mc(U, U′) ⊗ Mc(U′, U′′) −→ Mc(U, U′′) is a morphism of coalgebras, we note that (y ◦ x)(1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ (y ◦ x)(p) =  (y(1) ◦ x(1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ (y(p) ◦ x(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Combining with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='13), we see that the right vertical composition in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='12) may be described  explicitly as  ψ2(x ⊗ y)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ up) = (y(1) ◦ x(1))(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ (y(p) ◦ x(p))(up) = y(1)(x(1)(u1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ y(p)(x(p)(up))  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='14)  On the other hand, we note that the following diagram is commutative  Mc(U, U′) ⊗ Mc(U′, U′′)  (τ(U,U′)⊗τ(U′,U′′))  −−−−−−−−−−−−−−−−→  Mc(HH•(U), HH•(U′′))  (ϕc(HH•(U),HH•(U′))◦τ(U,U′))⊗  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�(ϕc(HH•(U′),HH•(U′′))◦τ(U′,U′′))  ϕc(HH•(U),HH•(U′′))  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�  Vectk(HH•(U), HH•(U′)) ⊗ Vectk(HH•(U′), HH•(U′′)) −−−−−−→  Vectk(HH•(U), HH•(U′′))  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='15)  From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='15), it follows that the left vertical composition in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='12) may be described explicitly as  ψ1(x ⊗ y)(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ up)  = y(x(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ up))  = y(1)(x(1)(u1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ y(p)(x(p)(up))  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='16)  From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='14) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='16), we see that ψ1 = ψ2 and hence the diagram (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='11) commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Similarly by considering the coalgebra k  and using the fact that the p-th iterated coproduct ∆p(1) = 1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ 1(p-times), we see that the following compositions are equal  k −→ Mc(HH•(U), HH•(U))  ϕc(HH•(U),HH•(U))  −−−−−−−−−−−−−−−→ Vectk(HH•(U), HH•(U))  k −→ Mc(U, U)  τ(U,U)  −−−−−→ Mc(HH•(U), HH•(U))  ϕc(HH•(U),HH•(U))  −−−−−−−−−−−−−−−→ Vectk(HH•(U), HH•(U))  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='17)  11  It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='17) that the following diagram commutes  k  �  �● Mc(HH•(U), HH•(U))  Mc(U, U)  τ(U,U)  �❧ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='18)  This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  □  5  Comodule measurings for SAYD modules  Let U = (U, AL, sL, tL, ∆L, ǫL, S ) be a Hopf algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' From now onwards, we set Ae := A ⊗k Aop and define  ηL : Ae = A ⊗k Aop  sL⊗tL  −−−−→ U ⊗ U −→ U  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1)  where the second arrow in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1) is the multilplication on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Following [14, § 2], we note that there are now four commuting  actions of A on U which are denoted as follows  a ⊲ u ⊳ b := sL(a)tL(b)u  a ◮ u ◭ b := usL(b)tL(a)  a, b ∈ A, u ∈ U  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2)  By Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1, we then have an A-coring  ∆L : U −→ U⊳ ⊗A ⊲U  ǫL : U −→ A  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3)  The left action ◮ of A on U may be treated as a right action of Aop on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Similarly, the right action ⊳ of A on U may be treated  as a left action by Aop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Accordingly, we may consider the tensor product  ◮U ⊗Aop U⊳ := U ⊗k U/span{a ◮ u ⊗ v − u ⊗ v ⊳ a | u, v ∈ U, a ∈ A}  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4)  There is now a Hopf-Galois map (see [5], [14], [19])  β(U) : ◮U ⊗Aop U⊳ −→ U⊳ ⊗A ⊲U  u ⊗Aop v �→ u(1) ⊗A u(2)v  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='5)  Since U is a Hopf algebroid, it follows (see [5, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2]) that the morphism β(U) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='5) is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Accordingly, in  the notation of [14], [19], we write  u+ ⊗Aop u− := β(U)−1(u ⊗A 1)  u ∈ U  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='6)  In this section, we will consider comodule measurings between stable anti-Yetter Drinfeld modules over Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For  this, we first recall the notion of comodule measuring between ordinary modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let R, R′ be rings and let P, P′ be modules  over R and R′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Then, a comodule measuring from P to P′ consists of a pair of maps (see [4], [12])  ψ : C −→ Vectk(R, R′)  ω : D −→ Vectk(P, P′)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7)  where C is a k-coalgebra, D is a right C-comodule, ψ : C −→ Vectk(R, R′) is a coalgebra measuring and  ω(y)(pr) = y(pr) = y(0)(p)y(1)(r) = ω(y(0))(p)ψ(y(1))(r)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='8)  for y ∈ D, p ∈ P and r ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For U = (U, AL, sL, tL, ∆L, ǫL, S ), we will now recall the notions of U-modules, U-comodules and  stable anti-Yetter Drinfeld modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' (see [14, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4]) Let U = (U, AL, sL, tL, ∆L, ǫL, S ) be a Hopf algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' A right U-module P is a right module  over the k-algebra U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Because of the ring homomorphism ηL : Ae −→ U, any right U-module P is also equipped with a right  Ae-module structure (or (A, A)-bimodule structure) given by  b ◮ p ◭ a = p(a ⊗ b) = pηL((a ⊗ 1)(1 ⊗ b)) = psL(a)tL(b)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='9)  for (a ⊗ b) ∈ Ae = A ⊗k Aop and p ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  12  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' (see [6], [8], [14], [18]) Let U = (U, AL, sL, tL, ∆L, ǫL, S ) be a Hopf algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' A left U-comodule P is a left  comodule over the A-coring (U, ∆L : U −→ U ⊗AL U, ǫL : U −→ AL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' In particular, a left U-comodule P is equipped with a left  A-module structure (a, p) �→ ap as well as a left A-module map  ∆P : P −→ U⊳ ⊗A P  p �→ p(−1) ⊗ p(0)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='10)  Following [14, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='5], we note that any left U-comodule P also carries a right A-module structure given by setting  pa := ǫL(p(−1)sL(a))p(0)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='11)  for p ∈ P, a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' This makes any left U-comodule P into a right Ae = A ⊗k Aop-module by setting  p(a ⊗ b) = bpa = bǫL(p(−1)sL(a))p(0)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='12)  for p ∈ P and (a ⊗ b) ∈ Ae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' (see [14, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7]) Let U = (U, AL, sL, tL, ∆L, ǫL, S ) be a Hopf algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' A stable anti-Yetter Drinfeld  module (or SAYD module) P over U consists of the following  (1) A right U-module structure on P denoted by (p, u) �→ pu for p ∈ P and u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  (2) A left U-comodule structure on P given by ∆P : P −→ U⊳ ⊗A P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  (3) The right Ae-module structure on P induced by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='9) coincides with the right Ae-module structure on P as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='12):  psL(a)tL(b) = b ◮ p ◭ a = bǫL(p(−1)sL(a))p(0)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='13)  (4) For u ∈ U and p ∈ P, one has  ∆P(pu) = u−p(−1)u+(1) ⊗A p(0)u+(2)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='14)  (5) Stability condition: for any p ∈ P, one has p(0)p(−1) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let R, R′ be k-algebras and let Re = R ⊗k Rop, R′e = R′ ⊗k R′op be their respective enveloping algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let C be  a cocommutative k-coalgebra and let ψ : C −→ Vectk(R, R′) be a measuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Then,  ψe : C −→ Vectk(Re, R′e)  ψe(c)(r ⊗ r′) = c(r1 ⊗ r2) = c(1)(r1) ⊗ c(2)(r2) = ψ(c(1))(r1) ⊗ ψ(c(2))(r2)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='15)  is a measuring of algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='15), it is immediate that c(1 ⊗ 1) = ǫC(c)(1 ⊗ 1), where ǫC is the counit on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Since C is cocommutative, we  have for (r1 ⊗ r2), (r3 ⊗ r4) ∈ Re  c((r1 ⊗ r2)(r3 ⊗ r4)) = c(r1r3 ⊗ r4r2)  = c(1)(r1r3) ⊗ c(2)(r4r2)  = c(1)(r1)c(2)(r3) ⊗ c(3)(r4)c(4)(r2)  = c(1)(r1)c(3)(r3) ⊗ c(4)(r4)c(2)(r2)  = (c(1)(r1) ⊗ c(2)(r2))(c(3)(r3) ⊗ c(4)(r4))  = c(1)(r1 ⊗ r2)c(2)(r3 ⊗ r4)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='16)  □  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let U = (U, S ) = (U, AL, sL, tL, ∆L, ǫL, S ) and U′ = (U′, S ′) = (U′, AL, s′  L, t′  L, ∆′  L, ǫ′  L, S ′) be Hopf algebroids over  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P′) be an SAYD-module over U (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let C be a cocommutative k-coalgebra and D be a right C-comodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Suppose that we are given the following data  Ψ : C −→ Vectk(U, U′)  ψ : C −→ Vectk(A, A′)  Ω : D −→ Vectk(P, P′)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='17)  such that  13  (1) (Ψ, ψ) is a measuring of Hopf algebroids from U to U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  (2) (Ψ, Ω) is a comodule measuring from the right U-module P to the right U′ module P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Then, we have:  (a) (ψe, Ω) is a comodule measuring from the right Ae-module P to the right A′e-module P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  (b) For each d ∈ D, the following morphism is well-defined  d : U⊳ ⊗A P −→ U′  ⊳ ⊗A′ P′  d(u ⊗A p) := Ψ(d(1))(u) ⊗A′ Ω(d(0))(p)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='18)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' (a) Since C is cocommutative, we already know from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4 that ψe : C −→ Vectk(Ae, A′e) is a coalgebra measuring  from Ae to A′e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We now consider (a ⊗ b) ∈ Ae = A ⊗k Aop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='9), we know that p(a ⊗ b) = psL(a)tL(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For any d ∈ D, we  now have  Ω(d)(p(a ⊗ b)) = Ω(d)(psL(a)tL(b))  = Ω(d(0))(p)Ψ(d(1))(sL(a)tL(b))  = Ω(d(0))(p)Ψ(d(1))(sL(a))Ψ(d(2))(tL(b))  = Ω(d(0))(p)s′  L(ψ(d(1))(a))t′  L(ψ(d(2))(b))  = Ω(d(0))(p)(ψ(d(1)(a)) ⊗ ψ(d(2)(b)))  = Ω(d(0))(p)(ψe(d(1))(a ⊗ b))  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='19)  (b) Since P and P′ are SAYD modules, it follows from the definition in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='9) and the condition in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='13) that  ap = ptL(a)  a′p′ = p′t′  L(a′)  a ∈ A, a′ ∈ A′, p ∈ P, p′ ∈ P′  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='20)  where the left hand side of the equalities in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='20) comes from the left A-module action on P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' the left A′-module action  on P′) appearing in the structure map ∆P : P −→ U⊳ ⊗A P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' the structure map ∆′  P′ : P′ −→ U′  ⊳ ⊗A′ P′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For a ∈ A, u ∈ U  and p ∈ P, we now see that  d(u ⊗A ap)  = Ψ(d(1))(u) ⊗A′ Ω(d(0))(ap)  = Ψ(d(1))(u) ⊗A′ Ω(d(0))(ptL(a))  (using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='20))  = Ψ(d(2))(u) ⊗A′ Ω(d(0))(p)Ψ(d(1))(tL(a))  = Ψ(d(2))(u) ⊗A′ Ω(d(0))(p)t′  L(ψ(d(1))(a))  = Ψ(d(2))(u) ⊗A′ ψ(d(1))(a)Ω(d(0))(p)  (using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='20))  = Ψ(d(2))(u) ⊳ ψ(d(1))(a) ⊗A′ Ω(d(0))(p)  = t′  L(ψ(d(1))(a))Ψ(d(2))(u) ⊗A′ Ω(d(0))(p)  (using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2))  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='21)  On the other hand, we also have  d(u ⊳ a ⊗A p)  = d(tL(a)u ⊗A p)  (using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2))  = Ψ(d(1))(tL(a)u) ⊗A′ Ω(d(0))(p)  = Ψ(d(1))(tL(a))Ψ(d(2))(u) ⊗A′ Ω(d(0))(p)  = t′  L(ψ(d(1))(a))Ψ(d(2))(u) ⊗A′ Ω(d(0))(p)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='22)  This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  □  We are now ready to introduce the notion of a comodule measuring between SAYD modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let U = (U, S ) = (U, AL, sL, tL, ∆L, ǫL, S ) and U′ = (U′, S ′) = (U′, AL, s′  L, t′  L, ∆′  L, ǫ′  L, S ′) be Hopf algebroids  over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P′) be an SAYD-module over U (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let C be a cocommutative coalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Then, a (right) measuring  comodule over (C, Ψ, ψ) from P to P′ consists of the following data  Ψ : C −→ Vectk(U, U′)  ψ : C −→ Vectk(A, A′)  Ω : D −→ Vectk(P, P′)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='23)  such that  (1) (Ψ, ψ) is a measuring of Hopf algebroids from U to U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  14  (2) (Ψ, Ω) is a comodule measuring from the right U-module P to the right U′ module P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  (3) For any d ∈ D, the following diagram commutes  P  ∆P  −−−−−−→ U⊳ ⊗A P  d:=Ω(d)  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�  d  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�  P′  ∆′  P′  −−−−−−→ U′  ⊳ ⊗A′ P′  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='24)  where the right vertical morphism is as defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='18)  We will now construct universal measuring comodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' By definition, the right comodules over a k-coalgebra C are coalgebras  over the comonad  ⊗k C : Vectk −→ Vectk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Accordingly, the forgetful functor Comod − C −→ Vectk from the category of  right C-comodules has a right adjoint (see, for instance, [7, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4]) that we denote by RC, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=', we have natural isomorphisms  Vectk(D, V) � Comod − C(D, RC(V))  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='25)  for any D ∈ Comod − C and V ∈ Vectk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let U = (U, AL, sL, tL, ∆L, ǫL, S ) and U′ = (U′, AL, s′  L, t′  L, ∆′  L, ǫ′  L, S ′) be Hopf algebroids over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P′)  be an SAYD-module over U (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let C be a cocommutative coalgebra and (Ψ, ψ) : C −→ V(U, U′) be a measuring of  Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Then, there exists a measuring (C, Ψ, ψ)-comodule (QC(P, P′), Θ : QC(P, P′) −→ Vectk(P, P′)) satisfying the following property:  given any measuring (C, Ψ, ψ)-comodule (D, Ω : D −→ Vectk(P, P′)) from P to P′, there exists a morphism χ : D −→ QC(P, P′)  of right C-comodules such that the following diagram is commutative  QC(P, P′)  Θ  � Vectk(P, P′)  D  χ  �❍❍❍❍❍❍❍❍❍  Ω  �t  t  t  t  t  t  t  t  t  t  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='26)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We put V := Vectk(P, P′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' By the adjunction in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='25), there is a canonical morphism ρ(V) : RC(V) −→ V of vector  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We set QC(P, P′) := � Q, where the sum is taken over all right C-subcomodules over RC(V) such that the restriction  ρ(V)|Q : Q −→ V = Vectk(P, P′) is a (C, Ψ, ψ)-comodule measuring from P to P′ in the sense of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' It is clear that  Θ : ρ(V)|QC(P, P′) : QC(P, P′) −→ V = Vectk(P, P′) is a (C, Ψ, ψ)-measuring comodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Additionally, given a measuring (C, Ψ, ψ)-comodule (D, Ω : D −→ Vectk(P, P′)) from P to P′, the adjunction in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='25) gives a  morphism χ : D −→ RC(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' But then we notice that ρ(V)|χ(D) : χ(D) −→ V is a measuring (C, Ψ, ψ)-comodule, whence it  follows that the image χ(D) ⊆ QC(P, P′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' The result is now clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  □  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let U = (U, AL, sL, tL, ∆L, ǫL, S ), U′ = (U′, AL, s′  L, t′  L, ∆′  L, ǫ′  L, S ′) and U′′ = (U′′, A′′  L, s′′  L, t′′  L , ∆′′  L, ǫ′′  L , S ′′) be Hopf  algebroids over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let P, P′ and P′′ be SAYD modules over U, U′ and U′′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Suppose that we have:  (1) Ψ : C −→ Vectk(U, U′), ψ : C −→ Vectk(A, A′) and Ω : D −→ Vectk(P, P′) giving the data of a measuring comodule from  P to P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  (2) Ψ′ : C′ −→ Vectk(U′, U′′), ψ′ : C′ −→ Vectk(A′, A′′) and Ω : D′ −→ Vectk(P′, P′′) giving the data of a measuring  comodule from P′ to P′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Then, the following  (Ψ′, ψ′) ◦ (Ψ, ψ) : C ⊗ C′ (Ψ,ψ)⊗(Ψ′,ψ′)  −−−−−−−−−−→ V(U, U′) ⊗ V(U′, U′′) −→ V(U, U′′)  Ω′ ◦ Ω : D ⊗ D′ Ω⊗Ω′  −−−−→ Vectk(P, P′) ⊗ Vectk(P′, P′′)  −→ Vectk(P, P′′)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='27)  gives the data of a measuring comodule from P to P′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' There is also a canonical morphism of right (C ⊗ C′)-comodules  QC(P, P′) ⊗ QC′(P′, P′′) −→ QC⊗C′(P, P′′)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='28)  15  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We know from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='6 that (Ψ′, ψ′) ◦ (Ψ, ψ) : C ⊗ C′ −→ V(U, U′′) is a measuring of Hopf algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' It  may also be directly verified that ((Ψ′, ψ′) ◦ (Ψ, ψ), Ω′ ◦ Ω) is a comodule measuring from the right U-module P to the right  U′′-module P′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' To check the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='24) in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='6, we observe that for any d ⊗ d′ ∈ D ⊗ D′, u ∈ U and p ∈ P:  (d ⊗ d′)(u ⊗A p) = (d ⊗ d′)(1)(u) ⊗A′′ (d ⊗ d′)(0)(p) = d′  (1)(d(1)(u)) ⊗A′′ d′  (0)(d(0)(p)) = d′(d(u ⊗A p)))  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='29)  Since the measurings (Ψ, ψ, Ω) and (Ψ′, ψ′, Ω′) both satisfy the condition in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='24), it is clear that so does (Ψ′◦Ψ, ψ′ ◦ψ, Ω′ ◦Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Hence, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='27) gives the data of a measuring comodule from P to P′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' By definition, QC(P, P′) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' QC′(P′, P′′)) is a measuring  comodule from P to P′ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' from P′ to P′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='27) it now follows that QC(P, P′) ⊗ QC′(P′, P′′) is a measuring comodule  from P to P′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' The morphism in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='28) is now obtained by the universal property of QC⊗C′(P, P′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  □  We now consider the “global category of comodules” Comodk whose objects are pairs (C, D), where C is a cocommutative k-  coalgebra and D is a right C-comodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' A morphism (f, g) : (C, D) −→ (C′, D′) in Comodk consists of a k-coalgebra morphism  f : C −→ C′ and a morphism g : D −→ D′ of C′-comodules, where D is treated as a C′-comodule by corestriction of scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  It is clear that putting (C, D) ⊗ (C′, D′) := (C ⊗ C′, D ⊗ D′) makes Comodk into a symmetric monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let S AYDk be the category given by:  (a) Objects: pairs (U, P), where U is a Hopf-algebroid and P is an S AYD-module over U  (b) Hom-objects: for pairs (U, P), (U′, P′) ∈ S AYDk, we set  S AYDk((U, P), (U′, P′)) := (Mc(U, U′), QMc(U,U′)(P, P′)) ∈ Comodk  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='30)  Then, S AYDk is enriched over the symmetric monoidal category Comodk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For any (U, P) ∈ S AYDk, the scalar multiples of the identity map give a morphism k −→ Mc(U, U) of k-coalgebras,  and along with the universal property in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7 give a morphism k −→ QMc(U,U)(P, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We now consider (U, P), (U′, P′),  (U′′, P′′) ∈ S AYDk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='8 with C = Mc(U, U′) and C′ = Mc(U′, U′′), we obtain a morphism QMc(U,U′)(P, P′) ⊗  QMc(U′,U′′)(P, P′) −→ QMc(U,U′)⊗Mc(U′,U′′)(P, P′′) of (Mc(U, U′) ⊗ Mc(U′, U′′))-comodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' From the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7, we  already have a morphism Mc(U, U′) ⊗ Mc(U′, U′′) −→ Mc(U, U′′) of k-coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Combining, we have a morphism  S AYDk((U, P), (U′, P′)) ⊗ S AYDk((U′, P′), (U′′, P′′)) −→ (Mc(U, U′′), QMc(U,U′)⊗Mc(U′,U′′)(P, P′′))  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='31)  in Comodk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' In (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='31), QMc(U,U′)⊗Mc(U′,U′′)(P, P′′) is treated as a Mc(U, U′′)-module via the morphism Mc(U, U′)⊗Mc(U′, U′′) −→  Mc(U, U′′) of k-coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' From the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7, we also know that the morphism Mc(U, U′) ⊗ Mc(U′, U′′) −→  Mc(U, U′′) arises from the universal property of Mc(U, U′′) applied to the measuring Mc(U, U′) ⊗ Mc(U′, U′′) −→ V(U, U′) ⊗  V(U′, U′′)  −→ V(U, U′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Hence, the canonical map QMc(U,U′)⊗Mc(U′,U′′)(P, P′′) −→ Vectk(P, P′′) gives a measuring when treated  as a Mc(U, U′′)-comodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' The universal property of QMc(U,U′′)(P, P′′) as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7 now yields a morphism  (Mc(U, U′′), QMc(U,U′)⊗Mc(U′,U′′)(P, P′′)) −→ (Mc(U, U′′), QMc(U,U′′)(P, P′′))  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='32)  in Comodk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Composing (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='32) with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='31), we obtain the required composition of Hom-objects S AYDk((U, P), (U′, P′)) ⊗  S AYDk((U′, P′), (U′′, P′′)) −→ S AYDk((U, P), (U′′, P′′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  □  6  Comodule measurings and morphisms on cyclic (co)homology  Throughout this section, we fix the following: let U = (U, AL, sL, tL, ∆L, ǫL, S ), and U′ = (U′, A′  L, s′  L, t′  L, ∆′  L, ǫ′  L, S ′) be Hopf  algebroids over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let P and P′ be SAYD modules over U and U′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let (Ψ, ψ) : C −→ V(U, U′) be a cocommutative  measuring and let Ω : D −→ Vectk(P, P′) be a (C, Ψ, ψ)-measuring comodule from P to P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Since U, U′ are Hopf algebroids, we have recalled in Section 5 that the morphisms β(U) : ◮U ⊗Aop U⊳ −→ U⊳ ⊗A ⊲U and  β(U′) : ◮U′ ⊗A′op U′  ⊳ −→ U′  ⊳ ⊗A′ ⊲U′ in the notation of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='5) are bijections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We now need the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  16  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For each x ∈ C, the following diagram commutes:  U⊳ ⊗A ⊲U  β(U)−1  −−−−−−→  ◮U ⊗Aop U⊳  x  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�  x  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�  U′  ⊳ ⊗A′ ⊲U′  β(U′)−1  −−−−−−→ ◮U′ ⊗A′op U′  ⊳  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1)  Here, the left vertical map is given by u1 ⊗A u2 �→ x(1)(u1) ⊗A′ x(2)(u2) and the right vertical map by u1 ⊗Aop u2 �→ x(1)(u1) ⊗A′op  x(2)(u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' It is easy to see that the vertical morphisms in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1) are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Further, since β(U) and β(U′) are invertible, it  suffices to check that the following diagram commutes  ◮U ⊗Aop U⊳  β(U)  −−−−−−→ U⊳ ⊗A ⊲U  x  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�x  ◮U′ ⊗A′op U′  ⊳  β(U′)  −−−−−−→ U′  ⊳ ⊗A′ ⊲U′  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2)  We now see that for u ⊗Aop v ∈ ◮U ⊗Aop U⊳ and x ∈ C, we have  x(β(U)(u ⊗Aop v)) = x(u(1) ⊗A u(2)v) = x(1)(u(1)) ⊗A x(2)(u(2))x(3)(v) = (x(1)(u))(1) ⊗A (x(1)(u))(2)x(2)(v) = β(U′)(x(1)(u) ⊗ x(2)(v))  This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  □  From Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1, it follows in the notation of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='6) that we have  x(1)(u+) ⊗A′op x(2)(u−) = x(u+ ⊗Aop u−) = β(U′)−1(x(u ⊗A 1)) = x(u)+ ⊗A′op x(u)−  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3)  for each u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We now recall from [14, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1] that the Hochschild homology groups HH•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' the cyclic  homology groups HC•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P)) of U with coefficients in the SAYD module P are obtained from the cyclic module C•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P) :=  P ⊗Aop (◮U⊳)⊗Aop• with operators as follows (where ¯u := u1 ⊗Aop ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗Aop un, p ∈ P)  di(p ⊗Aop ¯u) :=  \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3  p ⊗Aop u1 ⊗Aop · · · ⊗Aop un−1tL(ǫL(un))  if i = 0  p ⊗Aop u1 ⊗Aop · · · ⊗Aop un−iun−i+1 ⊗Aop .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  if 1 ≤ i ≤ n − 1  pu1 ⊗Aop u2 ⊗Aop · · · ⊗Aop un  if i = n  si(p ⊗Aop ¯u) :=  \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3  p ⊗Aop u1 ⊗Aop · · · ⊗Aop un ⊗Aop 1  if i = 0  p ⊗Aop · · · ⊗Aop un−i ⊗Aop 1 ⊗Aop un−i+1 ⊗Aop .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  if 1 ≤ i ≤ n − 1  p ⊗Aop 1 ⊗Aop u1 ⊗Aop · · · ⊗Aop un  if i = n  tn(p ⊗Aop ¯u) := p(0)u1  + ⊗Aop u2  + ⊗Aop · · · ⊗Aop un  + ⊗Aop un  − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='u1  −p(−1)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4)  We now have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For each y ∈ D, the family  Ωn(y) : Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P) −→ Cn(U′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P′)  p ⊗ u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un �→ y(p ⊗ u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un) = y(0)(p) ⊗ y(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ y(n)(un)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='5)  for n ≥ 0 gives a morphism of cyclic modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' In particular, we have induced morphisms  Ωhoc  (y) : HH•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P) −→ HH•(U′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P′)  Ωcy  (y) : HC•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P) −→ HC•(U′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P′)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='6)  on Hochschild and cyclic homologies for each y ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  17  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' From the fact that C is cocommutative and the conditions in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='6, it is clear that the morphisms Ωn(y) are well  defined, as well as the fact that they commute with the face maps and degeneracies appearing in the cyclic modules C•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P)  and C•(U′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P′) as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' To verify that the morphisms in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='5) also commute with the cyclic operators, we note that for  p ⊗Aop u1 ⊗Aop ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗Aop un ∈ Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P)  y(tn(p ⊗ u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un)) = y(p(0)u1  + ⊗ u2  + ⊗ · · · ⊗ un  + ⊗ un  − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' u1  −p(−1))  = y(0)(p(0))y(1)(u1  +) ⊗ y(2)(u2  +) ⊗ · · · ⊗ y(n)(un  +) ⊗ y(n+1)(un  −) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='y(2n)(u1  −)y(2n+1)(p(−1))  = y(0)(p(0))y(2)(u1  +) ⊗ y(4)(u2  +) ⊗ · · · ⊗ y(2n)(un  +) ⊗ y(2n+1)(un  −) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' y(3)(u1  −)y(1)(p(−1))  (since C is cocommutative)  = y(0)(p)(0)y(1)(u1  +) ⊗ y(3)(u2  +) ⊗ · · · ⊗ y(2n−1)(un  +) ⊗ y(2n)(un  −) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' y(2)(u1  −)y(0)(p)(−1)  (using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='24))  = y(0)(p)(0)y(1)(u1)+ ⊗ y(2)(u2)+ ⊗ · · · ⊗ y(n)(un)+ ⊗ y(n)(un)− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' y(1)(u1)−y(0)(p)(−1)  (using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3))  This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  □  We now come to cyclic cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For this, we recall that from [14, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1] that the Hochschild cohomology groups  HH•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' the cyclic cohomology groups HC•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P)) of U with coefficients in the SAYD module P are obtained from  the cocyclic module C•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P) := (⊲U⊳)⊗A• ⊗A P with operators as follows (where ¯u := u1 ⊗A ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗A un, p ∈ P)  δi(¯u ⊗A p)  =  \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3  1 ⊗A u1 ⊗A · · · ⊗A un ⊗A p  if i = 0  u1 ⊗A · · · ⊗A ∆L(ui) ⊗A · · · ⊗A un ⊗A p  if 1 ≤ i ≤ n  u1 ⊗A · · · ⊗A un ⊗A p(−1) ⊗A p(0)  if i = n + 1  δi(p)  =  � 1 ⊗A p  if j = 0  p(−1) ⊗A p(0)  if j = 1  σi(¯u ⊗A p)  = u1 ⊗A · · · ⊗A ǫL(ui+1) ⊗A · · · ⊗A un ⊗A p  0 ≤ i ≤ n − 1  τn(¯u ⊗A p)  = u1  −(1)u2 ⊗A · · · ⊗A u1  −(n−1)un ⊗A u1  −(n)p(−1) ⊗A p(0)u1  +  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7)  We now have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For each y ∈ D, the family  Ω  n(y) : Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P) −→ Cn(U′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P′)  u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un ⊗ p �→ y(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un ⊗ p) = y(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ y(n)(un) ⊗ y(0)(p)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='8)  for n ≥ 0 gives a morphism of cocyclic modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' In particular, we have induced morphisms  Ω hoc(y) : HH•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P) −→ HH•(U′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P′)  Ω cy(y) : HC•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P) −→ HC•(U′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P′)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='9)  on Hochschild and cyclic cohomologies for each y ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' It is clear that the morphisms in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='8) are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For y ∈ D and i = n + 1 in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7), we note that  y(δn+1(u1 ⊗ · · · ⊗ un ⊗ p))  = y(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' y(n)(un) ⊗ y(n+1)(p(−1)) ⊗ y(0)(p(0))  = y(2)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' y(n+1)(un) ⊗ y(1)(p(−1)) ⊗ y(0)(p(0))  (since C is cocommutative)  = y(1)(u1) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' y(n)(un) ⊗ (y(0)(p))(−1) ⊗ y(0)(p)(0)  (using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='24))  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='10)  Similarly, we may verify that the morphisms in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='8) commute with the face and degeneracy maps appearing in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' To show  that they also commute with the cyclic operators appearing in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='7), we note that for u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un ⊗ p ∈ Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P) and y ∈ D, we  have  y(τn(u1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' ⊗ un ⊗ p)) = y(u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='−(1)u2 ⊗A · · · ⊗A u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='−(n−1)un ⊗A u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='−(n)p(−1) ⊗A p(0)u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='+)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='= y(1)(u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='−(1))y(2)(u2) ⊗A · · · ⊗A y(2n−3)(u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='−(n−1))y(2n−2)(un) ⊗A y(2n−1)(u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='−(n))y(2n)(p(−1)) ⊗A y(0)(p(0)u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='+)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='= y(1)(u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='−(1))y(n+1)(u2) ⊗A · · · ⊗A y(n−1)(u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='−(n−1))y(2n−1)(un) ⊗A y(n)(u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='−(n))y(2n)(p(−1)) ⊗A y(0)(p(0)u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='+)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='= y(1)(u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='−)(1)y(2)(u2) ⊗A · · · ⊗A y(1)(u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='−)(n−1)y(n)(un) ⊗A y(1)(u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='−)(n)y(n+1)(p(−1)) ⊗A y(0)(p(0)u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='+)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='= y(2)(u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='−)(1)y(3)(u2) ⊗A · · · ⊗A y(2)(u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='−)(n−1)y(n+1)(un) ⊗A y(2)(u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='−)(n)y(n+2)(p(−1)) ⊗A y(0)(p(0))y(1)(u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='+)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='= y(1)(u1)−(1)y(2)(u2) ⊗A · · · ⊗A y(1)(u1)−(n−1)y(n)(un) ⊗A y(1)(u1)−(n)y(n+1)(p(−1)) ⊗A y(0)(p(0))y(1)(u1)+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='(using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3))  = y(1)(u1)−(1)y(2)(u2) ⊗A · · · ⊗A y(1)(u1)−(n−1)y(n)(un) ⊗A y(1)(u1)−(n)y(0)(p)(−1) ⊗A y(0)(p)(0)y(1)(u1)+  (using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='24))  This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  □  18  Finally, we recall from [14, § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3] that there are Hopf-Galois isomorphisms relating the modules C•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P) and C•(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P)  ξn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P) : Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P)  �  −→ Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P)  p ⊗ u1 ⊗ · · · ⊗ un �→ u1  (1) ⊗ u1  (2)u2  (1) ⊗ · · · ⊗ u1  (n)u2  (n−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='un−1  (2) un ⊗ p  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='11)  We will conclude this section by showing that the morphisms induced by comodule measurings of SAYD modules are compat-  ible with the Hopf-Galois isomorphisms in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let U = (U, AL, sL, tL, ∆L, ǫL, S ), and U′ = (U′, A′  L, s′  L, t′  L, ∆′  L, ǫ′  L, S ′) be Hopf algebroids over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let P and P′  be SAYD modules over U and U′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Let (Ψ, ψ) : C −→ V(U, U′) be a cocommutative measuring and let Ω : D −→  Vectk(P, P′) be a (C, Ψ, ψ)-measuring comodule from P to P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' Then, for each y ∈ D, the following diagram commutes  Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P)  ξn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='P)  −−−−−−→ Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P)  Ωn(y)  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�  \uf8e6\uf8e6\uf8e6\uf8e6\uf8e6�Ω  n(y)  Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P)  ξn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='P)  −−−−−−→ Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='12)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We set N := n(n − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' For y ∈ D and p ⊗ u1 ⊗ · · · ⊗ un ∈ Cn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P), we see that  Ω  n(y)(ξn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P)(p ⊗ u1 ⊗ · · · ⊗ un))  = Ω  n(y)(u1  (1) ⊗ u1  (2)u2  (1) ⊗ · · · ⊗ u1  (n)u2  (n−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' un−1  (2) un ⊗ p)  = y(1)(u1  (1)) ⊗ y(2)(u1  (2))y(3)(u2  (1)) ⊗ · · · ⊗ y(N+1)(u1  (n))y(N+2)(u2  (n−1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' y(N+n−1)(un−1  (2) )y(N+n)(un) ⊗ y(0)(p)  = y(1)(u1  (1)) ⊗ y(2)(u1  (2))y(n+1)(u2  (1)) ⊗ · · · ⊗ y(n)(u1  (n))y(2n−1)(u2  (n−1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='y(N+n−1)(un−1  (2) )y(N+n)(un) ⊗ y(0)(p)  = y(1)(u1)(1) ⊗ y(1)(u1)(2)y(2)(u2)(1) ⊗ · · · ⊗ y(1)(u1)(n)y(2)(u2)(n−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='y(n−1)(un−1)(2)y(n)(un) ⊗ y(0)(p)  = ξn(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' P)(Ωn(y)(p ⊗ u1 ⊗ · · · ⊗ un))  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='13)  □  7  Operads with multiplication, comp modules and morphisms on cyclic homology  We start the final section by recalling from Kowalzig [16] the following two notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' (see [16, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2]) A non-Σ operad O over k consists of the following:  (a) A collection of vector spaces O = {O(n)}n≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  (b) A family of k-linear operations ◦i : O(p)⊗O(q) −→ O(p+q−1) and an identity 1 ∈ O(1) satisfying the following conditions  (for φ ∈ O(p), ψ ∈ O(q), χ ∈ O(r))  φ ◦i ψ  = 0  if p < i or p = 0  (φ ◦i ψ) ◦ j χ  =  \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3  (φ ◦ j χ) ◦i+r−1 ψ  if j < i  φ ◦i (ψ ◦ j−i+1 χ)  if i ≤ j < q + i  (φ ◦ j−q+1 χ) ◦i ψ  if j ≥ q + i  φ ◦i 1  = 1 ◦1 φ = φ  for i ≤ p  (c) An operad multiplication µ ∈ O(2) and a unit e ∈ O(0) such that  µ ◦1 µ = µ ◦2 µ  µ ◦1 e = µ ◦2 e = 1  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1)  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' (see [16, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='1]) A cyclic unital comp module M over an operad O with multiplication consists of the  following data:  (a) A collection of vector spaces M = {M(n)}n≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  19  (b) A family of k-bilinear operations •i : O(p) ⊗ M(n) −→ M(n − p + 1), 0 ≤ i ≤ n + 1 − p satisfying the following conditions  for φ ∈ O(p), ψ ∈ O(q), x ∈ M(n)  φ •i (ψ • j x) =  \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3  ψ • j (φ •i+q−1 x)  j < i  (φ • j−i+1 ψ) •i x  if j − p < i ≤ j  ψ • j−p+1 (φ •i x)  if 1 ≤ i ≤ j − p  as well as 1 •i x = x for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  (c) A cyclic operator t : M(n) −→ M(n) for n ≥ 1 satisfying  t(φ •i x) = φ •i t(x)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='2)  for φ ∈ O(p), x ∈ M(n) and 0 ≤ i ≤ n − p as well as tn+1 = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  We take pairs (O, M) consisting of a non-linear Σ operad O and a cyclic unital comp module M over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We now consider  comodule measurings between such pairs  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' A comodule measuring from (O, M) to (O′, M′) consists of the following:  (a) A cocommutative coalgebra C and a family of morphisms {Φn : C −→ Vectk(O(n), O′(n))}n≥0 satisfying  Φp+q−1(x)(φ ◦i ψ) = Φp(x(1))(φ) ◦′  i Φq(x(2))(φ)  Φ2(x)(µ) = ǫ(x)µ′  Φ0(x)(e) = ǫ(x)e′  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3)  for φ ∈ O(p), ψ ∈ O(q) and any x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  (b) A comodule D over C and a family of morphisms {Ψn : D −→ Vectk(M(n), M′(n))}n≥0 satisfying  Ψn−p+1(φ •i x) = Ψp(y(0))(φ) •i Ψn(y(1))(x)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4)  for y ∈ D, φ ∈ O(p), x ∈ M(n), 0 ≤ i ≤ n + 1 − p and also  Ψn(y)(t(x)) = t′(Ψn(y)(x))  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='5)  for y ∈ D, x ∈ M(n), where t and t′ are respectively the cyclic operators on M and M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  We now recall from [16, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='5] that the cyclic homology of (O, M) is obtained from the cyclic module C•(O, M) :=  M(•) whose cyclic operators are t : M(n) −→ M(n) and whose face maps and degeneracies are given as follows:  di(x) := µ •i x, (0 ≤ i < n)  dn(x) := µ •0 t(x)  sj(x) := e • j+1 x, 0 ≤ j ≤ n  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='6)  The cyclic homologies of this cyclic module will be denoted by HC•(O, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We conclude with the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' If D is a C-measuring comodule from (O, M) to (O′, M′), then each y ∈ D induces a morphism Ψcy  (y) :  HC•(O, M) −→ HC•(O′, M′) on Hochschild homologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' We know from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='5) that the action of any y ∈ D commutes with the cyclic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' From the definitions in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='6)  and the conditions in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content='3), it is clear that the action also commutes with the degeneracies and face maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
+page_content=' The result is now  clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQfDxyZ/content/2301.08418v1.pdf'}
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+Constraining the Limitations of NEATM-like Models: A Case Study with Near-Earth
+Asteroid (285263) 1998 QE2
+Samuel A. Myers1
+, Ellen S. Howell1
+, Christopher Magri2
+, Ronald J. Vervack, Jr.3
+, Yanga R. Fernández4
+,
+Sean E. Marshall5
+, and Patrick A. Taylor6
+1 Lunar and Planetary Laboratory, University of Arizona, 1629 E. University Boulevard, Tucson, AZ 85721, USA; sammyers@lpl.arizona.edu
+2 University of Maine Farmington, 173 High Street, Farmington, ME 04938, USA
+3 Johns Hopkins Applied Physics Laboratory, 11100 John Hopkins Road, Laurel, MD 20723, USA
+4 University of Central Florida, 4111 Libra Drive, Orlando, FL 32816, USA
+5 Arecibo Observatory/University of Central Florida, HC-03 Box 53995, Arecibo, Puerto Rico 00612, USA
+6 National Radio Astronomy Observatory/Green Bank Observatory, 1180 Boxwood Estate Road, Charlottesville, VA 22903, USA
+Received 2022 August 10; revised 2022 November 28; accepted 2022 December 1; published 2023 January 10
+Abstract
+Near-Earth asteroids (NEAs) are a key test bed for investigations into planet formation, asteroid dynamics, and
+planetary defense initiatives. These studies rely on understanding NEA sizes, albedo distributions, and regolith
+properties. Simple thermal models are a commonly used method for determining these properties; however, they
+have inherent limitations owing to the simplifying assumptions they make about asteroid shapes and properties.
+With the recent collapse of the Arecibo Telescope and a decrease of direct size measurements, as well as future
+facilities such as LSST and NEO Surveyor coming online soon, these models will play an increasingly important
+role in our knowledge of the NEA population. Therefore, it is key to understand the limits of these models. In this
+work we constrain the limitations of simple thermal models by comparing model results to more complex
+thermophysical models, radar data, and other existing analyses. Furthermore, we present a method for placing
+tighter constraints on inferred NEA properties using simple thermal models. These comparisons and constraints are
+explored using the NEA (285263) 1998 QE2 as a case study. We analyze QE2 with a simple thermal model and
+data from both the NASA IRTF SpeX instrument and NEOWISE mission. We determine an albedo between 0.05
+and 0.10 and thermal inertia between 0 and 425J m−2 s−1/2 K−1. We find that overall the simple thermal model is
+able to well constrain the properties of QE2; however, we find that model uncertainties can be influenced by
+topography, viewing geometry, and the wavelength range of data used.
+Unified Astronomy Thesaurus concepts: Asteroids (72); Asteroid surfaces (2209); Near-Earth objects (1092)
+1. Introduction
+Asteroids were once derided by astronomers as the “vermin
+of the sky,” but they now form an important piece of our efforts
+to understand our own solar system. Understanding their sizes,
+albedo
+distributions,
+and
+regolith
+properties
+is
+key
+for
+investigations into many aspects of solar system science,
+including solar system formation, main belt asteroid orbital
+evolution, surface processes on airless bodies, and under-
+standing our meteorite collection. Near-Earth asteroids (NEAs),
+in particular, are excellent targets for these efforts owing to
+their proximity to Earth.
+In addition to understanding the albedos and regoliths of
+these objects, accurately measuring the sizes of NEAs is pivotal
+for planetary defense initiatives—the area of study focused on
+preventing catastrophic asteroid impacts with Earth. This is
+because the size of an object is directly related to the energy of
+impact (Morrison & Teller 1995), which determines the impact
+severity. Thus, observation and modeling techniques that
+provide estimates of these properties are key for understanding
+the NEA population.
+There are a few methods for obtaining size estimates and
+other physical properties from NEA observations. Radar
+images, detailed thermophysical models, and simple thermal
+models can all be used to obtain size estimates. All of these
+methods, along with light-curve measurements, can also place
+constraints on other physical properties of asteroids. Other
+methods, such as direct imaging (Dollfus 1971; Marchis et al.
+2006; Marchis & Vega 2014), stellar occultations (Millis &
+Dunham 1989; Arai et al. 2020), and spacecraft encounters
+exist (Belton et al. 1992, 1996; Veverka et al. 2000; Lauretta
+et al. 2019) but are only applicable in rare cases. Of the more
+common methods, radar images can provide a size estimate
+without other information (Ostro 1985). Radar observations
+can also be used to construct detailed models of the asteroid’s
+shape (Hudson & Ostro 1994; Magri et al. 2007, 2011; Nolan
+et al. 2013). Light-curve measurements can also produce shape
+models, although they are often less detailed than radar-derived
+shape models and do not include an absolute size scale (Ďurech
+et al. 2012 and references therein). These shape models can be
+coupled with thermal spectra to constrain other physical
+properties of the asteroid as well, such as thermal inertia or
+surface roughness (Marshall et al. 2017; Howell et al. 2018;
+Jones 2018; Hinkle et al. 2022).
+Historically, the Arecibo Telescope has been a source of
+numerous NEA radar observations. The Arecibo Telescope
+detected over 900 NEAs and made size estimates of roughly
+400 of those (Howell et al. 2020). However, with the recent
+loss of the Arecibo Telescope, there will be a lack of direct size
+and shape measurements of NEAs. (Although Goldstone is able
+to make radar measurements, it has a lower sensitivity and less
+availability for targets of opportunity.) As a result, in the future
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+https://doi.org/10.3847/PSJ/aca89d
+© 2023. The Author(s). Published by the American Astronomical Society.
+Original content from this work may be used under the terms
+of the Creative Commons Attribution 4.0 licence. Any further
+distribution of this work must maintain attribution to the author(s) and the title
+of the work, journal citation and DOI.
+1
+
+there will be a greater reliance on other methods to understand
+the
+physical
+properties
+of
+NEAs.
+These
+methods
+will
+necessarily be models, like simple thermal models, that assume
+asteroid shapes or use less well-constrained shape models.
+Simple thermal models, such as the Standard Thermal Model
+(Lebofsky et al. 1986; Lebofsky & Spencer 1989) and the
+Near-Earth
+Asteroid
+Thermophysical
+Model
+(NEATM;
+Harris 1998), are a convenient method for obtaining NEA
+sizes and physical properties in part because they are easy to
+run. They require only visible and thermal infrared data and are
+computationally
+fast.
+For
+this
+reason,
+they
+are
+already
+commonly used to analyze data collected by large survey
+missions like NEOWISE (Mainzer et al. 2011b) and Explor-
+eNEOs (Trilling et al. 2010). Due to the large volume of data
+collected by these types of surveys and the sparse amount of
+data collected on any single object, simple thermal models are
+often the only practical way to quickly interpret the data. In
+these cases, simple thermal models are used to identify both
+scientifically interesting and potentially dangerous NEAs (e.g.,
+Trilling et al. 2010).
+However, simple thermal models make simplifying assump-
+tions about the asteroid’s shape and surface that can result in
+inaccuracies and thus poor constraints of inferred NEA
+properties. This is especially relevant for determinations of
+asteroid sizes—values that are pivotal for planetary defense
+activities. Simple thermal models can only make direct
+determinations of asteroid sizes in specific cases. If absolute
+photometry in both the visible and infrared is available, size
+can be solved for directly. However, these estimates require
+assuming that the visible and infrared data were acquired at
+similar viewing geometries. This assumption is often made
+with models employing NEOWISE or ExploreNEOs observa-
+tions. Alternatively, if only normalized flux is available, then
+the size must be estimated from the modeled albedo in
+combination with the absolute magnitude, H. In this case, the
+estimates are subject to uncertainties in the magnitude (Bowell
+et al. 1989; Jurić et al. 2002; Vereš et al. 2015), as well as
+typically large error bars in the inferred albedo, producing poor
+constraints. In fact, recent work has shown that there are
+inconsistencies between sizes derived from NEOWISE data
+using these models and sizes derived using other methods
+(Howell et al. 2012; Taylor et al. 2014; Masiero et al. 2019;
+Taylor et al. 2019; Masiero et al. 2021).
+In this paper, we seek to better understand the limitations of
+simple thermal models, such as NEATM, by comparing simple
+thermal model results to more complex thermophysical models,
+radar data, and other existing analyses of a given object. We
+also present a method for placing tighter constraints on inferred
+NEA properties using these simple thermal models. We use a
+simple, NEATM-like model (Section 3) to model the observed
+NEA, and the consistency of the best-fit parameters is then
+checked by comparing the models to normalized flux data
+collected across multiple nights that represent a range of
+viewing geometries. We also compare the models to the
+absolute photometry collected by the NEOWISE spacecraft. By
+observing an object across multiple viewing geometries and
+combining normalized flux spectra with absolute photometry,
+we are able to place tight bounds on modeled NEA properties.
+These simple thermal model results are then compared to
+model results from SHERMAN (Magri et al. 2018), a complex
+thermophysical
+model;
+radar
+measurements;
+and
+other
+observations
+and
+analyses
+of
+the
+given
+object.
+These
+comparisons allow us to place constraints on the overall
+limitations of the simple thermal model and identify key factors
+that influence uncertainties in simple thermal model results.
+This analysis is performed on the well-studied NEA
+(285263) 1998 QE2 (hereafter referred to as QE2). QE2 is a
+spheroidal, binary NEA system, with an existing radar-derived
+shape model (Springmann et al. 2014). The secondary has a
+diameter ∼25% that of the primary (Springmann et al. 2014)
+and thus contributes only 6% of the total flux. Therefore, the
+primary object dominates the thermal emission from the
+system, and we neglect the secondary in our analysis. QE2 is
+an Xk-type asteroid in the Bus−DeMeo taxonomy, as derived
+from our SpeX prism spectra and a visible spectrum obtained
+by Hicks et al. (2013).
+As part of our investigation into the limitations of the
+NEATM-like model, we find a discrepancy in the currently
+accepted H-magnitude for QE2. We find that the current value
+is inconsistent with the size derived from the radar measure-
+ments of QE2. We investigate this discrepancy and discuss
+implications. As part of this investigation, we compare our
+results to previous studies to understand QE2ʼs composition
+and surface properties (Fieber-Beyer et al. 2020), as well as its
+spin state (Moskovitz et al. 2017). These comparisons allow us
+to further benchmark the uncertainties in the results of our
+method for placing tight constraints on NEA properties derived
+with simple thermal models.
+In Section 2 we discuss the data used for our analysis. In
+Section 3 we describe our simple, NEATM-like model, and in
+Section 4 we present the results for QE2 from this model. In
+Section 5 we describe our analysis of the uncertainties in these
+model results. We compare our simple, NEATM-like model
+results to model results from SHERMAN, radar data of QE2,
+and the results of other previous studies. We then discuss
+implications for the limitations of simple thermal models. We
+conclude with a summary of our results in Section 6.
+2. Spectral and Radar Data
+2.1. IRTF Observations
+The primary data used to constrain our models are normalized
+flux spectra obtained with the SpeX instrument at the NASA
+IRTF (Rayner et al. 2003). We use normalized flux, as it has
+smaller uncertainties relative to absolute photometry. These
+observations are carried out as part of our ongoing investigation
+into the physical properties of NEAs. We observed using both
+prism mode (0.8–2.5 μm) and Long-Wavelength Cross-Dispersed
+(LDX)1.9 mode (2.2–4.1 μm). Note that the observations of QE2
+presented here were done before the upgrade to SpeX that
+expanded the wavelength ranges of all settings.
+For QE2, observations were carried out over six nights, from
+2013 May 30 to 2013 July 10. Over this time, the solar phase
+angle of QE2 varied from 18°.0 to 39°.7, which let us observe
+different viewing geometries and illumination states. As a
+result, we see the thermal emission at different local times of
+day. This is important because it allows us to check the
+consistency of the fit parameters (Section 3). The various sub-
+Earth locations of QE2 that we observed are shown in Figure 1.
+A summary of the observational parameters for our six nights
+of SpeX data is shown in Table 1.
+2
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+All SpeX observations were done in pairs, nodding the
+telescope along a 15″ slit. We used exposure times of 15 s for
+our LXD data and 10–30 s for our prism data. The data were
+processed using the Spextool software package (Cushing et al.
+2004), and the spectra were extracted from summed images. In
+addition to the object, we observed solar-analog stars in a
+similar manner. At least one was a nearby G star within ∼5° of
+the object on the sky. All stars were compared to a well-
+characterized solar analog star on each night, and their spectra
+were corrected for slight spectral slope variations if necessary.
+Each asteroid–star pair was combined in a ratio after correcting
+each for atmospheric absorption lines. The spectra were then
+determined using a weighted average over all asteroid–star
+pairs and binned to form the final spectra. Bad data points were
+flagged and excluded from the fitting and averaging process.
+The detailed methods for this entire process are given in
+Howell et al. (2018).
+The data are broken up across each night into several
+independent sets of roughly 20–30 minutes each to sample
+different areas of the surface. QE2 has a rotation period of
+4.749 ± 0.002 hr (Springmann et al. 2014), meaning that each
+spectrum is separated by roughly 25°–40° of longitude at the
+equator. The sub-Earth latitudes and longitudes at the midtimes of
+the observations are shown in Figure 1. These sub-Earth
+coordinates are calculated using the shape model of Springmann
+et al. (2014). The LXD data for each of the six nights are shown in
+Figure 2. Each spectrum is normalized at 1.6 μm to give
+normalized flux. (Note that there is no significant thermal
+Figure 1. Sub-Earth locations on QE2 during observations as determined by a radar shape model (Springmann et al. 2014). (a) The pole solution with the “bumpy”
+topography in the northern hemisphere. (b) The pole solution with the topography partially in the southern hemisphere (Section 2.3). The range of sub-Earth locations
+observed indicates that QE2 was observed across multiple different viewing geometries. This range of observations is key for constraining QE2ʼs parameters using our
+NEATM-like model.
+3
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+Sub-EarthLocations
+17
+A)
+16
+Latitude (degree)
+Date
+15
+30May
+02 Jun
+4
+08 Jun
+十
+15Jun
+区
+18Jun
+13
+米
+10Jul
+12
+11
+B)
+米
+atitude(degree)
+5
+Date
+30May
+02 Jun
+区
+08 Jun
+15Jun
+18Jun
+-10
+米
+10Jul
+-15
+0
+60
+120
+180
+240
+300
+360
+Longitude(degree)contamination at this wavelength.) We use normalized flux
+because the relative uncertainties are much smaller than for
+absolutely calibrated photometry. We cover the range from
+completely reflected to thermally dominated to ensure that our
+simple thermal model is well constrained in both regimes. This
+technique has the advantage of being more flexible but the
+disadvantage that the data are highly correlated in wavelength.
+2.2. NEOWISE Observations
+In addition to our SpeX data, we fit our simple thermal
+model to data collected by NEOWISE. Unlike the SpeX data,
+which measure normalized flux, NEOWISE measures absolute
+photometry. Thus, fitting our simple thermal model to the
+NEOWISE data allows us to check that the best-fit parameters
+are consistent with both the spectrum shape and calibrated flux
+values. This provides an additional independent check on the
+consistency of the simple thermal model and allows us to
+identify any potential issues with the model not observed when
+fitting normalized flux data alone.
+We retrieve the NEOWISE data and associated uncertainties
+from the NASA/IPAC Infrared Science Archive (Mainzer
+et al. 2011a, 2014).7 We do not use the raw images, but instead
+retrieve processed data that list the magnitudes and uncertain-
+ties for channels W1 (effective wavelength 3.4 μm) and W2
+(effective wavelength 4.6 μm) for each time the object was
+observed. We remove data points that are flagged for potential
+contamination, such as by cosmic-ray hits, and average
+together all remaining observations. The uncertainty in the
+NEOWISE data is dominated by systematic errors and not
+statistical noise. All observations, except one, have similar
+uncertainties. We thus take a weighted average of the
+observations and adopt the variance of the overall data set,
+divided by the square root of the number of observations minus
+one, as our 1σ uncertainties. For QE2, all observations were
+taken over a short time interval such that the change in QE2ʼs
+orbital position was minimal. Therefore, we averaged together
+all available observations, resulting in one averaged set of data
+points from eight individual observations that span roughly 29
+hr and approximately six rotation periods. The individual
+observations are evenly distributed across the rotation phase. A
+summary of the observational parameters for the averaged
+observation is given in Table 1. A list of the individual
+observations is given in Table 2.
+After retrieval, the data are then converted from NEOWISE
+magnitudes to Fλ units following the procedures outlined in the
+WISE Data Processing Handbook (Wright et al. 2010; Cutri
+et al. 2012). For this process we apply a final blackbody color
+correction corresponding to a 221 K object. This blackbody
+temperature is determined by fitting ideal blackbody curves to
+the NEOWISE data in an iterative process until the corrected
+NEOWISE data and ideal blackbody curves converge. The
+blackbody temperature used for the initial correction is
+calculated using the theoretical blackbody temperature relation
+T
+L
+A
+r
+1
+16
+,
+1
+H
+sb
+4
+2
+(
+)
+( )
+
+s
+p
+=
+-
+where Le is the solar luminosity, A is the Bond albedo, rH is the
+object–Sun distance, and σsb is the Stefan–Boltzmann constant.
+Table 1
+Summary of Observations, Including Values Input Directly into the NEATM-like Model
+Date
+Set
+Midtime
+rH (au)
+Δ (au)
+α (deg)
+Instrument
+2013 May 30
+A
+06:46:50
+1.046 8
+0.040 3
+34.3
+SpeX
+2013 May 30
+B
+07:22:08
+1.046 8
+0.040 3
+34.2
+SpeX
+2013 May 30
+C
+08:36:57
+1.049 8
+0.040 2
+33.9
+SpeX
+2013 Jun 02
+A
+06:51:57
+1.052 2
+0.040 1
+18.3
+SpeX
+2013 Jun 02
+B
+07:08:19
+1.052 2
+0.040 1
+18.3
+SpeX
+2013 Jun 02
+C
+07:17:50
+1.052 2
+0.040 1
+18.3
+SpeX
+2013 Jun 02
+D
+07:34:17
+1.052 2
+0.040 1
+18.2
+SpeX
+2013 Jun 08
+A
+08:12:16
+1.067 1
+0.060 5
+30.0
+SpeX
+2013 Jun 08
+B
+09:25:01
+1.067 2
+0.060 8
+30.1
+SpeX
+2013 Jun 08
+C
+09:37:14
+1.067 2
+0.060 8
+30.1
+SpeX
+2013 Jun 08
+D
+10:38:10
+1.067 4
+0.061 0
+30.2
+SpeX
+2013 Jun 08
+E
+10:50:40
+1.067 4
+0.061 1
+30.2
+SpeX
+2013 Jun 15
+A
+11:06:28
+1.091 0
+0.098 8
+38.8
+SpeX
+2013 Jun 15
+B
+12:16:11
+1.091 2
+0.099 1
+38.8
+SpeX
+2013 Jun 18
+A
+13:07:51
+1.103 3
+0.116 9
+39.7
+SpeX
+2013 Jul 10
+A
+10:23:08
+1.218 8
+0.256 2
+34.0
+SpeX
+2013 Jul 10
+B
+10:29:19
+1.218 9
+0.256 2
+34.0
+SpeX
+2013 Jul 10
+C
+11:10:20
+1.219 0
+0.256 4
+34.0
+SpeX
+2013 Jul 10
+D
+11:49:53
+1.219 2
+0.256 6
+34.0
+SpeX
+2013 Jul 10
+E
+13:09:29
+1.219 6
+0.257 0
+34.0
+SpeX
+2017 Jul 01
+A
+10:51:35
+1.767 6
+1.445 3
+35.1
+NEOWISE
+Note. Set refers to different data sets on a given night. Midtime is the midtime of observation for the data set in UTC time. (Each SpeX observation spans roughly
+20–30 minutes, while the NEOWISE observation spans 29 hr. Thus, each SpeX spectrum is separated by roughly 25°–40° of longitude.) rH is the Sun–object distance,
+Δ is the Earth–object distance, and α is the solar phase angle. Note that the observations are carried out across a range of solar phase angles and viewing geometries.
+7
+https://www.ipac.caltech.edu/doi/irsa/10.26131/IRSA144
+4
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+Figure 2. Processed LXD data sets for each night of observations with SpeX and NEOWISE. (a–f) SpeX data for each of the six nights. The different letters within
+each panel indicate different data sets collected each night (Table 1). The y-axis is normalized flux, normalized to 1.6 μm. (Note that there is no significant thermal
+contamination at this wavelength.) (g) NEOWISE data in absolute flux density. Note that the NEOWISE data are plotted over a different wavelength range. We plot
+both the 1σ and 3σ uncertainties. (h) The “A“ data set for each night of SpeX data. These spectra highlight how different viewing geometries across the different nights
+produce a range of spectral slopes. We see that changes in viewing geometry produce changes in the spectra shape both within nights and across all nights of
+observations. Modeling these differences allows us to place tighter constraints on NEA properties.
+5
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+125
+A)
+B)
+100
+xnI
+2013 May 30
+2013 Jun 02
+Normalized 
+75
+ＡB
+50
+CD
+25
+0
+125
+C
+D)
+100
+ Flux
+Normalized
+75
+2013 Jun 15
+2013 Jun 08
+A
+A
+50
+BCDE
++
+25
+0
+125
+F)
+E)
+100
+Normalized
+75
+2013 Jul 10
+2013 Jun 18
+AB-
++A
++
+50
++
+CDE
++
+25
+3.2
+3.7
+4.2
+7
+3.2
+3.7
+4.2
+Wavelength (microns)
+Wavelength (microns)
+1e-20
+125
+G)
+H)
+_wn
+8e-21
+100
+SpeX, All Nights
+2017 Jul 01
+Normalized I
+2013 May 30
+6e-21
+75
+2013 Jun 02
+3g
+2013 Jun 08
++
+2013 Jun 15
+4e-21
+50
+2013 Jun 18
+2013 Jul 10
+25
+2e-21
+0e+00
+3.2
+2.5
+3.5
+4.5
+3.7
+4.2 
+3.0
+4.0
+5.0
+Wavelength (microns)
+Wavelength (microns)The Bond albedo is estimated according to the method
+described in Lebofsky & Spencer (1989):
+A
+G p
+0.29
+0.684
+,
+2
+(
+)
+( )
+=
++
+where G is the slope parameter in the HG magnitude system
+(Bowell et al. 1989) and p is the visual geometric albedo. The
+standard assumption of G = 0.15 is used, and p is taken from
+the model fits to the SpeX data. Note that since the fitting
+process is iterative, choices of the initial guess parameters do
+not strongly affect the final result. The end products of this
+conversion process are flux densities reported in units of W
+cm−2 μm−1, which match the units of our simple thermal
+model output. The final NEOWISE data for QE2 are shown in
+Figure 2. We show the data with both 1σ and 3σ uncertainties.
+2.3. Radar Shape Model
+As part of our investigation into the limitations of simple
+thermal models, we compare the results of our NEATM-like
+models to many other data sources and models, including radar
+images and a radar shape model. The radar image is a direct
+measurement of the size that only depends on the viewing
+geometry and the speed of light. A spheroidal object, such as
+QE2, shows a radius in radar range at nearly all aspects and is a
+robust size estimate. We compare the radar size to sizes derived
+from our NEATM-like model, based on the magnitude and
+albedo. We emphasize that this information is not used as an
+input of our NEATM-like model and is only used to compare
+with our NEATM-like model results.
+The radar shape model for QE2 is described by Springmann
+et al. (2014). The model is constructed using observations from
+the Arecibo Observatory and Goldstone. Data used were
+collected between 2013 May 31 and June 9, during QE2ʼs close
+approach to Earth. These radar images are used to derive a
+shape model as described in Magri et al. (2011). A nonlinear
+iterative process is used to adjust synthetic radar images to
+match the observations by minimizing the difference between
+them. This process is described in detail in several papers for
+other objects (Magri et al. 2011; Nolan et al. 2013). The shape
+model of QE2 is preliminary, and the complete analysis is
+beyond the scope of this paper. However, the derived diameter
+of the principal axes of QE2 is robust and reliable as a
+comparison to values obtained here. This analysis gives a
+diameter for QE2 of 3.2 ± 0.3 km and a diameter of the
+secondary of 800 ± 80 m. QE2 is spheroidal, with a few
+dominant surface features.
+Springmann
+et
+al.
+(2014)
+find
+a
+rotation
+rate
+of
+4.749 ± 0.002 hr for QE2 and two possible pole solutions,
+both of which are prograde. One of these solutions, which we
+refer to as the A solution, places most of the “bumpy”
+topography of QE2 in the northern hemisphere. This solution
+has a pole position of λ = 119° and β = 55°, where λ is the
+ecliptic pole longitude and β is the ecliptic pole latitude. The
+second solution, which we refer to as the B solution, places the
+“bumpy” topography partially in the southern hemisphere. This
+solution has a pole position of λ = 158° and β = 41°. Both
+solutions are shown in Figure 3.
+3. NEATM-like Model
+The simple thermal model we use to fit the data is based on
+the
+Standard
+Thermal
+Model
+(Lebofsky
+et
+al.
+1986;
+Lebofsky & Spencer 1989) and NEATM (Harris 1998). Our
+Figure 3. Sky views of QE2 on 2013 July 10 that show the radar shape model from Springmann et al. (2014). The arrows indicate the pole and spin direction. Left: the
+A solution with a pole position of λ = 119° and β = 55°. Right: the B solution with a pole position of λ = 158° and β = 41°.
+Table 2
+List of Individual NEOWISE Observations Used to Obtain the Single
+Averaged NEOWISE Data Set
+Date
+Midtime
+m1 (mag)
+σ1 (mag)
+m2 (mag)
+σ2 (mag)
+2017 Jun 30
+19:32:22
+16.768
+0.468
+13.609
+0.136
+2017 Jun 30
+22:41:03
+16.256
+0.253
+13.844
+0.158
+2017 Jul 01
+03:23:49
+16.625
+0.392
+13.944
+0.172
+2017 Jul 01
+06:32:19
+16.966
+0.535
+13.658
+0.131
+2017 Jul 01
+15:58:00
+16.449
+0.338
+14.193
+0.292
+2017 Jul 01
+19:06:30
+16.927
+0.522
+13.694
+0.135
+2017 Jul 01
+22:15:00
+16.557
+0.476
+13.801
+0.161
+2017 Jul 02
+01:23:41
+16.539
+0.375
+13.770
+0.124
+2017 Jul 01
+10:51:35
+16.528
+0.092
+13.758
+0.051
+Note. Midtime is the midtime of observation in UTC time. m1 and m2 are the
+NEOWISE reported magnitudes for W1 (effective wavelength 3.4 μm) and W2
+(effective wavelength 4.6 μm), respectively. σ1 and σ2 are the NEOWISE
+reported magnitude uncertainties for W1 and W2, respectively. The last row is
+the averaged observation.
+6
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+model is a variation of these models that we call our NEATM-
+like model (Howell et al. 2018). Like these models, for a given
+set of asteroid parameters, our NEATM-like model produces a
+theoretical thermal emission spectrum of the object that can be
+fit to any subset of the visible to near-IR spectra of an asteroid.
+However, our model also utilizes a simple incorporation of the
+rotation rate of the object that allows us to model the thermal
+inertia. The thermal inertia is a measurement of how well the
+object’s surface retains heat energy from the Sun and is
+measured in J m−2 s−1/2 K−1 (hereafter referred to as TIU for
+thermal inertia units). By determining the thermal inertia, in
+combination with the rotation rate, our NEATM-like model is
+able to account for differences across the day and night sides of
+an object. Thus, when incorporating many different observa-
+tions of a single object, taken at different viewing geometries,
+we are able to model how changes in thermal inertia affect the
+thermal emission of an object. Overall, this incorporation
+allows us to get a more robust picture of the properties of the
+object. We note that other than this addition, this model is
+functionally similar to the standard NEATM model.
+In addition to incorporating these parameters, our model also
+makes the typical assumption of a spherical shape for the
+asteroid. It also assumes subsolar and subobserver points on the
+asteroid’s equator and prograde rotation at a fixed rotation rate.
+(The NEATM-like model does not account for shape effects,
+and the radar-derived shape model of Springmann et al. 2014 is
+only used to compare to the NEATM-like model results to
+investigate the limitations of the NEATM-like model.) The
+model also incorporates a free-floating beaming parameter—a
+scaling factor between the observed and predicted flux from the
+asteroid. This factor accounts for additional effects not included
+in the model, such as surface roughness, deviations from a
+spherical shape, local shadowing, and nonzero obliquity. The
+beaming parameter generally ranges between ∼0.5 and 2.0,
+with higher values usually occurring at higher phase angles or
+for more irregularly shaped asteroids.
+Overall, our model includes three free-floating parameters:
+the visual geometric albedo, thermal inertia, and beaming
+parameter. The output of each run is a model spectrum of the
+asteroid, based on the input parameters, for each combination
+of the free-floating parameters. Thus, identifying best-fit
+parameters requires inspecting the model results and making
+direct comparisons to the data.
+For a given object, the consistency of these fit parameters
+can be checked by comparing the results to thermal infrared
+data collected across multiple nights that represent a range of
+viewing geometries. This is important because many combina-
+tions of albedo, thermal inertia, and beaming parameter can fit
+any individual observation. By comparing model results for a
+single object to data taken at multiple different viewing
+geometries of that object, we can thus identify consistent values
+of albedo and thermal inertia that fit every observation,
+breaking degeneracies in the solution. The beaming parameter
+is allowed to vary, as it is expected to change in value across
+each observation. Thus, across multiple different viewing
+geometries, only a tight range of albedo and thermal inertia
+values will fit every observation. This is true even when the
+beaming parameter is allowed to vary, as more extreme
+deviations in albedo or thermal inertia would require increas-
+ingly extreme values of the beaming parameter to fit the
+observations, and realistic beaming parameters are generally
+constrained to the range of ∼0.5–2.0 (Delbó et al. 2003). Note
+that these comparisons are done solely to constrain the
+parameter fits of the NEATM-like model and are separate
+from the comparisons done as part of our investigation into the
+limitations of the NEATM-like model (Section 5).
+The fixed model inputs for our NEATM-like model are the
+object's rotation period, a visible-to-near-IR reflectance ratio,
+Earth–object and Sun–object distances, solar phase angle,
+emissivity, and spherical equivalent diameter. For QE2, we use
+a rotation period of 4.749 ± 0.002 hr that was used by a
+previously derived radar shape model (Springmann et al. 2014).
+We also use a spherical equivalent diameter of 3.2 km from the
+same shape model. We note that since the shape of QE2 is very
+close to spherical, the assumption of spherical shape by the
+NEATM-like model is a very good assumption. The visible-to-
+near-IR reflectance ratio is estimated to be 1.127 using our
+SpeX prism spectra and a visible spectrum obtained by Hicks
+et al. (2013). This is a color correction factor used to relate the
+visible albedo to the near-infrared albedo at 1.6 μm, chosen as
+the normalization wavelength of the spectra. Earth–object and
+Sun–object distances, as well as solar phase angle, are
+calculated for each observation using JPL Horizons8 based
+on the midtime of observation for each data set. These values
+are listed in Table 1. The emissivity is set to 0.9.
+4. NEATM-like Model Results
+We generate NEATM-like models for each of our normal-
+ized flux SpeX data sets and our single absolute photometry
+NEOWISE data set. Models are generated across a wide range
+of albedos, thermal inertias, and beaming parameters. Models
+are then compared to the data using an objective function to
+constrain QE2ʼs properties. For any given data set, models of
+varying parameters change monotonically (Figure 4). These
+models are sorted by calculating a reduced χ2 between the
+model and the data. When performing this calculation, we only
+consider data points between 3.00 and 4.05 μm, as this is the
+region of strongest thermal emission without significant
+overlap with atmospheric water vapor lines. For the NEOWISE
+data set, both NEOWISE data points are used.
+It is important to note that the reduced χ2 value we calculate
+is not a formal χ2, as it does not reach a minimum at unity and
+does not go up by a value of 1 when the model is 1σ away from
+the data. This is because the uncertainties in the data are
+dominated by systematic effects, not statistical errors. The data
+points are not independent, as they are strongly correlated in
+wavelength and are affected by changing effects such as
+atmospheric conditions on different days, viewing geometry,
+and rotational changes of the asteroid. As a result, this
+calculation can be used to sort the goodness of fit of models for
+a given data set but cannot be used to compare models across
+data sets. Thus, for each data set, we use this method to identify
+the range of albedos and thermal inertias that produce models
+that lie within the 1σ uncertainties of the data. Figure 5 shows
+the variation in models that were accepted to fit the data for
+each data set. (Note that for the NEOWISE data we also
+examine the models that fit within the 3σ uncertainties. This
+range is also shown for the NEOWISE data.) Any models
+within the shown region are considered to fit the data. All other
+models for the given data set are discarded, as they are poor fits
+to the data.
+8
+https://ssd.jpl.nasa.gov/horizons/
+7
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+For each data set, we then have a range of albedos and
+thermal inertias that can be said to fit that given data set. These
+individual fit spaces are shown in Figure 6. (For the NEOWISE
+data, we show the models that fit the 1σ uncertainties.) Overall,
+we have 21 such data sets: 20 data sets spread across six nights
+of IRTF SpeX observations, and 1 set of NEOWISE data. To
+identify the range of albedos and thermal inertias that describe
+QE2 overall, we then search for the region of overlap between
+each of these 21 different model sets. These results are shown
+in Figure 7. There is a clear section in the parameter space of
+models that fit nearly every data set. We define this region as
+the best-fit space.
+All models within this space are consistent with the SpeX
+data, but do not fit the NEOWISE data with 1σ uncertainties.
+We then examine models that fit the NEOWISE data with 3σ
+uncertainties, and find that all models within the best-fit space
+are consistent with the NEOWISE data. This could be because
+the NEOWISE observations were taken at a much higher Sun–
+object distance than the SpeX data. As a result, QE2 was much
+colder at the time of these observations which may be
+introducing complexities to the thermal emission that our
+simple thermal model is not able to capture. Such effects may
+be better understood using a more complex thermophysical
+model, however a full investigation of this effect is beyond the
+scope of this work.
+Overall, our analysis gives best-fit ranges of 0.05–0.10 for
+the visual geometric albedo and 0–425 TIU for the thermal
+inertia. Note that there is a correlation such that higher thermal
+inertias require lower albedos. Results are summarized in
+Table 3.
+In general, we find a preference for lower beaming
+parameters of ∼0.55–0.80. Beaming parameter results are
+shown in Figure 8. We remind the reader that we expect the
+beaming parameter to change across observations, and so we
+do not attempt to fit for a single overall value of the beaming
+parameter. These values are calculated by taking the best-fit
+beaming parameter value for a fixed albedo of 0.07 and a fixed
+thermal inertia of 150 TIU. These values are chosen because
+they are near the center of the best-fit region. The NEOWISE
+beaming parameters are calculated using the 3σ uncertainties as
+they are the results consistent with the SpeX data. As expected,
+the beaming parameter is generally higher for higher phase
+angles. The exceptions to this trend are July 10 and the
+NEOWISE data, both of which have substantially greater rH
+and Δ values than the other nights. These larger distances also
+explain the noisier data observed on July 10.
+5. Limits of the NEATM-like Model
+In calculating our best-fit model ranges, we compared our
+model results across many data sets taken at different viewing
+geometries of QE2 (Figure 1). These comparisons have
+allowed us to place tighter constraints on our modeled albedo
+and thermal inertia than would be possible with single
+observations. These albedos and thermal inertias can then be
+compared to results from more complex thermophysical
+models, radar data, and other observations to identify how
+accurately the NEATM-like model was able to constrain the
+properties of QE2. Our model results also provide us with a
+range of beaming parameter values that change as a function of
+viewing geometry. Analyzing these changes in beaming
+parameter can allow us to identify the unmodeled factors
+limiting the accuracy of our NEATM-like model. Overall, by
+comparing our model results to previous studies of QE2, we
+can gain insight into the limitations of simple thermal models
+as applied to a single object. In the subsections below we walk
+through comparisons of our simple thermal model results to
+various other models and data sets. For each comparison, we
+discuss in what ways our simple thermal model results differ
+and discuss implications for the factors affecting the uncertain-
+ties of simple thermal model results.
+5.1. Albedo, Size, and H-magnitude
+Our modeled visual geometric albedo for QE2 of 0.05–0.10
+is higher than but overlaps with previously published values of
+0.03 0.02
+0.03
+-
++
+(Moskovitz et al. 2017) and 0.04 ± 0.01 (Fieber-
+Beyer et al. 2020). We can use our modeled albedo, in
+combination with a radar-derived size, to estimate QE2ʼs H-
+magnitude. This is given by the relationship
+⎛
+⎝
+⎞
+⎠
+H
+p
+D
+5 log
+1329 km ,
+3
+10
+( )
+= -
+where p is the albedo and D is the object diameter in kilometers
+(Pravec & Harris 2007, Equation (3)). Using the diameter of
+3.2 ± 0.3 km given by Springmann et al. (2014) and our
+modeled albedo range of 0.05–0.10, we get an H-magnitude of
+15.4–16.6. This value is lower than (but partially overlaps with)
+previously given H-magnitude values of 16.4 (Trilling et al.
+2010) and 17.3 (Moskovitz et al. 2017) for QE2.
+However, the radar shape model constrains the diameter with
+high accuracy. The radar-derived shape can be considered a
+true constraint on QE2ʼs size, as size can be measured directly
+from a radar image (Ostro 1985). Figure 9 shows a radar image
+of QE2 taken by the Arecibo Telescope on 2013 June 10. The
+vertical extent of the image shows distance from the observer to
+the terminator of the object. Thus, the resolution of the pixels,
+combined with knowledge of the speed of light, directly gives
+the object’s radius. In this image, QE2 covers 210 pixels in the
+vertical extent at 7.5 m pixel−1, giving an apparent radius of
+1575 m or a diameter of 3.15 km. However, using an H-
+magnitude of 17.3 and albedos of 0.05–0.10 gives a diameter
+Figure 4. A range of NEATM-like models compared to one of our SpeX data
+sets. As either the albedo or thermal inertia changes monotonically, the models
+correspondingly change monotonically across the data. This property allows us
+to identify a range of models that fit the data and is typical to all of our data
+sets. All models that fall within the 1σ error bars of the data would be
+considered good fits to the data. As such, in this case only the pV = 0.06 and
+Γ = 100 TIU model would be considered a good fit. Models shown here all
+have η = 0.86. Changes in beaming parameter can also monotonically affect
+how the models fit to the data.
+8
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+Variation in NEATM-like Models
+125
+十
+30 May A
+100
+pv =0.03,  =100
+Normalized Flux
+pv =0.06, I =100
+pv =0.09, I =100
+75
+pv =0.06, I =0
+王
+pv =0.06, I =200
+王
+王
+王
+正
+50
+25
+3.2
+3.7
+4.2
+Wavelength (microns)Figure 5. The variation in NEATM-like models that were accepted to fit the data for each data set. Any models within the shaded region are considered to fit the data.
+All other models for the given data set are discarded, as they are poor fits to the data. An objective function is used to identify which models fall within the shown
+region (Section 4). (a–f) SpeX data. The y-axis is normalized flux. The spectra are offset for clarity. (g) NEOWISE data in absolute flux density. Note that the
+NEOWISE data are plotted over a different wavelength range. For the NEOWISE data we examine models that fit within both the 1σ and 3σ uncertainties. Both
+regions are shown.
+9
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+250
+A)
+B)
+200
+xn
+2013 Jun 02
+2013 May 30
+ABCD
+ABC
+Normali
+100
+50
+0
+250
+C)
+D)
+2013 Jun 08
+2013 Jun 15
+A
+BCDE
+A
+Normal
+B
+100
+50
+0
+250
+E)
+F)
+200
+lux
+2013 Jul 10
+2013 Jun 18
+十+
+ABCDE
++ A
+100
+50
+0
+2.7
+3.2
+3.7
+4.2
+3.2
+3.7
+4.2
+Wavelength (microns)
+Wavelength (microns)
+1e-20
+G)
+wn
+9e-21
+8e-21
+2017 Jul 01
+7e-21
+6e-21
+1g
+3g
+5e-21
+4e-21
+3e-21
+2e-21
+1e-21
+0e+00
+2.5
+3.0
+3.5
+4.0
+4.5
+5.0
+Wavelength (microns)Figure 6. Reduced χ2 maps for each of the spectra as fit by our simple, NEATM-like model. Warmer colors mean higher values (worse fits), and cooler colors mean
+lower values (better fits). Note that different max values are used for different spectra, as the reduced χ2 are not directly comparable across different spectra
+(Section 4). Each χ2 map is equivalent to showing the range of models that fit a given data set.The fit space of the NEOWISE data corresponds to the 1σ uncertainties
+10
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+0.18
+0.16
+X
+30 May A
+30 May B
+7.00
+30 May C
+6
+0.14
+5.25
+0.12
+3.50
+1.75
+0.00
+0.06
+0.04
+0.02
+0.18
+0.16
+X
+02 Jun A
+02 Jun B
+02 Jun C
+0.14
+2
+0.12
+0.06
+0.04
+0.02
+0.18
+0.16
+x
+02 Jun D
+08 Jun A
+08 Jun B
+0.14
+6
+4
+0.12
+2
+2
+0
+0.06
+0.04
+0.02
+0.18
+0.16
+X
+08 Jun C
+08 Jun D
+08 Jun E
+0.14
+3
+0.12
+2
+0 0.08
+0.06
+0.04
+0.02
+0.18
+0.16
+x
+15 Jun A
+15 Jun B
+18 Jun A
+6
+6
+12
+0.14
+4
+0.12
+2
+?
+0.06
+0.04
+0.02
+0.18
+0.16
+10 Jul A
+10 Jul B
+10 Jul C
+0.14
+3
+0.12
+2
+0.06
+0.04
+0.02
+0.18
+0.16
+10 Jul D
+10 Jul E
+3.00
+0.14
+5.25
+.25
+0.12
+3.50
+.50
+1.75
+0.75
+0.00
+1.2
+0.00
+0.9
+0.6
+0.06
+0.3
+0.04
+NEOWISE
+0.0
+0.02
+0 50 100 150 200 250 300 350 400 450 500 550 050 100 150 200 250 300 350 400 450 500 550 050 100 150 200 250 300 350 400 450 500 550
+Thermal Inertia (TIU)
+Thermal Inertia (TIU)
+Thermal Inertia (TIU)between 1.5 and 2.1 km, well outside of the 1σ errors of the
+radar measurement.
+We investigate this unusually large discrepancy in the H-
+magnitude by looking at existing observations. Using an H-
+magnitude value and an assumed G value, we can calculate
+predicted apparent magnitudes. These predicted apparent
+magnitudes can then be compared to observed apparent
+magnitudes reported to the Minor Planet Center (MPC).9
+Ephemeris values are calculated for QE2 using JPL Horizons10
+at 1-day intervals throughout 2013. We then calculate predicted
+apparent magnitudes for the H-magnitude consistent with the
+radar-determined
+size and our modeled albedo, the H-
+magnitude used by Moskovitz et al. (2017), and a range of G
+values from 0 to 0.15. This was done following the procedure
+in Bowell et al. (1989). These predicted apparent magnitudes
+are then compared to all the apparent magnitudes listed in the
+MPC. The results are shown in Figure 10. We see that H-
+magnitudes of neither 16.0 nor 17.3 perfectly match the data,
+but instead provide an upper and lower bound, respectively.
+However, we note that an H-magnitude of 16.0 appears to
+provide a more reasonable fit than an H-magnitude of 17.3.
+So what could be causing these H-magnitude differences?
+One possible explanation is related to the G parameter. The G
+parameter is often assumed to be 0.15 and is not fitted directly.
+Figure 10 shows that for H = 16.0 lower G values fit better,
+while for H = 17.3 higher G values fit better. For QE2, we
+would expect a lower G value, as lower G values are generally
+preferred
+for
+low-albedo
+objects
+owing
+to
+the
+smaller
+opposition effect. However, we note that the differences do
+not exceed ∼0.5 mag and thus cannot fully explain the
+discrepancy.
+Another possible explanation is related to color effects;
+however, the color of QE2 is very close to solar, and thus this is
+also unlikely to be a large factor in this case. The discrepancy
+could also be due to the secondary contributing to the
+magnitude. Using the radar shape model (Springmann et al.
+2014), we can calculate the effective diameter of the combined
+primary and secondary to be 3.3 ± 0.3 km. Using our modeled
+albedo range, this gives an H-magnitude difference of only
+∼0.1 and thus is an ignorable contribution to the uncertainty.
+Therefore, none of these effects by themselves can fully explain
+the observed differences. Overall, given the accuracy of the
+radar measurement of the diameter and our tightly constrained
+albedo range, the true H-magnitude cannot be as high as 17.3.
+It is therefore likely that a better estimate of the H-magnitude
+lies somewhere in between 16.0 and 17.3. Furthermore, our
+analysis shows that higher H-magnitudes and thus lower
+albedos are
+likely favored
+for QE2, potentially
+further
+constraining the results from our simple thermal model.
+5.2. Wavelength Range of Observations
+We can also analyze our results by leveraging the large
+wavelength range of our observations. Our observations span
+0.8–4.1 μm, and thus we are able to observe both the thermally
+dominated region of the spectra (3.0 μm) and the thermal tail
+(∼2.0–2.5 μm). We are therefore able to compare our model
+fits to both regions. This is notable because many studies (e.g.,
+Moskovitz et al. 2017) rely only on the tail region. We show
+this comparison for a selection of our data sets in Figure 11.
+We find that in nearly all cases the models that best fit the
+thermally dominated region also fit the tail region. However,
+for some dates (such as some data sets for 2013 July 10), an
+albedo increase of ∼0.02 relative to the model that fits the
+thermally dominated region is required to fit the tail region.
+This implies that QE2 may have an inhomogeneous surface and
+that we may be observinglocal thermal variations. Such
+variations could impart a wavelength-dependent change in the
+flux, thus creating the observed discrepancy. Another possibi-
+lity is that some other effect, such as surface roughness, that our
+NEATM-like model does not account for may be causing this
+mismatch. This result is important because it shows the dangers
+of relying on only a limited spectral region to derive surface
+properties such as albedo.
+5.3. Surface Topography
+The potential effects of a surface inhomogeneity can be
+investigated by comparing our NEATM-like model results to
+results from a more complex thermal model. In addition to our
+NEATM-like models, we generate models using SHERMAN.
+SHERMAN is a more complex thermophysical model that
+Figure 7. Final best-fit space for the visible geometric albedo and thermal
+inertia for QE2 using our simple, NEATM-like model. The color of the points
+represents the number of data sets that are fit by the associated parameter
+values. A cooler color means that the given parameters are consistent with more
+data sets. White indicates models consistent with £ 2 data sets. The black line
+outlines the region of best fit. This region corresponds to the region of overlap
+between all the individual model ranges found to fit each individual data set
+(Section 4). There is a correlation such that higher thermal inertias require
+lower albedos. All models were run with the same fixed model inputs listed in
+Section 3 and using ephemeris inputs listed in Table 1. This figure is generated
+using the results from the 1σ uncertainties on the NEOWISE data.
+Table 3
+Best-fit Model Ranges for the Three Free-floating Parameters of Our NEATM-
+like Model
+Parameter
+Range
+Albedo
+0.05–0.10
+Thermal inertia
+0–425 TIU
+Beaming parameter
+∼0.55–0.80
+Note. Albedo is visual geometric albedo. We expect the albedo and thermal
+inertia to be consistent across all data sets, and thus the ranges given represent
+the uncertainty in our model results. However, we expect the range of
+acceptable beaming parameters to change across observations, and thus the
+range given represents the range of values observed across all data sets.
+9
+https://minorplanetcenter.net/
+10 https://ssd.jpl.nasa.gov/horizons/
+11
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+Number of Data Sets Fit per Mode
+Geometric Albedo
+0.18
+0.16
+Number
+0.14
+20
+0.12
+12
+0.10
+3
+0.08
+0.06
+Visual
+0.04
+0.02
+0.00
+0
+50100150200250300350400450500550
+Thermal Inertia (TIU)takes account of the object’s shape and that can separate the
+effects of obliquity and self-shadowing. (For a full description
+of SHERMAN, see Magri et al. 2018.) We give SHERMAN
+the radar-derived shape model of QE2 (Springmann et al.
+2014), as well as our SpeX thermal infrared data. We also input
+a reflectance spectrum from our prism data, as well as a Hapke
+Figure 8. Plot of fitted beaming parameters as a function of solar phase angle adjusted so that 0° corresponds to QE2ʼs minimum phase angle during its close approach
+to Earth. We also compare our beaming parameters to those found by Moskovitz et al. (2017). Note the introduction of negative phase angles to differentiate between
+observations taken before (positive values) and after (negative values) opposition. The error bars represent the range of beaming parameters. The range is calculated by
+identifying models that fit the data with fixed albedo and thermal inertia (Section 4). Moskovitz et al. (2017) values are taken from their Figure 3. We see that our data
+exhibit roughly the same trend where the data overlap, but that our beaming values are significantly offset from the Moskovitz et al. (2017) values.
+Figure 9. Radar image of QE2 taken by the Arecibo Telescope on 2013 June 10. The vertical extent of the image shows distance from the observer to the terminator of
+the object. The horizontal extent shows Doppler shift, with blueshift to redshift going left to right. The resolution of the pixels, combined with knowledge of the speed
+of light, directly gives the object’s radius. In this image, QE2 covers 210 pixels in the vertical extent at 7.5 m pixel−1, giving an apparent radius of 1575 m or a
+diameter of 3.15 km.
+12
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+Beaming Parameter vs. Solar Phase Angle
+2.0
+1.9
+1.8
+1.7
+Moskovitz et al. (2017)
+1.6
+This Work
+May11
+1.5
+1.4
+1.3
+n
+1.2
+May 30巫
+Jun15
+巫 Jun 08
+1.1
+Jul5
+Jun 02
+1.0
+0.9
+0.8
+Jun18Jun15
+May 30 巫
+0.7
+Jul 10巫
+巫 Jun 08
+巫 Jun 02
+0.6
+NEOWISE 巫
+0.5
+-30
+-20
+-10
+0
+10
+20
+30
+40
+50
+60
+Adiusted αscattering function. SHERMAN has three free-floating para-
+meters: visual geometric albedo, thermal inertia, and crater
+fraction. The crater fraction is a proxy for surface roughness
+and describes the fraction of each model facet covered with
+hemispherical craters, following the method of Lagerros
+(1998). SHERMAN outputs a modeled thermal spectrum that
+we then compare with our thermal infrared data.
+SHERMAN is a forward model, so we generate many
+models across different values of the free-floating parameters to
+match to our data. Some preliminary model results are shown
+in Figure 12. We find that an albedo of 0.053, thermal inertia of
+200 TIU, and crater fraction of 70% can roughly match the
+data. These values are also consistent with the results of the
+NEATM-like model.
+The SHERMAN results also show that the topography of
+QE2 is affecting the thermal emission.Using SHERMAN, we
+run models using both possible pole solutions. The results
+show slight differences in the model fits to the data between
+these solutions, with a clear preference for the B solution,
+implying that these features are most likely located in QE2ʼs
+southern hemisphere (Figure 12). Thus, topography is likely
+playing a role for QE2 and is likely affecting the uncertainties
+in the simple thermal model results. Furthermore, topography
+may be one of the effects being captured by variations in our
+NEATM-like model’s beaming parameter.
+5.4. Beaming Parameter Trends
+The NEATM-like model’s beaming parameter is a scaling
+factor that accounts for additional effects not included in the
+model. As such, we can analyze the trend in our measured
+beaming parameters across each night of observation to
+understand the limitations of our NEATM-like model. We find
+beaming parameters that range from 0.54 to 0.78. These values
+therefore differ significantly from the value of η = 1.2 predicted
+by Harris (1998) for NEAs. Our modeled beaming parameters
+are instead much closer to the η ≈ 0.75 value predicted by
+Lebofsky et al. (1986) for main belt objects. Since the beaming
+parameter accounts for additional factors not incorporated into
+the NEATM-like model, we can use these differences to
+identify potential properties affecting QE2ʼs thermal emission.
+QE2 is a particularly good target for this analysis owing to its
+extremely spherical shape. Therefore, shape effects are likely a
+very small contributor to changes in the beaming parameter.
+One potential method for investigating beaming parameters
+is by looking for trends as a function of solar phase angle.
+Moskovitz et al. (2017) previously applied this method to QE2.
+Using beaming parameter as a proxy for thermal emission,
+Moskovitz et al. (2017) identified QE2 as a prograde rotator.
+We investigate this trend by showing the phase angle for QE2,
+which has a minimum value of 17°.1 on June 3, along with the
+fitted beaming parameters for the best-fit NEATM models for
+each night. We compare our results to those found by
+Moskovitz et al. (2017) in Figure 8.
+We find that our beaming parameter values do exhibit
+roughly the same trend as the Moskovitz et al. (2017) data but
+are significantly offset from the Moskovitz et al. (2017) data.
+We find much lower beaming parameter values than the
+Moskovitz et al. (2017) values of ∼1.1–1.4. We also find a
+range of thermal inertias that is overlapping with, but lower
+than their estimated range of ∼200–400 TIU done by
+comparing their NEATM results to more complex models.
+This is not unexpected, as our beaming parameter has been
+Figure 10. Plot of predicted apparent magnitudes for QE2 compared to all magnitudes reported to the MPC. All observations are from 2013 during QE2ʼs close
+approach to Earth. The predicted apparent magnitudes were calculated using ephemeris from JPL Horizons at 1-day intervals throughout 2013. We used H = 16.0 (a
+value from our modeled H-magnitude range) and H = 17.3 (the H-magnitude from Moskovitz et al. 2017), as well as a range of G values. We see that a lower H-
+magnitude, more consistent with our modeled range, agrees with the data for low G values.
+13
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+Predicted Apparent Magnitudes Compared to MPC
+Observations of QE2 in 2013
+10
+H,G
+MPc Observations
+12
+H=16, G=0
+H=16, G=0.05
+H=16, G=0.15
+H=17.3, G=0
+H=17.3, G=0.05
+14
+H=17.3, G=0.15
+18
+20
+Jan
+Mar
+Jun
+Sep
+Dec
+Timeseparated from the thermal inertia. The Moskovitz et al. (2017)
+beaming parameter must account for all the effects of thermal
+inertia, as they do not model thermal inertia explicitly, unlike
+our NEATM-like model, which does incorporate thermal
+inertia.
+Another possible explanation for why we observe different
+beaming parameters is because of our expanded wavelength
+range (Section 5.2). We incorporate data up to 4.05 μm in our
+NEATM-like model, while Moskovitz et al. (2017) only
+incorporate data up to 2.5 μm. As shown in Figure 11,
+Figure 11. Plot of NEATM-like models with varying visual geometric albedos across a selected range of dates. The data sets shown for May 30 and June 15 are the
+“A” data sets. All models shown have thermal inertia and beaming parameters that are within the best-fit ranges for the given date. Each row is a different data set. The
+left panels show the tail region, and the right panels show the thermally dominated region. We see that for July 10 A the models that fit the thermally dominated region
+do not fit the tail region and vice versa. An increase in albedo of ∼0.02 is required to fit the tail region for July 10 A. This is indicative of some kind of surface
+inhomogeneity.
+14
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+4
+125
+Normalized Flux
+100
+3
+75
+30 May
+30 May
+pv =0.07
+pv =0.07
+50
+25
+0
+0
+125
+4
+100
+3
+Normalized Flux
+L
+75
++ 15 Jun
+15 Jun
+Normali
+pv =0.08
+pv =0.08
+50
+25
+0
+125
+4
+100
+3
+Normalized Flux
+L
+10 Jul A
+ 10 Jul A
+pv =0.07
+75
+pv =0.07
+pv =0.08
+pv =0.08
+Normali
+pv =0.09
+pv =0.09
+50
+25
+0
+0
+125
+4
+100
+3
+FI
+Normalized 
+Normalized 
+75
+10 Jul D
+12
+10 Jul D
+pv =0.09
+pv =0.09
+50
+25
+0
+2.0
+2.2
+2.4
+2.6
+2.8
+3.0
+3.2
+3.7
+4.2
+Wavelength (microns)
+Wavelength (microns)mismatches in model fits between the thermally dominated
+region and tail region of the spectra are possible. We check this
+by comparing the Moskovitz et al. (2017) fits to our data at
+longer wavelengths (Figure 13). As expected, we see that
+although the Moskovitz et al. (2017) models fit the tail region,
+they do not fit the thermally dominated region.
+The differences in measured beaming parameters could also
+be related to the illumination geometry of QE2. The technique
+used by Moskovitz et al. (2017) relies on assuming that the
+observations of the asteroids were made with equatorial
+illumination and thus may not be as robust when viewing an
+object with a different illumination geometry. (Moskovitz et al.
+2017 also recognize this possibility.) Although the observations
+of QE2 are made at low sub-Earth latitudes, it is possible that
+the discrepancy in the beaming parameters could arise from
+high subsolar latitudes. For QE2 these can range from ∼30° to
+∼45° for the A pole solution or from ∼10° to ∼15° for the B
+pole solution. Thus, because QE2 is not being observed
+looking directly at its equator, this means that self-shadowing
+from topographical features on the asteroid’s surface is likely to
+be important. Even for the more equatorial illuminated B pole
+solution, self-shadowing could still be playing a significant
+role, as QE2 does not have an equatorial ridge and thus still has
+topographical variation at the equator. This agrees with our
+SHERMAN results that show the importance of topography on
+QE2, which may be contributing to observed temperature
+differences (Section 5.3). Thus, this may further explain why
+our beaming parameter results differ from those of Moskovitz
+et al. (2017).
+6. Summary and Conclusions
+We present simple thermal model fits using our NEATM-
+like model for the NEA (285263) 1998 QE2. Furthermore,
+we compare these model results to more complex thermo-
+physical models, radar data, and other existing analyses of
+QE2 to understand the key factors affecting the uncertainties
+in simple thermal model results. For our simple thermal
+model fits, QE2 was observed with the SpeX instrument on
+the NASA IRTF on six nights in 2013, representing a range
+of viewing and illumination geometries. Additional data were
+acquired by the NEOWISE spacecraft in 2017. A visual
+geometric albedo between 0.05 and 0.10 and thermal inertia
+between 0 and 425 TIU are found to be consistent with all six
+nights of SpeX data. These results are also consistent with the
+NEOWISE absolute photometry at the 3σ level. These
+constraints are more robust than they would be using
+NEOWISE observations alone, due to the larger uncertainties
+on absolute photometry. The general model agreement with
+both
+absolute
+flux
+and
+normalized
+flux
+measurements
+increases our confidence in our model results, while also
+allowing us to benefit from the smaller uncertainties on
+normalized flux data. This is possible because of our
+incorporation of data representing a range of viewing
+geometries. As a result, we are able to break degeneracies
+in model results based on a single night of observations.
+In order to constrain the limits of simple thermal models as
+applied to a single object, we compare our results to more
+complex thermophysical models and previous observations.
+We find that our modeled albedo values are higher than but
+overlap with previously published values (Moskovitz et al.
+2017; Fieber-Beyer et al. 2020) and are consistent with results
+from the complex thermophysical model SHERMAN. We also
+identify a discrepancy in the resulting H-magnitude value when
+using the radar-derived size measurement (Springmann et al.
+2014). Based on the tight constraints we place on QE2ʼs albedo
+and the tighter constraints Springmann et al. (2014) place on
+QE2ʼs diameter, we believe that the true H-magnitude value
+must be brighter than current measurements suggest. As a
+result, the true albedo is likely toward the lower end of the
+range we identify using our NEATM-like model.
+We also leverage the wide wavelength range of our data set
+to compare our best-fit model results to both the tail region and
+thermally dominated region of our spectra. We find that for
+some dates, although our models fit the thermally dominated
+region well, they require a higher albedo to fit the tail region.
+This highlights the need to incorporate data across a wide
+wavelength range when modeling asteroid surface properties.
+We posit that these differences may be due tolocal thermal
+variations, but a full investigation is beyond the scope of
+this work.
+In addition to these discrepancies, we also find differences
+between our modeled beaming parameters and existing models.
+The most likely source of these differences may be the
+orientation of QE2 and wavelength range of data used.
+Observing these differences has also allowed us to infer that
+topography may play a significant role in determining the
+thermal emission of QE2. Thus, in this case, the inability to
+model self-shadowing effects from topographical variations
+may be a key limiting aspect of the simple thermal models.
+Furthermore, this analysis again shows the importance of
+incorporating data from a wide wavelength range when
+working with simple thermal models.
+Overall, our work has demonstrated our ability to place
+tighter constraints on the results of simple thermal models by
+comparing
+data
+taken
+across
+multiple
+different
+viewing
+geometries. By combining normalized flux with absolute
+photometry, we are able to place tighter constraints than would
+be possible with absolute photometry alone. Finally, we are
+able to place some constraints on the limits of simple thermal
+models as applied to single objects, finding that topography,
+viewing geometry, and the wavelength range of data used can
+all affect simple thermal model results. This work is important
+Figure 12. SHERMAN model results for June 8 and July 10 using both the A
+and B pole solutions. All models have a visual geometric albedo of 0.053,
+thermal inertia of 200 TIU, and crater fraction of 70%. We see a clear
+preference for the B pole solution in the June 8 data and a slight preference for
+the B pole solution in the July 10 data. Thus, we see that QE2ʼs topography
+may be playing a role in shaping its thermal emission. We also note that the
+albedo and thermal inertia are consistent with our NEATM-like model.
+15
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+SHERMAN Models for o8 Jun
+and 10 Ju
+125
+Normalized Flux
+08 Jun
+100
+08 Jun - A
+08 Jun - B
+75
+ 10 Jul
+10 Jul- A
+10 Jul - B
+50
+25
+0
+2.7
+3.2
+3.7
+4.2
+Wavelength (microns)for diagnosing cases (such as QE2) where more detailed
+analysis of an object may be required to fully understand its
+properties.
+Being able to extract more information from simple thermal
+models, like our NEATM-like model, will be critical as we
+move into the future of large survey missions such as LSST and
+NEO Surveyor. The large data volumes produced by these
+missions will necessitate the use of simple models to make full
+use of the data. Using these data as efficiently as possible will
+require further insights into the limitations of simple thermal
+models. As this work shows, although these models are reliable
+for statistical measurements of large groups of objects, the
+results
+for
+individual
+objects
+may
+be
+subject
+to
+great
+uncertainties. Addressing these issues will therefore allow us
+to make full use of these models and gain even greater insights
+into fields such as planet formation, asteroid dynamics, and
+planetary defense.
+This work was partially funded by the NASA YORPD
+program (NASA grant 80NSSC21K0658) and NSF AST
+1856411. S.A.M. was supported by the University of Arizona,
+Lunar and Planetary Laboratory, Lieutenant Colonel Kenneth
+Rondo Carson and Virginia Bryan Carson Graduate Fellow-
+ship. This material is based on work supported by the National
+Science Foundation Graduate Research Fellowship Program
+under grant No. DGE-2137419. Any opinions, findings, and
+conclusions or recommendations expressed in this material are
+those of the author(s) and do not necessarily reflect the views of
+the National Science Foundation. S.E.M. was supported by
+NASA’s Near-Earth Object Observations Program through
+grant 80NSSC19K0523.
+ORCID iDs
+Samuel A. Myers
+https://orcid.org/0000-0001-8500-6601
+Ellen S. Howell
+https://orcid.org/0000-0002-7683-5843
+Figure 13. Plot of best-fit models from Moskovitz et al. (2017) compared to our longer-wavelength data. These models are generated using our simple, NEATM-like
+model. All data sets shown are the “A” data set for the given date. All models shown have zero thermal inertia and albedos of 0.03, as per the Moskovitz et al. (2017)
+fits. The shown η values correspond to the ranges reported for each date by Moskovitz et al. (2017). The left panels show the tail region, and the right panels show the
+thermally dominated region. We see that the models fit the tail region well, as expected. However, we note that these models do not fit the thermally dominated region.
+This discrepancy may explain why we find different modeled beaming parameters than Moskovitz et al. (2017).
+16
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+125
+4
+100
+3
+Normalized Flux
+30 May
+30 May
+n=1.10
+n =1.10
+Normalized |
+n =1.15
+75
+n =1.15
+n =1.20
+n
+=1.20
+2
+50
+25
+0
+0
+125
+4
+100
+Normalized Flux 
+Normalized 
+02 Jun
+75
+02 Jun
+n =1.05
+n =1.05
+2
+n =1.10
+n =1.10
+n =1.15
+50
+王全
+25
+0
+0
+125
+4
+100
+Normalized Flux 
+Normalized
+15 Jun
+15 Jun
+75
+n =1.10
+n =1.10
+乡乡乡乡多
+n =1.15
+n =1.15
+n =1.20
+n =1.20
+50
+25
+0
+2.0
+2.2
+2.4
+2.6
+2.8
+3.0
+3.2
+3.7
+4.2
+Wavelength (microns
+Wavelength (microns)Christopher Magri
+https://orcid.org/0000-0002-2200-4622
+Ronald J. Vervack, Jr.
+https://orcid.org/0000-0002-
+8227-9564
+Yanga R. Fernández
+https://orcid.org/0000-0003-1156-9721
+Sean E. Marshall
+https://orcid.org/0000-0002-8144-7570
+Patrick A. Taylor
+https://orcid.org/0000-0002-2493-943X
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+Myers et al.
+
diff --git a/39E4T4oBgHgl3EQfbQyO/content/tmp_files/load_file.txt b/39E4T4oBgHgl3EQfbQyO/content/tmp_files/load_file.txt
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+page_content='Constraining the Limitations of NEATM-like Models: A Case Study with Near-Earth  Asteroid (285263) 1998 QE2  Samuel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Myers1  , Ellen S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Howell1  , Christopher Magri2  , Ronald J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Vervack, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3  , Yanga R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Fernández4  ,  Sean E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Marshall5  , and Patrick A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Taylor6  1 Lunar and Planetary Laboratory, University of Arizona, 1629 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' University Boulevard, Tucson, AZ 85721, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' sammyers@lpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='arizona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='edu  2 University of Maine Farmington, 173 High Street, Farmington, ME 04938, USA  3 Johns Hopkins Applied Physics Laboratory, 11100 John Hopkins Road, Laurel, MD 20723, USA  4 University of Central Florida, 4111 Libra Drive, Orlando, FL 32816, USA  5 Arecibo Observatory/University of Central Florida, HC-03 Box 53995, Arecibo, Puerto Rico 00612, USA  6 National Radio Astronomy Observatory/Green Bank Observatory, 1180 Boxwood Estate Road, Charlottesville, VA 22903, USA  Received 2022 August 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' revised 2022 November 28;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' accepted 2022 December 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' published 2023 January 10  Abstract  Near-Earth asteroids (NEAs) are a key test bed for investigations into planet formation, asteroid dynamics, and  planetary defense initiatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These studies rely on understanding NEA sizes, albedo distributions, and regolith  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Simple thermal models are a commonly used method for determining these properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' however, they  have inherent limitations owing to the simplifying assumptions they make about asteroid shapes and properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  With the recent collapse of the Arecibo Telescope and a decrease of direct size measurements, as well as future  facilities such as LSST and NEO Surveyor coming online soon, these models will play an increasingly important  role in our knowledge of the NEA population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Therefore, it is key to understand the limits of these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' In this  work we constrain the limitations of simple thermal models by comparing model results to more complex  thermophysical models, radar data, and other existing analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Furthermore, we present a method for placing  tighter constraints on inferred NEA properties using simple thermal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These comparisons and constraints are  explored using the NEA (285263) 1998 QE2 as a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We analyze QE2 with a simple thermal model and  data from both the NASA IRTF SpeX instrument and NEOWISE mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We determine an albedo between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='05  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='10 and thermal inertia between 0 and 425J m−2 s−1/2 K−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We find that overall the simple thermal model is  able to well constrain the properties of QE2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' however, we find that model uncertainties can be influenced by  topography, viewing geometry, and the wavelength range of data used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Unified Astronomy Thesaurus concepts: Asteroids (72);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Asteroid surfaces (2209);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Near-Earth objects (1092)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Introduction  Asteroids were once derided by astronomers as the “vermin  of the sky,” but they now form an important piece of our efforts  to understand our own solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Understanding their sizes,  albedo  distributions,  and  regolith  properties  is  key  for  investigations into many aspects of solar system science,  including solar system formation, main belt asteroid orbital  evolution, surface processes on airless bodies, and under-  standing our meteorite collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Near-Earth asteroids (NEAs),  in particular, are excellent targets for these efforts owing to  their proximity to Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  In addition to understanding the albedos and regoliths of  these objects, accurately measuring the sizes of NEAs is pivotal  for planetary defense initiatives—the area of study focused on  preventing catastrophic asteroid impacts with Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This is  because the size of an object is directly related to the energy of  impact (Morrison & Teller 1995), which determines the impact  severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Thus, observation and modeling techniques that  provide estimates of these properties are key for understanding  the NEA population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  There are a few methods for obtaining size estimates and  other physical properties from NEA observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Radar  images, detailed thermophysical models, and simple thermal  models can all be used to obtain size estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' All of these  methods, along with light-curve measurements, can also place  constraints on other physical properties of asteroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Other  methods, such as direct imaging (Dollfus 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Marchis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Marchis & Vega 2014), stellar occultations (Millis &  Dunham 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Arai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2020), and spacecraft encounters  exist (Belton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 1992, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Veverka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Lauretta  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2019) but are only applicable in rare cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Of the more  common methods, radar images can provide a size estimate  without other information (Ostro 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Radar observations  can also be used to construct detailed models of the asteroid’s  shape (Hudson & Ostro 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Magri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2007, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Nolan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Light-curve measurements can also produce shape  models, although they are often less detailed than radar-derived  shape models and do not include an absolute size scale (Ďurech  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2012 and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These shape models can be  coupled with thermal spectra to constrain other physical  properties of the asteroid as well, such as thermal inertia or  surface roughness (Marshall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Jones 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Hinkle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Historically, the Arecibo Telescope has been a source of  numerous NEA radar observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The Arecibo Telescope  detected over 900 NEAs and made size estimates of roughly  400 of those (Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' However, with the recent  loss of the Arecibo Telescope, there will be a lack of direct size  and shape measurements of NEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (Although Goldstone is able  to make radar measurements, it has a lower sensitivity and less  availability for targets of opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=') As a result, in the future  The Planetary Science Journal, 4:5 (17pp), 2023 January  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3847/PSJ/aca89d  © 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The Author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Published by the American Astronomical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Original content from this work may be used under the terms  of the Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0 licence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Any further  distribution of this work must maintain attribution to the author(s) and the title  of the work, journal citation and DOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  1  there will be a greater reliance on other methods to understand  the  physical  properties  of  NEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  These  methods  will  necessarily be models, like simple thermal models, that assume  asteroid shapes or use less well-constrained shape models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Simple thermal models, such as the Standard Thermal Model  (Lebofsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Lebofsky & Spencer 1989) and the  Near-Earth  Asteroid  Thermophysical  Model  (NEATM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Harris 1998), are a convenient method for obtaining NEA  sizes and physical properties in part because they are easy to  run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' They require only visible and thermal infrared data and are  computationally  fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  For  this  reason,  they  are  already  commonly used to analyze data collected by large survey  missions like NEOWISE (Mainzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2011b) and Explor-  eNEOs (Trilling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Due to the large volume of data  collected by these types of surveys and the sparse amount of  data collected on any single object, simple thermal models are  often the only practical way to quickly interpret the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' In  these cases, simple thermal models are used to identify both  scientifically interesting and potentially dangerous NEAs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=',  Trilling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  However, simple thermal models make simplifying assump-  tions about the asteroid’s shape and surface that can result in  inaccuracies and thus poor constraints of inferred NEA  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This is especially relevant for determinations of  asteroid sizes—values that are pivotal for planetary defense  activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Simple thermal models can only make direct  determinations of asteroid sizes in specific cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' If absolute  photometry in both the visible and infrared is available, size  can be solved for directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' However, these estimates require  assuming that the visible and infrared data were acquired at  similar viewing geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This assumption is often made  with models employing NEOWISE or ExploreNEOs observa-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Alternatively, if only normalized flux is available, then  the size must be estimated from the modeled albedo in  combination with the absolute magnitude, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' In this case, the  estimates are subject to uncertainties in the magnitude (Bowell  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Jurić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Vereš et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2015), as well as  typically large error bars in the inferred albedo, producing poor  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' In fact, recent work has shown that there are  inconsistencies between sizes derived from NEOWISE data  using these models and sizes derived using other methods  (Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Masiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Masiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  In this paper, we seek to better understand the limitations of  simple thermal models, such as NEATM, by comparing simple  thermal model results to more complex thermophysical models,  radar data, and other existing analyses of a given object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We  also present a method for placing tighter constraints on inferred  NEA properties using these simple thermal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We use a  simple, NEATM-like model (Section 3) to model the observed  NEA, and the consistency of the best-fit parameters is then  checked by comparing the models to normalized flux data  collected across multiple nights that represent a range of  viewing geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We also compare the models to the  absolute photometry collected by the NEOWISE spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' By  observing an object across multiple viewing geometries and  combining normalized flux spectra with absolute photometry,  we are able to place tight bounds on modeled NEA properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  These simple thermal model results are then compared to  model results from SHERMAN (Magri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2018), a complex  thermophysical  model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  radar  measurements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  and  other  observations  and  analyses  of  the  given  object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  These  comparisons allow us to place constraints on the overall  limitations of the simple thermal model and identify key factors  that influence uncertainties in simple thermal model results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  This analysis is performed on the well-studied NEA  (285263) 1998 QE2 (hereafter referred to as QE2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' QE2 is a  spheroidal, binary NEA system, with an existing radar-derived  shape model (Springmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The secondary has a  diameter ∼25% that of the primary (Springmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2014)  and thus contributes only 6% of the total flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Therefore, the  primary object dominates the thermal emission from the  system, and we neglect the secondary in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' QE2 is  an Xk-type asteroid in the Bus−DeMeo taxonomy, as derived  from our SpeX prism spectra and a visible spectrum obtained  by Hicks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  As part of our investigation into the limitations of the  NEATM-like model, we find a discrepancy in the currently  accepted H-magnitude for QE2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We find that the current value  is inconsistent with the size derived from the radar measure-  ments of QE2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We investigate this discrepancy and discuss  implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' As part of this investigation, we compare our  results to previous studies to understand QE2ʼs composition  and surface properties (Fieber-Beyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2020), as well as its  spin state (Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These comparisons allow us  to further benchmark the uncertainties in the results of our  method for placing tight constraints on NEA properties derived  with simple thermal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  In Section 2 we discuss the data used for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' In  Section 3 we describe our simple, NEATM-like model, and in  Section 4 we present the results for QE2 from this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' In  Section 5 we describe our analysis of the uncertainties in these  model results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We compare our simple, NEATM-like model  results to model results from SHERMAN, radar data of QE2,  and the results of other previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We then discuss  implications for the limitations of simple thermal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We  conclude with a summary of our results in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Spectral and Radar Data  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' IRTF Observations  The primary data used to constrain our models are normalized  flux spectra obtained with the SpeX instrument at the NASA  IRTF (Rayner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We use normalized flux, as it has  smaller uncertainties relative to absolute photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These  observations are carried out as part of our ongoing investigation  into the physical properties of NEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We observed using both  prism mode (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='8–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='5 μm) and Long-Wavelength Cross-Dispersed  (LDX)\uf0a01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='9 mode (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='1 μm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Note that the observations of QE2  presented here were done before the upgrade to SpeX that  expanded the wavelength ranges of all settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  For QE2, observations were carried out over six nights, from  2013 May 30 to 2013 July 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Over this time, the solar phase  angle of QE2 varied from 18°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0 to 39°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='7, which let us observe  different viewing geometries and illumination states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' As a  result, we see the thermal emission at different local times of  day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This is important because it allows us to check the  consistency of the fit parameters (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The various sub-  Earth locations of QE2 that we observed are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  A summary of the observational parameters for our six nights  of SpeX data is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  2  The Planetary Science Journal, 4:5 (17pp), 2023 January  Myers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  All SpeX observations were done in pairs, nodding the  telescope along a 15″ slit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We used exposure times of 15 s for  our LXD data and 10–30 s for our prism data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The data were  processed using the Spextool software package (Cushing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  2004), and the spectra were extracted from summed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' In  addition to the object, we observed solar-analog stars in a  similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' At least one was a nearby G star within ∼5° of  the object on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' All stars were compared to a well-  characterized solar analog star on each night, and their spectra  were corrected for slight spectral slope variations if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Each asteroid–star pair was combined in a ratio after correcting  each for atmospheric absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The spectra were then  determined using a weighted average over all asteroid–star  pairs and binned to form the final spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Bad data points were  flagged and excluded from the fitting and averaging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  The detailed methods for this entire process are given in  Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  The data are broken up across each night into several  independent sets of roughly 20–30 minutes each to sample  different areas of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' QE2 has a rotation period of  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='749 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='002 hr (Springmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2014), meaning that each  spectrum is separated by roughly 25°–40° of longitude at the  equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The sub-Earth latitudes and longitudes at the midtimes of  the observations are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These sub-Earth  coordinates are calculated using the shape model of Springmann  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The LXD data for each of the six nights are shown in  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Each spectrum is normalized at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='6 μm to give  normalized flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (Note that there is no significant thermal  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Sub-Earth locations on QE2 during observations as determined by a radar shape model (Springmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (a) The pole solution with the “bumpy”  topography in the northern hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (b) The pole solution with the topography partially in the southern hemisphere (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The range of sub-Earth locations  observed indicates that QE2 was observed across multiple different viewing geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This range of observations is key for constraining QE2ʼs parameters using our  NEATM-like model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  3  The Planetary Science Journal, 4:5 (17pp), 2023 January  Myers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Sub-EarthLocations  17  A)  16  Latitude (degree)  Date  15  30May  02 Jun  4  08 Jun  十  15Jun  区  18Jun  13  米  10Jul  12  11  B)  米  atitude(degree)  5  Date  30May  02 Jun  区  08 Jun  15Jun  18Jun  10  米  10Jul  15  0  60  120  180  240  300  360  Longitude(degree)contamination at this wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=') We use normalized flux  because the relative uncertainties are much smaller than for  absolutely calibrated photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We cover the range from  completely reflected to thermally dominated to ensure that our  simple thermal model is well constrained in both regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This  technique has the advantage of being more flexible but the  disadvantage that the data are highly correlated in wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' NEOWISE Observations  In addition to our SpeX data, we fit our simple thermal  model to data collected by NEOWISE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Unlike the SpeX data,  which measure normalized flux, NEOWISE measures absolute  photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Thus, fitting our simple thermal model to the  NEOWISE data allows us to check that the best-fit parameters  are consistent with both the spectrum shape and calibrated flux  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This provides an additional independent check on the  consistency of the simple thermal model and allows us to  identify any potential issues with the model not observed when  fitting normalized flux data alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  We retrieve the NEOWISE data and associated uncertainties  from the NASA/IPAC Infrared Science Archive (Mainzer  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2011a, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='7 We do not use the raw images, but instead  retrieve processed data that list the magnitudes and uncertain-  ties for channels W1 (effective wavelength 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='4 μm) and W2  (effective wavelength 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='6 μm) for each time the object was  observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We remove data points that are flagged for potential  contamination, such as by cosmic-ray hits, and average  together all remaining observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The uncertainty in the  NEOWISE data is dominated by systematic errors and not  statistical noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' All observations, except one, have similar  uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We thus take a weighted average of the  observations and adopt the variance of the overall data set,  divided by the square root of the number of observations minus  one, as our 1σ uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' For QE2, all observations were  taken over a short time interval such that the change in QE2ʼs  orbital position was minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Therefore, we averaged together  all available observations, resulting in one averaged set of data  points from eight individual observations that span roughly 29  hr and approximately six rotation periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The individual  observations are evenly distributed across the rotation phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' A  summary of the observational parameters for the averaged  observation is given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' A list of the individual  observations is given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  After retrieval, the data are then converted from NEOWISE  magnitudes to Fλ units following the procedures outlined in the  WISE Data Processing Handbook (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Cutri  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' For this process we apply a final blackbody color  correction corresponding to a 221 K object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This blackbody  temperature is determined by fitting ideal blackbody curves to  the NEOWISE data in an iterative process until the corrected  NEOWISE data and ideal blackbody curves converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The  blackbody temperature used for the initial correction is  calculated using the theoretical blackbody temperature relation  T  L  A  r  1  16  ,  1  H  sb  4  2  (  )  ( )  \uf065  s  p  = where Le is the solar luminosity, A is the Bond albedo, rH is the  object–Sun distance, and σsb is the Stefan–Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Table 1  Summary of Observations, Including Values Input Directly into the NEATM-like Model  Date  Set  Midtime  rH (au)  Δ (au)  α (deg)  Instrument  2013 May 30  A  06:46:50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='046 8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='040 3  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3  SpeX  2013 May 30  B  07:22:08  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='046 8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='040 3  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  SpeX  2013 May 30  C  08:36:57  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='049 8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='040 2  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='9  SpeX  2013 Jun 02  A  06:51:57  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='052 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='040 1  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3  SpeX  2013 Jun 02  B  07:08:19  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='052 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='040 1  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3  SpeX  2013 Jun 02  C  07:17:50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='052 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='040 1  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3  SpeX  2013 Jun 02  D  07:34:17  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='052 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='040 1  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  SpeX  2013 Jun 08  A  08:12:16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='067 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='060 5  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0  SpeX  2013 Jun 08  B  09:25:01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='067 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='060 8  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='1  SpeX  2013 Jun 08  C  09:37:14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='067 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='060 8  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='1  SpeX  2013 Jun 08  D  10:38:10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='067 4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='061 0  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  SpeX  2013 Jun 08  E  10:50:40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='067 4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='061 1  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  SpeX  2013 Jun 15  A  11:06:28  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='091 0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='098 8  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='8  SpeX  2013 Jun 15  B  12:16:11  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='091 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='099 1  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='8  SpeX  2013 Jun 18  A  13:07:51  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='103 3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='116 9  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='7  SpeX  2013 Jul 10  A  10:23:08  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='218 8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='256 2  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0  SpeX  2013 Jul 10  B  10:29:19  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='218 9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='256 2  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0  SpeX  2013 Jul 10  C  11:10:20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='219 0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='256 4  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0  SpeX  2013 Jul 10  D  11:49:53  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='219 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='256 6  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0  SpeX  2013 Jul 10  E  13:09:29  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='219 6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='257 0  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0  SpeX  2017 Jul 01  A  10:51:35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='767 6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='445 3  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='1  NEOWISE  Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Set refers to different data sets on a given night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Midtime is the midtime of observation for the data set in UTC time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (Each SpeX observation spans roughly  20–30 minutes, while the NEOWISE observation spans 29 hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Thus, each SpeX spectrum is separated by roughly 25°–40° of longitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=') rH is the Sun–object distance,  Δ is the Earth–object distance, and α is the solar phase angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Note that the observations are carried out across a range of solar phase angles and viewing geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  7  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='edu/doi/irsa/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='26131/IRSA144  4  The Planetary Science Journal, 4:5 (17pp), 2023 January  Myers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Processed LXD data sets for each night of observations with SpeX and NEOWISE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (a–f) SpeX data for each of the six nights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The different letters within  each panel indicate different data sets collected each night (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The y-axis is normalized flux, normalized to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='6 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (Note that there is no significant thermal  contamination at this wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=') (g) NEOWISE data in absolute flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Note that the NEOWISE data are plotted over a different wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We plot  both the 1σ and 3σ uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (h) The “A“ data set for each night of SpeX data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These spectra highlight how different viewing geometries across the different nights  produce a range of spectral slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We see that changes in viewing geometry produce changes in the spectra shape both within nights and across all nights of  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Modeling these differences allows us to place tighter constraints on NEA properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  5  The Planetary Science Journal, 4:5 (17pp), 2023 January  Myers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  125  A)  B)  100  xnI  2013 May 30  2013 Jun 02  Normalized  75  ＡB  50  CD  25  0  125  C  D)  100  Flux  Normalized  75  2013 Jun 15  2013 Jun 08  A  A  50  BCDE  +  25  0  125  F)  E)  100  Normalized  75  2013 Jul 10  2013 Jun 18  AB-  +A  +  50  +  CDE  +  25  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  Wavelength (microns)  Wavelength (microns)  1e-20  125  G)  H)  _wn  8e-21  100  SpeX, All Nights  2017 Jul 01  Normalized I  2013 May 30  6e-21  75  2013 Jun 02  3g  2013 Jun 08  +  2013 Jun 15  4e-21  50  2013 Jun 18  2013 Jul 10  25  2e-21  0e+00  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0  Wavelength (microns)  Wavelength (microns)The Bond albedo is estimated according to the method  described in Lebofsky & Spencer (1989):  A  G p  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='684  ,  2  (  )  ( )  =  +  where G is the slope parameter in the HG magnitude system  (Bowell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 1989) and p is the visual geometric albedo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The  standard assumption of G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='15 is used, and p is taken from  the model fits to the SpeX data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Note that since the fitting  process is iterative, choices of the initial guess parameters do  not strongly affect the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The end products of this  conversion process are flux densities reported in units of W  cm−2 μm−1, which match the units of our simple thermal  model output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The final NEOWISE data for QE2 are shown in  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We show the data with both 1σ and 3σ uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Radar Shape Model  As part of our investigation into the limitations of simple  thermal models, we compare the results of our NEATM-like  models to many other data sources and models, including radar  images and a radar shape model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The radar image is a direct  measurement of the size that only depends on the viewing  geometry and the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' A spheroidal object, such as  QE2, shows a radius in radar range at nearly all aspects and is a  robust size estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We compare the radar size to sizes derived  from our NEATM-like model, based on the magnitude and  albedo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We emphasize that this information is not used as an  input of our NEATM-like model and is only used to compare  with our NEATM-like model results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  The radar shape model for QE2 is described by Springmann  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The model is constructed using observations from  the Arecibo Observatory and Goldstone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Data used were  collected between 2013 May 31 and June 9, during QE2ʼs close  approach to Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These radar images are used to derive a  shape model as described in Magri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' A nonlinear  iterative process is used to adjust synthetic radar images to  match the observations by minimizing the difference between  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This process is described in detail in several papers for  other objects (Magri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Nolan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The shape  model of QE2 is preliminary, and the complete analysis is  beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' However, the derived diameter  of the principal axes of QE2 is robust and reliable as a  comparison to values obtained here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This analysis gives a  diameter for QE2 of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3 km and a diameter of the  secondary of 800 ± 80 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' QE2 is spheroidal, with a few  dominant surface features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Springmann  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  (2014)  find  a  rotation  rate  of  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='749 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='002 hr for QE2 and two possible pole solutions,  both of which are prograde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' One of these solutions, which we  refer to as the A solution, places most of the “bumpy”  topography of QE2 in the northern hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This solution  has a pole position of λ = 119° and β = 55°, where λ is the  ecliptic pole longitude and β is the ecliptic pole latitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The  second solution, which we refer to as the B solution, places the  “bumpy” topography partially in the southern hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This  solution has a pole position of λ = 158° and β = 41°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Both  solutions are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' NEATM-like Model  The simple thermal model we use to fit the data is based on  the  Standard  Thermal  Model  (Lebofsky  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Lebofsky & Spencer 1989) and NEATM (Harris 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Our  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Sky views of QE2 on 2013 July 10 that show the radar shape model from Springmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The arrows indicate the pole and spin direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Left: the  A solution with a pole position of λ = 119° and β = 55°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Right: the B solution with a pole position of λ = 158° and β = 41°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Table 2  List of Individual NEOWISE Observations Used to Obtain the Single  Averaged NEOWISE Data Set  Date  Midtime  m1 (mag)  σ1 (mag)  m2 (mag)  σ2 (mag)  2017 Jun 30  19:32:22  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='768  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='468  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='609  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='136  2017 Jun 30  22:41:03  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='256  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='253  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='844  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='158  2017 Jul 01  03:23:49  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='625  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='392  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='944  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='172  2017 Jul 01  06:32:19  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='966  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='535  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='658  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='131  2017 Jul 01  15:58:00  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='449  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='338  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='193  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='292  2017 Jul 01  19:06:30  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='927  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='522  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='694  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='135  2017 Jul 01  22:15:00  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='557  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='476  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='801  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='161  2017 Jul 02  01:23:41  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='539  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='375  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='770  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='124  2017 Jul 01  10:51:35  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='528  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='092  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='758  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='051  Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Midtime is the midtime of observation in UTC time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' m1 and m2 are the  NEOWISE reported magnitudes for W1 (effective wavelength 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='4 μm) and W2  (effective wavelength 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='6 μm), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' σ1 and σ2 are the NEOWISE  reported magnitude uncertainties for W1 and W2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The last row is  the averaged observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  6  The Planetary Science Journal, 4:5 (17pp), 2023 January  Myers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  model is a variation of these models that we call our NEATM-  like model (Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Like these models, for a given  set of asteroid parameters, our NEATM-like model produces a  theoretical thermal emission spectrum of the object that can be  fit to any subset of the visible to near-IR spectra of an asteroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  However, our model also utilizes a simple incorporation of the  rotation rate of the object that allows us to model the thermal  inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The thermal inertia is a measurement of how well the  object’s surface retains heat energy from the Sun and is  measured in J m−2 s−1/2 K−1 (hereafter referred to as TIU for  thermal inertia units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' By determining the thermal inertia, in  combination with the rotation rate, our NEATM-like model is  able to account for differences across the day and night sides of  an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Thus, when incorporating many different observa-  tions of a single object, taken at different viewing geometries,  we are able to model how changes in thermal inertia affect the  thermal emission of an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Overall, this incorporation  allows us to get a more robust picture of the properties of the  object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We note that other than this addition, this model is  functionally similar to the standard NEATM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  In addition to incorporating these parameters, our model also  makes the typical assumption of a spherical shape for the  asteroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' It also assumes subsolar and subobserver points on the  asteroid’s equator and prograde rotation at a fixed rotation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  (The NEATM-like model does not account for shape effects,  and the radar-derived shape model of Springmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2014 is  only used to compare to the NEATM-like model results to  investigate the limitations of the NEATM-like model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=') The  model also incorporates a free-floating beaming parameter—a  scaling factor between the observed and predicted flux from the  asteroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This factor accounts for additional effects not included  in the model, such as surface roughness, deviations from a  spherical shape, local shadowing, and nonzero obliquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The  beaming parameter generally ranges between ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0,  with higher values usually occurring at higher phase angles or  for more irregularly shaped asteroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Overall, our model includes three free-floating parameters:  the visual geometric albedo, thermal inertia, and beaming  parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The output of each run is a model spectrum of the  asteroid, based on the input parameters, for each combination  of the free-floating parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Thus, identifying best-fit  parameters requires inspecting the model results and making  direct comparisons to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  For a given object, the consistency of these fit parameters  can be checked by comparing the results to thermal infrared  data collected across multiple nights that represent a range of  viewing geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This is important because many combina-  tions of albedo, thermal inertia, and beaming parameter can fit  any individual observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' By comparing model results for a  single object to data taken at multiple different viewing  geometries of that object, we can thus identify consistent values  of albedo and thermal inertia that fit every observation,  breaking degeneracies in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The beaming parameter  is allowed to vary, as it is expected to change in value across  each observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Thus, across multiple different viewing  geometries, only a tight range of albedo and thermal inertia  values will fit every observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This is true even when the  beaming parameter is allowed to vary, as more extreme  deviations in albedo or thermal inertia would require increas-  ingly extreme values of the beaming parameter to fit the  observations, and realistic beaming parameters are generally  constrained to the range of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='5–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0 (Delbó et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Note  that these comparisons are done solely to constrain the  parameter fits of the NEATM-like model and are separate  from the comparisons done as part of our investigation into the  limitations of the NEATM-like model (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content="  The fixed model inputs for our NEATM-like model are the  object's rotation period, a visible-to-near-IR reflectance ratio,  Earth–object and Sun–object distances, solar phase angle,  emissivity, and spherical equivalent diameter." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' For QE2, we use  a rotation period of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='749 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='002 hr that was used by a  previously derived radar shape model (Springmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  We also use a spherical equivalent diameter of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2 km from the  same shape model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We note that since the shape of QE2 is very  close to spherical, the assumption of spherical shape by the  NEATM-like model is a very good assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The visible-to-  near-IR reflectance ratio is estimated to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='127 using our  SpeX prism spectra and a visible spectrum obtained by Hicks  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This is a color correction factor used to relate the  visible albedo to the near-infrared albedo at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='6 μm, chosen as  the normalization wavelength of the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Earth–object and  Sun–object distances, as well as solar phase angle, are  calculated for each observation using JPL Horizons8 based  on the midtime of observation for each data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These values  are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The emissivity is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' NEATM-like Model Results  We generate NEATM-like models for each of our normal-  ized flux SpeX data sets and our single absolute photometry  NEOWISE data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Models are generated across a wide range  of albedos, thermal inertias, and beaming parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Models  are then compared to the data using an objective function to  constrain QE2ʼs properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' For any given data set, models of  varying parameters change monotonically (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These  models are sorted by calculating a reduced χ2 between the  model and the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' When performing this calculation, we only  consider data points between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='00 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='05 μm, as this is the  region of strongest thermal emission without significant  overlap with atmospheric water vapor lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' For the NEOWISE  data set, both NEOWISE data points are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  It is important to note that the reduced χ2 value we calculate  is not a formal χ2, as it does not reach a minimum at unity and  does not go up by a value of 1 when the model is 1σ away from  the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This is because the uncertainties in the data are  dominated by systematic effects, not statistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The data  points are not independent, as they are strongly correlated in  wavelength and are affected by changing effects such as  atmospheric conditions on different days, viewing geometry,  and rotational changes of the asteroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' As a result, this  calculation can be used to sort the goodness of fit of models for  a given data set but cannot be used to compare models across  data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Thus, for each data set, we use this method to identify  the range of albedos and thermal inertias that produce models  that lie within the 1σ uncertainties of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Figure 5 shows  the variation in models that were accepted to fit the data for  each data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (Note that for the NEOWISE data we also  examine the models that fit within the 3σ uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This  range is also shown for the NEOWISE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=') Any models  within the shown region are considered to fit the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' All other  models for the given data set are discarded, as they are poor fits  to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  8  https://ssd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='jpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='gov/horizons/  7  The Planetary Science Journal, 4:5 (17pp), 2023 January  Myers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  For each data set, we then have a range of albedos and  thermal inertias that can be said to fit that given data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These  individual fit spaces are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (For the NEOWISE  data, we show the models that fit the 1σ uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=') Overall,  we have 21 such data sets: 20 data sets spread across six nights  of IRTF SpeX observations, and 1 set of NEOWISE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' To  identify the range of albedos and thermal inertias that describe  QE2 overall, we then search for the region of overlap between  each of these 21 different model sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These results are shown  in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' There is a clear section in the parameter space of  models that fit nearly every data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We define this region as  the best-fit space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  All models within this space are consistent with the SpeX  data, but do not fit the NEOWISE data with 1σ uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  We then examine models that fit the NEOWISE data with 3σ  uncertainties, and find that all models within the best-fit space  are consistent with the NEOWISE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This could be because  the NEOWISE observations were taken at a much higher Sun–  object distance than the SpeX data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' As a result, QE2 was much  colder at the time of these observations which may be  introducing complexities to the thermal emission that our  simple thermal model is not able to capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Such effects may  be better understood using a more complex thermophysical  model, however a full investigation of this effect is beyond the  scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Overall, our analysis gives best-fit ranges of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='05–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='10 for  the visual geometric albedo and 0–425 TIU for the thermal  inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Note that there is a correlation such that higher thermal  inertias require lower albedos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Results are summarized in  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  In general, we find a preference for lower beaming  parameters of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='55–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Beaming parameter results are  shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We remind the reader that we expect the  beaming parameter to change across observations, and so we  do not attempt to fit for a single overall value of the beaming  parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These values are calculated by taking the best-fit  beaming parameter value for a fixed albedo of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='07 and a fixed  thermal inertia of 150 TIU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These values are chosen because  they are near the center of the best-fit region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The NEOWISE  beaming parameters are calculated using the 3σ uncertainties as  they are the results consistent with the SpeX data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' As expected,  the beaming parameter is generally higher for higher phase  angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The exceptions to this trend are July 10 and the  NEOWISE data, both of which have substantially greater rH  and Δ values than the other nights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These larger distances also  explain the noisier data observed on July 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Limits of the NEATM-like Model  In calculating our best-fit model ranges, we compared our  model results across many data sets taken at different viewing  geometries of QE2 (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These comparisons have  allowed us to place tighter constraints on our modeled albedo  and thermal inertia than would be possible with single  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These albedos and thermal inertias can then be  compared to results from more complex thermophysical  models, radar data, and other observations to identify how  accurately the NEATM-like model was able to constrain the  properties of QE2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Our model results also provide us with a  range of beaming parameter values that change as a function of  viewing geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Analyzing these changes in beaming  parameter can allow us to identify the unmodeled factors  limiting the accuracy of our NEATM-like model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Overall, by  comparing our model results to previous studies of QE2, we  can gain insight into the limitations of simple thermal models  as applied to a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' In the subsections below we walk  through comparisons of our simple thermal model results to  various other models and data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' For each comparison, we  discuss in what ways our simple thermal model results differ  and discuss implications for the factors affecting the uncertain-  ties of simple thermal model results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Albedo, Size, and H-magnitude  Our modeled visual geometric albedo for QE2 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='05–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='10  is higher than but overlaps with previously published values of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='03 +  (Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2017) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='01 (Fieber-  Beyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We can use our modeled albedo, in  combination with a radar-derived size, to estimate QE2ʼs H-  magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This is given by the relationship  ⎛  ⎝  ⎞  ⎠  H  p  D  5 log  1329 km ,  3  10  ( )  = -  where p is the albedo and D is the object diameter in kilometers  (Pravec & Harris 2007, Equation (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Using the diameter of  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3 km given by Springmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2014) and our  modeled albedo range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='05–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='10, we get an H-magnitude of  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='4–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This value is lower than (but partially overlaps with)  previously given H-magnitude values of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='4 (Trilling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  2010) and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3 (Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2017) for QE2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  However, the radar shape model constrains the diameter with  high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The radar-derived shape can be considered a  true constraint on QE2ʼs size, as size can be measured directly  from a radar image (Ostro 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Figure 9 shows a radar image  of QE2 taken by the Arecibo Telescope on 2013 June 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The  vertical extent of the image shows distance from the observer to  the terminator of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Thus, the resolution of the pixels,  combined with knowledge of the speed of light, directly gives  the object’s radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' In this image, QE2 covers 210 pixels in the  vertical extent at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='5 m pixel−1, giving an apparent radius of  1575 m or a diameter of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='15 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' However, using an H-  magnitude of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3 and albedos of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='05–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='10 gives a diameter  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' A range of NEATM-like models compared to one of our SpeX data  sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' As either the albedo or thermal inertia changes monotonically, the models  correspondingly change monotonically across the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This property allows us  to identify a range of models that fit the data and is typical to all of our data  sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' All models that fall within the 1σ error bars of the data would be  considered good fits to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' As such, in this case only the pV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='06 and  Γ = 100 TIU model would be considered a good fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Models shown here all  have η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Changes in beaming parameter can also monotonically affect  how the models fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  8  The Planetary Science Journal, 4:5 (17pp), 2023 January  Myers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Variation in NEATM-like Models  125  十  30 May A  100  pv =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='03,  =100  Normalized Flux  pv =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='06, I =100  pv =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='09, I =100  75  pv =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='06, I =0  王  pv =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='06, I =200  王  王  王  正  50  25  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  Wavelength (microns)Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The variation in NEATM-like models that were accepted to fit the data for each data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Any models within the shaded region are considered to fit the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  All other models for the given data set are discarded, as they are poor fits to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' An objective function is used to identify which models fall within the shown  region (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (a–f) SpeX data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The y-axis is normalized flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The spectra are offset for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (g) NEOWISE data in absolute flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Note that the  NEOWISE data are plotted over a different wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' For the NEOWISE data we examine models that fit within both the 1σ and 3σ uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Both  regions are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  9  The Planetary Science Journal, 4:5 (17pp), 2023 January  Myers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  250  A)  B)  200  xn  2013 Jun 02  2013 May 30  ABCD  ABC  Normali  100  50  0  250  C)  D)  2013 Jun 08  2013 Jun 15  A  BCDE  A  Normal  B  100  50  0  250  E)  F)  200  lux  2013 Jul 10  2013 Jun 18  十+  ABCDE  + A  100  50  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  Wavelength (microns)  Wavelength (microns)  1e-20  G)  wn  9e-21  8e-21  2017 Jul 01  7e-21  6e-21  1g  3g  5e-21  4e-21  3e-21  2e-21  1e-21  0e+00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0  Wavelength (microns)Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Reduced χ2 maps for each of the spectra as fit by our simple, NEATM-like model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Warmer colors mean higher values (worse fits), and cooler colors mean  lower values (better fits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Note that different max values are used for different spectra, as the reduced χ2 are not directly comparable across different spectra  (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Each χ2 map is equivalent to showing the range of models that fit a given data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
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+page_content='04  NEOWISE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='02  0 50 100 150 200 250 300 350 400 450 500 550 050 100 150 200 250 300 350 400 450 500 550 050 100 150 200 250 300 350 400 450 500 550  Thermal Inertia (TIU)  Thermal Inertia (TIU)  Thermal Inertia (TIU)between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='1 km, well outside of the 1σ errors of the  radar measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  We investigate this unusually large discrepancy in the H-  magnitude by looking at existing observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Using an H-  magnitude value and an assumed G value, we can calculate  predicted apparent magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These predicted apparent  magnitudes can then be compared to observed apparent  magnitudes reported to the Minor Planet Center (MPC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='9  Ephemeris values are calculated for QE2 using JPL Horizons10  at 1-day intervals throughout 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We then calculate predicted  apparent magnitudes for the H-magnitude consistent with the  radar-determined  size and our modeled albedo, the H-  magnitude used by Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017), and a range of G  values from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This was done following the procedure  in Bowell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These predicted apparent magnitudes  are then compared to all the apparent magnitudes listed in the  MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The results are shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We see that H-  magnitudes of neither 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0 nor 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3 perfectly match the data,  but instead provide an upper and lower bound, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  However, we note that an H-magnitude of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0 appears to  provide a more reasonable fit than an H-magnitude of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  So what could be causing these H-magnitude differences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  One possible explanation is related to the G parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The G  parameter is often assumed to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='15 and is not fitted directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Figure 10 shows that for H = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0 lower G values fit better,  while for H = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3 higher G values fit better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' For QE2, we  would expect a lower G value, as lower G values are generally  preferred  for  low-albedo  objects  owing  to  the  smaller  opposition effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' However, we note that the differences do  not exceed ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='5 mag and thus cannot fully explain the  discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Another possible explanation is related to color effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  however, the color of QE2 is very close to solar, and thus this is  also unlikely to be a large factor in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The discrepancy  could also be due to the secondary contributing to the  magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Using the radar shape model (Springmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  2014), we can calculate the effective diameter of the combined  primary and secondary to be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Using our modeled  albedo range, this gives an H-magnitude difference of only  ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='1 and thus is an ignorable contribution to the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Therefore, none of these effects by themselves can fully explain  the observed differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Overall, given the accuracy of the  radar measurement of the diameter and our tightly constrained  albedo range, the true H-magnitude cannot be as high as 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  It is therefore likely that a better estimate of the H-magnitude  lies somewhere in between 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0 and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Furthermore, our  analysis shows that higher H-magnitudes and thus lower  albedos are  likely favored  for QE2, potentially  further  constraining the results from our simple thermal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Wavelength Range of Observations  We can also analyze our results by leveraging the large  wavelength range of our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Our observations span  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='8–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='1 μm, and thus we are able to observe both the thermally  dominated region of the spectra (\uf0893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0 μm) and the thermal tail  (∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='5 μm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We are therefore able to compare our model  fits to both regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This is notable because many studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=',  Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2017) rely only on the tail region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We show  this comparison for a selection of our data sets in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  We find that in nearly all cases the models that best fit the  thermally dominated region also fit the tail region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' However,  for some dates (such as some data sets for 2013 July 10), an  albedo increase of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='02 relative to the model that fits the  thermally dominated region is required to fit the tail region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  This implies that QE2 may have an inhomogeneous surface and  that we may be observing\uf0a0local thermal variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Such  variations could impart a wavelength-dependent change in the  flux, thus creating the observed discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Another possibi-  lity is that some other effect, such as surface roughness, that our  NEATM-like model does not account for may be causing this  mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This result is important because it shows the dangers  of relying on only a limited spectral region to derive surface  properties such as albedo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Surface Topography  The potential effects of a surface inhomogeneity can be  investigated by comparing our NEATM-like model results to  results from a more complex thermal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' In addition to our  NEATM-like models, we generate models using SHERMAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  SHERMAN is a more complex thermophysical model that  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Final best-fit space for the visible geometric albedo and thermal  inertia for QE2 using our simple, NEATM-like model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The color of the points  represents the number of data sets that are fit by the associated parameter  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' A cooler color means that the given parameters are consistent with more  data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' White indicates models consistent with £ 2 data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The black line  outlines the region of best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This region corresponds to the region of overlap  between all the individual model ranges found to fit each individual data set  (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' There is a correlation such that higher thermal inertias require  lower albedos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' All models were run with the same fixed model inputs listed in  Section 3 and using ephemeris inputs listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This figure is generated  using the results from the 1σ uncertainties on the NEOWISE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Table 3  Best-fit Model Ranges for the Three Free-floating Parameters of Our NEATM-  like Model  Parameter  Range  Albedo  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='05–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='10  Thermal inertia  0–425 TIU  Beaming parameter  ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='55–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='80  Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Albedo is visual geometric albedo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We expect the albedo and thermal  inertia to be consistent across all data sets, and thus the ranges given represent  the uncertainty in our model results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' However, we expect the range of  acceptable beaming parameters to change across observations, and thus the  range given represents the range of values observed across all data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  9  https://minorplanetcenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='net/  10 https://ssd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='jpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='gov/horizons/  11  The Planetary Science Journal, 4:5 (17pp), 2023 January  Myers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Number of Data Sets Fit per Mode  Geometric Albedo  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='16  Number  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='14  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='12  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='10  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='06  Visual  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='00  0  50100150200250300350400450500550  Thermal Inertia (TIU)takes account of the object’s shape and that can separate the  effects of obliquity and self-shadowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (For a full description  of SHERMAN, see Magri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=') We give SHERMAN  the radar-derived shape model of QE2 (Springmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  2014), as well as our SpeX thermal infrared data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We also input  a reflectance spectrum from our prism data, as well as a Hapke  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Plot of fitted beaming parameters as a function of solar phase angle adjusted so that 0° corresponds to QE2ʼs minimum phase angle during its close approach  to Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We also compare our beaming parameters to those found by Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Note the introduction of negative phase angles to differentiate between  observations taken before (positive values) and after (negative values) opposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The error bars represent the range of beaming parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The range is calculated by  identifying models that fit the data with fixed albedo and thermal inertia (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017) values are taken from their Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We see that our data  exhibit roughly the same trend where the data overlap, but that our beaming values are significantly offset from the Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Radar image of QE2 taken by the Arecibo Telescope on 2013 June 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The vertical extent of the image shows distance from the observer to the terminator of  the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The horizontal extent shows Doppler shift, with blueshift to redshift going left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The resolution of the pixels, combined with knowledge of the speed  of light, directly gives the object’s radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' In this image, QE2 covers 210 pixels in the vertical extent at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='5 m pixel−1, giving an apparent radius of 1575 m or a  diameter of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='15 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  12  The Planetary Science Journal, 4:5 (17pp), 2023 January  Myers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Beaming Parameter vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Solar Phase Angle  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='7  Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='6  This Work  May11  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3  n  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  May 30巫  Jun15  巫 Jun 08  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='1  Jul5  Jun 02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='8  Jun18Jun15  May 30 巫  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='7  Jul 10巫  巫 Jun 08  巫 Jun 02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='6  NEOWISE 巫  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='5  30  20  10  0  10  20  30  40  50  60  Adiusted αscattering function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' SHERMAN has three free-floating para-  meters: visual geometric albedo, thermal inertia, and crater  fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The crater fraction is a proxy for surface roughness  and describes the fraction of each model facet covered with  hemispherical craters, following the method of Lagerros  (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' SHERMAN outputs a modeled thermal spectrum that  we then compare with our thermal infrared data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  SHERMAN is a forward model, so we generate many  models across different values of the free-floating parameters to  match to our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Some preliminary model results are shown  in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We find that an albedo of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='053, thermal inertia of  200 TIU, and crater fraction of 70% can roughly match the  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These values are also consistent with the results of the  NEATM-like model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  The SHERMAN results also show that the topography of  QE2 is affecting the thermal emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='\uf0a0Using SHERMAN, we  run models using both possible pole solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The results  show slight differences in the model fits to the data between  these solutions, with a clear preference for the B solution,  implying that these features are most likely located in QE2ʼs  southern hemisphere (Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Thus, topography is likely  playing a role for QE2 and is likely affecting the uncertainties  in the simple thermal model results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Furthermore, topography  may be one of the effects being captured by variations in our  NEATM-like model’s beaming parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Beaming Parameter Trends  The NEATM-like model’s beaming parameter is a scaling  factor that accounts for additional effects not included in the  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' As such, we can analyze the trend in our measured  beaming parameters across each night of observation to  understand the limitations of our NEATM-like model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We find  beaming parameters that range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='54 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These values  therefore differ significantly from the value of η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2 predicted  by Harris (1998) for NEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Our modeled beaming parameters  are instead much closer to the η ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='75 value predicted by  Lebofsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (1986) for main belt objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Since the beaming  parameter accounts for additional factors not incorporated into  the NEATM-like model, we can use these differences to  identify potential properties affecting QE2ʼs thermal emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  QE2 is a particularly good target for this analysis owing to its  extremely spherical shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Therefore, shape effects are likely a  very small contributor to changes in the beaming parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  One potential method for investigating beaming parameters  is by looking for trends as a function of solar phase angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017) previously applied this method to QE2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Using beaming parameter as a proxy for thermal emission,  Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017) identified QE2 as a prograde rotator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  We investigate this trend by showing the phase angle for QE2,  which has a minimum value of 17°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='1 on June 3, along with the  fitted beaming parameters for the best-fit NEATM models for  each night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We compare our results to those found by  Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017) in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  We find that our beaming parameter values do exhibit  roughly the same trend as the Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017) data but  are significantly offset from the Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  We find much lower beaming parameter values than the  Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017) values of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='1–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We also find a  range of thermal inertias that is overlapping with, but lower  than their estimated range of ∼200–400 TIU done by  comparing their NEATM results to more complex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  This is not unexpected, as our beaming parameter has been  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Plot of predicted apparent magnitudes for QE2 compared to all magnitudes reported to the MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' All observations are from 2013 during QE2ʼs close  approach to Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The predicted apparent magnitudes were calculated using ephemeris from JPL Horizons at 1-day intervals throughout 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We used H = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0 (a  value from our modeled H-magnitude range) and H = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3 (the H-magnitude from Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2017), as well as a range of G values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We see that a lower H-  magnitude, more consistent with our modeled range, agrees with the data for low G values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  13  The Planetary Science Journal, 4:5 (17pp), 2023 January  Myers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Predicted Apparent Magnitudes Compared to MPC  Observations of QE2 in 2013  10  H,G  MPc Observations  12  H=16, G=0  H=16, G=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='05  H=16, G=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='15  H=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3, G=0  H=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3, G=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='05  14  H=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3, G=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='15  18  20  Jan  Mar  Jun  Sep  Dec  Timeseparated from the thermal inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017)  beaming parameter must account for all the effects of thermal  inertia, as they do not model thermal inertia explicitly, unlike  our NEATM-like model, which does incorporate thermal  inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Another possible explanation for why we observe different  beaming parameters is because of our expanded wavelength  range (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We incorporate data up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='05 μm in our  NEATM-like model, while Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017) only  incorporate data up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='5 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' As shown in Figure 11,  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Plot of NEATM-like models with varying visual geometric albedos across a selected range of dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The data sets shown for May 30 and June 15 are the  “A” data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' All models shown have thermal inertia and beaming parameters that are within the best-fit ranges for the given date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Each row is a different data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The  left panels show the tail region, and the right panels show the thermally dominated region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We see that for July 10 A the models that fit the thermally dominated region  do not fit the tail region and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' An increase in albedo of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='02 is required to fit the tail region for July 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This is indicative of some kind of surface  inhomogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  14  The Planetary Science Journal, 4:5 (17pp), 2023 January  Myers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  4  125  Normalized Flux  100  3  75  30 May  30 May  pv =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='07  pv =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='07  50  25  0  0  125  4  100  3  Normalized Flux  L  75  + 15 Jun  15 Jun  Normali  pv =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='08  pv =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='08  50  25  0  125  4  100  3  Normalized Flux  L  10 Jul A  10 Jul A  pv =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='07  75  pv =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='07  pv =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='08  pv =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='08  Normali  pv =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='09  pv =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='09  50  25  0  0  125  4  100  3  FI  Normalized  Normalized  75  10 Jul D  12  10 Jul D  pv =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='09  pv =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='09  50  25  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  Wavelength (microns)  Wavelength (microns)mismatches in model fits between the thermally dominated  region and tail region of the spectra are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We check this  by comparing the Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017) fits to our data at  longer wavelengths (Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' As expected, we see that  although the Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017) models fit the tail region,  they do not fit the thermally dominated region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  The differences in measured beaming parameters could also  be related to the illumination geometry of QE2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The technique  used by Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017) relies on assuming that the  observations of the asteroids were made with equatorial  illumination and thus may not be as robust when viewing an  object with a different illumination geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  2017 also recognize this possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=') Although the observations  of QE2 are made at low sub-Earth latitudes, it is possible that  the discrepancy in the beaming parameters could arise from  high subsolar latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' For QE2 these can range from ∼30° to  ∼45° for the A pole solution or from ∼10° to ∼15° for the B  pole solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Thus, because QE2 is not being observed  looking directly at its equator, this means that self-shadowing  from topographical features on the asteroid’s surface is likely to  be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Even for the more equatorial illuminated B pole  solution, self-shadowing could still be playing a significant  role, as QE2 does not have an equatorial ridge and thus still has  topographical variation at the equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This agrees with our  SHERMAN results that show the importance of topography on  QE2, which may be contributing to observed temperature  differences (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Thus, this may further explain why  our beaming parameter results differ from those of Moskovitz  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Summary and Conclusions  We present simple thermal model fits using our NEATM-  like model for the NEA (285263) 1998 QE2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Furthermore,  we compare these model results to more complex thermo-  physical models, radar data, and other existing analyses of  QE2 to understand the key factors affecting the uncertainties  in simple thermal model results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' For our simple thermal  model fits, QE2 was observed with the SpeX instrument on  the NASA IRTF on six nights in 2013, representing a range  of viewing and illumination geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Additional data were  acquired by the NEOWISE spacecraft in 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' A visual  geometric albedo between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='10 and thermal inertia  between 0 and 425 TIU are found to be consistent with all six  nights of SpeX data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These results are also consistent with the  NEOWISE absolute photometry at the 3σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These  constraints are more robust than they would be using  NEOWISE observations alone, due to the larger uncertainties  on absolute photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The general model agreement with  both  absolute  flux  and  normalized  flux  measurements  increases our confidence in our model results, while also  allowing us to benefit from the smaller uncertainties on  normalized flux data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This is possible because of our  incorporation of data representing a range of viewing  geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' As a result, we are able to break degeneracies  in model results based on a single night of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  In order to constrain the limits of simple thermal models as  applied to a single object, we compare our results to more  complex thermophysical models and previous observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  We find that our modeled albedo values are higher than but  overlap with previously published values (Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Fieber-Beyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' 2020) and are consistent with results  from the complex thermophysical model SHERMAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We also  identify a discrepancy in the resulting H-magnitude value when  using the radar-derived size measurement (Springmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Based on the tight constraints we place on QE2ʼs albedo  and the tighter constraints Springmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2014) place on  QE2ʼs diameter, we believe that the true H-magnitude value  must be brighter than current measurements suggest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' As a  result, the true albedo is likely toward the lower end of the  range we identify using our NEATM-like model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  We also leverage the wide wavelength range of our data set  to compare our best-fit model results to both the tail region and  thermally dominated region of our spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We find that for  some dates, although our models fit the thermally dominated  region well, they require a higher albedo to fit the tail region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  This highlights the need to incorporate data across a wide  wavelength range when modeling asteroid surface properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  We posit that these differences may be due to\uf0a0local thermal  variations, but a full investigation is beyond the scope of  this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  In addition to these discrepancies, we also find differences  between our modeled beaming parameters and existing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  The most likely source of these differences may be the  orientation of QE2 and wavelength range of data used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Observing these differences has also allowed us to infer that  topography may play a significant role in determining the  thermal emission of QE2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Thus, in this case, the inability to  model self-shadowing effects from topographical variations  may be a key limiting aspect of the simple thermal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Furthermore, this analysis again shows the importance of  incorporating data from a wide wavelength range when  working with simple thermal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Overall, our work has demonstrated our ability to place  tighter constraints on the results of simple thermal models by  comparing  data  taken  across  multiple  different  viewing  geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' By combining normalized flux with absolute  photometry, we are able to place tighter constraints than would  be possible with absolute photometry alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Finally, we are  able to place some constraints on the limits of simple thermal  models as applied to single objects, finding that topography,  viewing geometry, and the wavelength range of data used can  all affect simple thermal model results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This work is important  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' SHERMAN model results for June 8 and July 10 using both the A  and B pole solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' All models have a visual geometric albedo of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='053,  thermal inertia of 200 TIU, and crater fraction of 70%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We see a clear  preference for the B pole solution in the June 8 data and a slight preference for  the B pole solution in the July 10 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Thus, we see that QE2ʼs topography  may be playing a role in shaping its thermal emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We also note that the  albedo and thermal inertia are consistent with our NEATM-like model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  15  The Planetary Science Journal, 4:5 (17pp), 2023 January  Myers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  SHERMAN Models for o8 Jun  and 10 Ju  125  Normalized Flux  08 Jun  100  08 Jun - A  08 Jun - B  75  10 Jul  10 Jul- A  10 Jul - B  50  25  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='2  Wavelength (microns)for diagnosing cases (such as QE2) where more detailed  analysis of an object may be required to fully understand its  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  Being able to extract more information from simple thermal  models, like our NEATM-like model, will be critical as we  move into the future of large survey missions such as LSST and  NEO Surveyor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The large data volumes produced by these  missions will necessitate the use of simple models to make full  use of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Using these data as efficiently as possible will  require further insights into the limitations of simple thermal  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' As this work shows, although these models are reliable  for statistical measurements of large groups of objects, the  results  for  individual  objects  may  be  subject  to  great  uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Addressing these issues will therefore allow us  to make full use of these models and gain even greater insights  into fields such as planet formation, asteroid dynamics, and  planetary defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  This work was partially funded by the NASA YORPD  program (NASA grant 80NSSC21K0658) and NSF AST  1856411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' was supported by the University of Arizona,  Lunar and Planetary Laboratory, Lieutenant Colonel Kenneth  Rondo Carson and Virginia Bryan Carson Graduate Fellow-  ship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' This material is based on work supported by the National  Science Foundation Graduate Research Fellowship Program  under grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' DGE-2137419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Any opinions, findings, and  conclusions or recommendations expressed in this material are  those of the author(s) and do not necessarily reflect the views of  the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' was supported by  NASA’s Near-Earth Object Observations Program through  grant 80NSSC19K0523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  ORCID iDs  Samuel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Myers  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='org/0000-0001-8500-6601  Ellen S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Howell  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='org/0000-0002-7683-5843  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Plot of best-fit models from Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017) compared to our longer-wavelength data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' These models are generated using our simple, NEATM-like  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' All data sets shown are the “A” data set for the given date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' All models shown have zero thermal inertia and albedos of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='03, as per the Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017)  fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The shown η values correspond to the ranges reported for each date by Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' The left panels show the tail region, and the right panels show the  thermally dominated region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' We see that the models fit the tail region well, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' However, we note that these models do not fit the thermally dominated region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='  This discrepancy may explain why we find different modeled beaming parameters than Moskovitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
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+page_content=' Vervack, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
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+page_content='org/0000-0002-  8227-9564  Yanga R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content=' Fernández  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
+page_content='org/0000-0003-1156-9721  Sean E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E4T4oBgHgl3EQfbQyO/content/2301.05071v1.pdf'}
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+1
+TinyVers: A Tiny Versatile System-on-chip with
+State-Retentive eMRAM for ML Inference at the
+Extreme Edge
+Vikram Jain, Sebastian Giraldo, Jaro De Roose, Linyan Mei, Bert Boons, and Marian Verhelst
+Abstract—Extreme edge devices or Internet-of-thing nodes
+require both ultra-low power always-on processing as well as
+the ability to do on-demand sampling and processing. Moreover,
+support for IoT applications like voice recognition, machine
+monitoring, etc., requires the ability to execute a wide range
+of ML workloads. This brings challenges in hardware design to
+build flexible processors operating in ultra-low power regime.
+This paper presents TinyVers, a tiny versatile ultra-low power
+ML system-on-chip to enable enhanced intelligence at the Ex-
+treme Edge. TinyVers exploits dataflow reconfiguration to enable
+multi-modal support and aggressive on-chip power management
+for duty-cycling to enable smart sensing applications. The SoC
+combines a RISC-V host processor, a 17 TOPS/W dataflow
+reconfigurable ML accelerator, a 1.7 µW deep sleep wake-up
+controller, and an eMRAM for boot code and ML parameter
+retention. The SoC can perform up to 17.6 GOPS while achieving
+a power consumption range from 1.7 µW-20 mW. Multiple ML
+workloads aimed for diverse applications are mapped on the SoC
+to showcase its flexibility and efficiency. All the models achieve
+1-2 TOPS/W of energy efficiency with power consumption below
+230 µW in continuous operation. In a duty-cycling use case for
+machine monitoring, this power is reduced to below 10 µW.
+Index Terms—Extreme edge, tinyML, machine learning accel-
+erators, ultra-low power, system-on-chip.
+I. INTRODUCTION
+E
+Xtreme edge devices [1] or Internet-of-Things (IoT)
+nodes mostly perform non-vision tasks and can achieve
+good accuracy, even with small and lightweight neural network
+(NN) models [2]. This is in contrast to more traditional tasks
+designed for processing image data and contain millions to
+billions of parameters and operations with high hardware re-
+source demands. Consider the Google voice assistant as an ex-
+ample, which needs only 14 kilo bytes (kB) of NN parameters
+to run a keyword-spotting application on edge devices [3]. The
+insight that not all applications require maximum accuracy,
+large and complex NN models, has resulted in a new paradigm
+of ML application development, called tinyML or ML at the
+extreme edge [4]. This trend, at its core, has been driven by the
+V. Jain, L. Mei, and M. Verhelst are with the Department of Electrical
+Engineering - MICAS, KU Leuven, Belgium.
+S. Giraldo was with the Department of Electrical Engineering - MICAS,
+KU Leuven, Belgium. He is now with B12 Consulting, Belgium.
+J. De Roose and B. Boons were with the Department of Electrical
+Engineering - MICAS, KU Leuven, Belgium. They are now with Magics
+Technologies, Belgium.
+© 2023 IEEE. Personal use of this material is permitted. Permission from
+IEEE must be obtained for all other uses, in any current or future media,
+including reprinting/republishing this material for advertising or promotional
+purposes, creating new collective works, for resale or redistribution to servers
+or lists, or reuse of any copyrighted component of this work in other works.
+requirements imposed by battery-operated, performance- and
+power-constrained IoT nodes. Most IoT sensor nodes consist
+of a microcontroller unit (MCU) with a subset of sensors, a
+memory for storing acquired data, a CPU and a wireless data
+transceiver. The presence of these MCUs for data collection
+provides opportunities to process data very close to the sensor
+when the NN model is small, and avoids the high penalty of
+raw data transmission to more powerful edge or cloud units.
+Yet, this local ML processing, brings several new chal-
+lenges: 1.) As these nodes are battery-operated, the system is
+typically severely power or energy constrained requiring ultra-
+low power operation, with the ability to idle. 2.) the MCU,
+moreover, has limited compute power and memory space,
+resulting in a critical trade-off between model size, execution
+performance and hardware complexity; 3.) despite the need
+for efficiency, the system should also be flexible enough
+to support different classes of NN models across different
+applications, and 4.) it should have a small footprint. Several
+hardware for ML have been proposed in the recent literature
+and can be divided into three main categories: 1) extremely
+specialized edgeML accelerators designed for ultra-low power
+operation with little to no flexibility at low performance [5]–
+[8], 2) multi-modal edgeML accelerators providing medium
+level of flexibility with high performance at medium to high
+power consumption [9]–[13], and, 3) commercial-off-the-shelf
+(COTS) MCUs delivering higher flexibility but at low perfor-
+mance and medium power consumption [14]–[16]. Most of
+these hardware designs do not meet all the requirements of an
+extreme edge device. An exception is Vega [17] which presents
+a complete SoC, however, the specialized accelerator of Vega
+does not have the flexibility to handle all DNN workloads.
+Thus, a new class of flexible ultra-low power (ULP) platforms
+towards extreme edge deployment is needed.
+In this context, this work presents TinyVers [18], a highly
+adaptive SoC platform which significantly enhances the trade-
+off between energy efficiency and flexibility needed in extreme
+edge devices, through the use of: A.) a RISC-V proces-
+sor extended with a flexible ML accelerator (FlexML) with
+dataflow reconfiguration supporting diverse ML workloads and
+support for efficient zero-skipping in block structured sparsity
+and deconvolution; B.) an embedded magnetoresistive random
+access memory (eMRAM) for non-volatile storage enabling
+standalone operation with efficient power-down (or idling);
+C.) a programmable wake-up controller (WuC) supporting
+different power-on and idle modes to enable both always-on
+inference as well as on-demand and duty-cycled smart sensing
+arXiv:2301.03537v1  [cs.AR]  9 Jan 2023
+
+2
+and computation used in typical tinyML IoT applications. The
+SoC provides users flexibility not only in mapping diverse
+ML workloads for diverse tinyML applications, but also in
+supporting various use cases such as duty-cycling and smart
+sensing. We demonstrate TinyVers’ capabilities and improve-
+ments over state-of-the-art (SotA) on diverse applications in
+machine monitoring, anomaly detection, audio signal analysis,
+and image classification through the use of both deep learning
+as well as traditional ML workloads.
+The rest of the paper is organized as follows. The basics of
+ML compute kernels is introduced in Section II. Section III
+discusses the architecture overview of TinyVers, followed by
+Section IV providing further details of the FlexML accelerator.
+Section V provides details on how the software stack for
+ML deployment on TinyVers is undertaken. Subsequently,
+Section VI presents the experimental results of mapping
+different workloads and application use cases. Finally, Sec-
+tion VII compares TinyVers’ performance with related works
+and Section VIII concludes the paper.
+II. ALGORITHMIC BACKGROUND
+ML applications heavily exploit deep neural networks
+(DNN) with traditional convolutional (CNN) and fully con-
+nected (FC) layers. However, a plethora of new NN layer
+topologies are emerging. Some examples of these are the use
+of temporal convolutional networks (TCN) used in audio tasks
+like keyword spotting [19]–[21], or auto-encoders (AE) using
+convolution and deconvolution pairs in machine monitoring
+and anomaly detection tasks [22]–[24]. Morever, also machine
+learning models not relying on neural network layers are still
+used in extreme edge IoT nodes, such as support vector ma-
+chines (SVM) [25] used in novelty and anomaly detection ap-
+plications. The execution efficiency of all these workloads can
+can be improved with orders of magnitude when deployed on
+specialized accelerators. Yet, the wide variety in the compute
+kernels of interest complicates their efficient mapping on a
+single hardware platform. The following subsections deal with
+the different ML operation characteristics, their categorization
+into mathematical operations, and their hardware implications.
+A. Convolution and Dense Operation
+Convolutional and dense layers are the most common com-
+pute kernels used in DNNs and they can be decomposed
+into matrix-matrix multiplication (MMM) and matrix-vector
+multiplication (MVM) resp.. These two matrix operations can
+be represented mathematically as nested for loops as shown in
+Fig. 1. Most ML compute kernels can be categorized into one
+of these two mathematical operations, with some special layers
+requiring extra hardware changes. One such kernel is the TCN
+layer which can be represented as a 1D CNN and requires extra
+support for programmable dilation which is similar to strides
+in a convolution. Recurrent neural networks (RNN) like long
+short-term memory (LSTM) and gated recurrent unit (GRU)
+can be decomposed to MVM with need for extra hardware
+for activation functions. These hardware changes would be
+discussed further in Section IV.
+C
+C
+IY
+FY
+OY
+OX
+IX
+FX
+K
+*
+K
+Convolution Operation = MMM
+Input FMAP
+Weights
+Output FMAP
+C
+C
+*
+K
+Dense Operation = MVM
+Input FMAP
+Weights
+Output FMAP
+TCN
+CNN
+GAN
+AE
+LSTM
+FC
+SVM
+K
+for(y=0 to Y-1); for each output row
+ for(x=0 to X/N-1); for each output column
+  for(k=0 to K/N-1); for each output channel
+   for(c=0 to C-1); for each input channel
+    for(fy=0 to Fy-1); for each filter row
+     for(fx=0 to Fx-1); for each filter column
+      o[k][x][y] += i[c][x+fx][y+fy]*w[k][c][fx][fy]
+for(k=0 to K/N-1); for each output channel
+ for(c=0 to C/N-1); for each input channel
+  o[k] += i[c]*w[k][c]
+PE
+Spatial Unrolling X
+Temporal
+ Unrolling
+Spatial Unrolling Y
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+Fig. 1. Different ML models and their mathematical representation in terms
+of MMM and MVM. The nested for loop representation can be mapped onto
+specialized accelerators through spatial and temporal unrolling.
+When mapping MMMs and MVMs on specialized hardware
+accelerators, the nested for loops can be unrolled spatially
+and temporally, which is called dataflow in literature [26].
+On a 2D processing element (PE) array, two for loops can
+be spatially unrolled, i.e., the loops can be parallelized along
+the X and Y dimensions, as shown in Fig. 1. In the rest
+of the paper, this spatial unrolling is represented as (Spatial
+Unrolling X)|(Spatial Unrolling Y). The remaining for loops
+are temporally unrolled, i.e., sequential execution. Depending
+on the available parallelism and available re-usability, the
+spatial unrolling (X and Y) needs to be configurable, to be
+able to efficiently map all workloads, detailed in Section IV-B.
+B. Deconvolution
+Autoencoders used in many machine monitoring applica-
+tions consist of an encoder and a decoder pair, which tries
+to reconstruct the input data. After training on normal data,
+a reconstruction error signals an anomaly in the test data.
+Deconvolution or transposed convolution are used in these
+autoencoders and are built by combining the convolution and
+upsampling into a single operation. Deconvolution can be
+mapped as a convolution (MMM) but needs extra hardware
+to support zero-skipping of input for efficient mapping. Hard-
+ware modification can improve the mapping efficiency of this
+operation, and better exploit its inherent sparsity, as will be
+discussed in Section IV-C.
+C. Support Vector Machines (SVMs)
+SVMs are ML algorithms used for classification and re-
+gression tasks. When classification of input data between
+normal behavior and an anomaly is required, a binary classifier
+called a one-class support vector machine (OC-SVM) can be
+
+3
+used [27], [28]. The decision function of a OC-SVM using the
+radial basis function (RBF) kernel is given by the equation (1).
+For the Laplacian kernel, the L2 norm is replaced by L1 norm.
+f(x) =
+N
+�
+i=0
+αi · exp
+−∥x−svi∥2
+2σ2
+− b
+(1)
+where x is the input vector with length D, sv are the support
+vectors with length D, N is the number of support vectors,
+σ the standard deviation, α the Lagrange multiplier, and b the
+bias. The number of support vectors N, in combination with
+the vector length D, can become large in these workloads,
+making the L1 and L2 norm calculation complex, and their
+deployment can gain orders of magnitude in performance
+when deployed on specialized accelerators. The D and N
+dimensions of the norm operations can be treated similar to
+C and K dimensions of a dense layer (MVM) and can be
+spatially unrolled on the PE array. In addition to unrolling
+the norms, extra hardware to support squaring, subtraction,
+rounding and absolute operation needs to be added to each
+PE. The result of the norm calculation can then be used by a
+CPU core to compute the overall kernel.
+D. Structured Sparsity
+Exploiting sparsity in DNNs can help to reduce the com-
+putational complexity and memory requirements, by skipping
+zeros and compressing the NN parameters. However, random
+pruning or unstructured sparsity tends to be hard to efficiently
+map on hardware and requires special logic for zero-skipping
+and load balancing [29]–[31]. The structure of sparsity (gran-
+ularity of pruning) has high impact on hardware efficiency and
+prediction accuracy. Some works have found that unstructured
+sparsity achieves better prediction accuracy than structured
+sparsity but structured sparsity tends to be more hardware
+amenable and improves computational efficiency [30]. Thus, a
+structured sparse model could be trained with more iterations
+to revert back closer to the same prediction accuracy achieving
+similar overall efficiency/cost. Moreover, more coarse-grained
+sparsity can reduce the additional memory requirements im-
+posed for storing indices of non-sparse data.
+With all of these diverse ML workloads and their charac-
+teristics in mind, a platform which can efficiently map all of
+the above, needs to be designed.
+III. TINYVERS HARDWARE ARCHITECTURE
+TinyVers, as shown in Fig. 2, is a heterogeneous SoC
+consisting of a single core RISC-V processor, a flexible ML
+accelerator called FlexML, a 512 kB shared level-2 (L2)
+SRAM memory, a micro-DMA (uDMA) for data movement
+between peripherals/memory, a 512 kB eMRAM for non-
+volatile storage, and a WuC for power management. The SoC
+development is rooted in the PULPissimo platform [32]. It
+embeds a 2 kB read-only memory (ROM), which acts as the
+first stage boot loader (FSBL) and also controls boot from
+JTAG, external SPI flash or the eMRAM. Two communication
+busses are used: 1.) a logarithmic interconnect, which enables
+a tightly-coupled data memory (TCDM) providing single cycle
+eMRAM
+(512 KB)
+ROM
+Shared Memory L2 (512 kB)
+GPIO UART
+SPI
+I2C
+I2S
+CPI
+JTAG
+SCAN 
+CHAINS
+eMRAM 
+CNTL
+LP Data acq. Memory L2 
+(64 kB)
+TCDM interconnect
+uDMA
+DMA
+Source
+Source
+Sink
+RISC-V
+APB
+WuC (RTC
+& 
+Power FSM)
+2D SIMD
+Array
+8x8
+Weight L1 
+Memory
+Instruction 
+Memory
+Activation L1 
+Memory
+DMA
+Control 
+Registers
+Logic PD
+LP Data 
+Acq. Mem 
+Data Acq. 
+Mem PD
+L1 PD
+UDMA 
+PD
+AON PD
+MRAM 
+PD
+Power 
+Modes
+PD= Power Domain
+Boot
+OFF
+OFF
+OFF
+OFF
+OFF
+OFF
+OFF
+OFF
+OFF
+OFF
+OFF
+OFF
+ON/OFF
+OFF
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+Active
+Data Acq.
+LP Data Acq.
+Deep Sleep
+% VDD WAKE
+* VDD SCL
+** VDD SRAM
+^^ VDD MRAM
+# VCS MRAM
+ V+ bias
+V- bias
+^^ 
+ 
+# 
+% 
+% 
+Data 
+Mover 
+FSM
+FlexML Accelerator
+* 
+* 
+* 
+* 
+* 
+* 
+* 
+* 
+** 
+** 
+** 
+** 
+** 
+FlexML 
+Control 
+Unit
+Fig. 2. Overview of the complete TinyVers SoC showing the different power
+domains (PD) with their constituting modules and the power modes supported.
+access to the shared L2, and 2.) the APB standard bus, which
+is used for controlling different memory mapped modules.
+The interface between the SoC and FlexML accelerator is
+based on the HWPE framework presented in [33]. Using the
+streamers from [33], data is moved to-and-from the shared L2
+memory with the help of FlexML’s DMA engine which is a
+FSM controlling the data (un)loading of its private memories
+and double buffering operation. Several peripheral interface
+protocols are supported by the SoC including UART, SPI, I2C,
+I2S, and CPI, in addition to having 32 general purpose IOs
+(GPIO). Separate clocks are used for the main core logic,
+the peripheral interfaces, and the always-on domain which
+includes the WuC and the IO pads.
+A. Smart Sensing Modes for TinyML
+IoT tinyML applications typically operate by collecting data
+across a specified time window through an array of sensors,
+after which the collected data can be processed to make
+decisions. In many applications, the time window across which
+the data needs to be collected before processing can start,
+can vary from a few ms to sec. Moreover, during the sensor
+data collection, many modules of the MCU are not used since
+no heavy processing is done yet. This brings opportunities in
+improving power saving in many tinyML applications: during
+data collection, only the modules necessary for moving the
+windowed data from the sensor peripheral interfaces to the
+memory need to remain active, while e.g. the CPU can be
+put to sleep. Furthermore, in applications which work on
+time series data like audio, the memory requirement for the
+windowed data is small (< 64 kB), such that also a large part
+of the main memory of the MCU can be powered-down to
+avoid leakage power of the unused memory section.
+To this end, TinyVers introduces two tinyML optimized
+data acquisition power modes: 1.) ‘Data acq.’ and 2.) ‘LP
+data acq.’, as shown in Fig. 2. The data acq. mode, targeted
+towards applications with large sample data like vision, keeps
+the uDMA module and the complete shared L2 memory (512
+
+4
+Full Active
+Data Acq
+LP Data Acq
+0
+100
+200
+300
+31
+20
+8
+325
+77
+10
+356
+97
+18
+Power(µW)
+Dynamic
+Leakage
+Total
+Fig. 3.
+Power simulation of post-synthesis netlist undertaken in Cadence
+Genus tool for the three power modes. In all the three modes, I2S data is
+collected at a sampling frequency of 44.1 kHz for a window of 2 seconds.
+Full active power reported includes configuration of uDMA by RISC-V core
+and interrupt handling procedure, in addition to data collection.
+Power
+uDMA
+Power
+OFF
+Power
+ON
+Switch
+Power 1
+Switch
+Power 2
+Power
+OFF
+Reset
+Isolate
+Clk
+enable
+Power
+ON
+Top level FSM
+Bottom level FSM
+Power
+Logic & 
+L1
+Power
+MRAM
+Power
+L2 &
+L2 udma
+Fig. 4. Flow diagram showing the hierarchical FSM used in the WuC.
+kB) powered up. In contrast to that, the LP data acq. mode
+only keeps part of the shared L2 memory (64 kB) powered up,
+in addition to the uDMA. This mode is specifically targeted
+towards applications which needs time series and audio data
+like keyword spotting, machine monitoring, biosignal analysis,
+etc. Fig. 3 shows an estimation of the power saving that can
+be achieved when moving from a full active mode to the
+two tinyML sensing modes, with almost 3.5× improvement
+between the full active and data acq. modes and 5.5× between
+data acq. and LP data acq. modes.
+B. Power Management
+Aggressive power management is pursued in TinyVers on
+top of standard low power design. The SoC is divided into 6
+switchable power domains and 1 always-on domain (AON),
+as shown in Fig. 2. Each switchable power domain consists
+of multiple power gating switches, which isolate the VDD
+of the power domain from the global VDD supply. These
+power gating switches are controlled by control signals driven
+from the WuC of the AON domain. All interconnect crossings
+between the power domains are equipped with bidirectional
+level shifters and isolation cells, such that the individual supply
+voltages of the domains can be controlled independently.
+The smart WuC is in charge of this power management
+control, relying on a real-time counter (RTC). The counter
+can be programmed by the RISC-V core with millisecond
+granularity. The RISC-V core can instruct the WuC to bring
+the SoC into one of the five supported power modes shown in
+Fig. 2. To this end, the WuC encompasses hierarchical finite-
+state machines (FSM) driven by the RTC, as shown in Fig. 4,
+controlling the power-up and power-down of the complete SoC
+and the different power domains. The top level FSM controls
+the sequence of power-up/down of the different power domains
+and the bottom level FSMs control the fine-grain sequence to
+(de)activate the isolation cells and the power gating switches
+of the individual power domains.
+Emerging memories like ReRAM, MRAM, FeRAM, PCM,
+etc. [34], [35], have shown promise in building cost-effective
+embedded non-volatile memories (NVM) targeting applica-
+tions in edge computing for automotive or industry 4.0. NVM
+memories can be used as the storage space for boot code
+and other parameters that need to be stored. This enables
+two things: 1.) Duty-cycling can be used as a means of
+reducing power consumption in applications which do not
+require always-on operation; and 2.) the SoC does not need
+to go to a central cloud server in order to fetch its boot codes
+and NN parameters when it is power-cycled. Moreover, the
+availability of the NVM embedded on-chip, avoids the high
+energy cost of fetching data from off-chip.
+MRAM promoted as a universal memory, uses magnetic
+polarity to store data in its bitcells [36]. Being non-volatile
+and almost as dense as traditional SRAM, they are a good fit
+for tinyML applications using extreme edge SoCs. With this
+in mind, TinyVers integrates a 512 kB embedded MRAM on-
+chip, enabling extreme power management strategies for smart
+sensing and on-demand computation. In the SoC, the eMRAM
+acts as a non-volatile storage for the boot code that the RISC-
+V needs to wake-up and start processing, and can also store the
+NN parameters of the mapped ML workloads. The eMRAM
+can, finally, also be used as a non-volatile scratchpad space
+for storing windowed data in smart sensing applications. The
+interface between eMRAM and the shared L2 memory uses
+the uDMA unit and the design is based on the work of [17].
+IV. FLEXML ACCELERATOR
+This section firstly describes the architecture overview of the
+FlexML accelerator, followed by the dataflow reconfiguration
+used for flexible mapping, efficient zero-skipping used for
+deconvolution and structured sparsity, and finally the hardware
+for supporting SVM, as briefly discussed in Section II.
+A. FlexML Architecture Overview
+The FlexML accelerator is TinyVers’ specialized, versatile
+hardware accelerator. FlexML is designed to efficiently support
+the large diversity in ML workloads for tinyML applications,
+while exploiting the data reuse present in individual layer
+characteristics. This is achieved through a zero-latency runtime
+dataflow reconfiguration, discussed in Section IV-B. As shown
+in Fig. 5, FlexML encompasses an 8×8 single instruction
+multiple data (SIMD) array of processing elements (PE),
+wherein each processing element consists of a precision-
+scalable multiply-accumulate (MAC) unit with support for INT
+8/4/2 [37], shown in Fig. 6. As a result of the precision-
+scalability, the SIMD array can be reconfigured to be a
+8×8/16/32 array of INT8/4/2 MAC units, resp.. Each PE per-
+forms 1/2/4 MAC operations per cycle based on the selected
+precision (INT8/4/2) and the results are accumulated in a 32-
+bit register with full/partial output stationarity, reducing the
+movement cost of the large bit-width partial sums. The final
+output is passed through a ReLU function (if enabled), fol-
+lowed by re-quantization to the selected precision and written
+
+5
+DMA Engine
+Inst. 
+Mem
+Cntl
+FSM
+Sparsity 
+Mem
+2x2 kB
+Weight 
+Mem (L1)
+2x32 kB
+Adder Trees
+Act 
+Mem (L1)
+2x32 kB
+NLFG
+&
+Max
+Pool
+Input FIFO (L0)
+SIMD PE Array 8x8
+IX
+Layer Type
+ucode 
+Inst.
+K
+Fx
+Fy
+Input 
+pointer
+Weight
+Pointer
+.........
+IY
+C
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+Fig. 5. FlexML accelerator architecture overview with ucode instruction.
+*
+>>
+ABS
+REG
+REG
+Round
+Sub
+ReLU
+Overflow
+Control
+Input 
+Activation
+Output 
+Activation
+1
+0
+0
+0
+Input 
+Weight
+From Neighbor
+PE
+-
++
+8b
+20b
+8b
+16b
+…
+… …
+…
+… …
+unused
+4b
+4b
+Gated
+4b
+4b
+Gated
+12b
+9b
+2b 2b 2b
+…
+… …
+…
+… …
+Gated
+2b
+Gated
+2b
+2b
+2b
+2b
+8b
+6b
+Fig. 6.
+Block diagram of the processing elements used in the flexML
+accelerator, showing the precision-scalable MAC unit and the additional
+hardware to support SVM.
+back to the activation L1. Mixed precision quantization can
+help in improving performance of DNN models when moving
+below 8 bits precision. However, the hardware overhead of
+mixed precision can reduce the overall efficiency of PEs
+due to varying bandwidth and serialized dataflow [38]. Thus,
+FlexML only supports symmetric precision for its weights and
+activation. In addition, a simple shift and ReLU is used for
+normalization of output, which also keeps hardware overhead
+low. In order to maintain accuracy of the models, a hardware
+aware training framework, mentioned in Section V, is used.
+Supporting the SIMD PE array, are private level-1 (L1)
+SRAM based memories for storing both weights (64 kB)
+and activations (64 kB). Both the weight L1 and activation
+L1 are composed of two 32 kB banks operating in a ping-
+pong manner to overlap data writing and reading, improving
+the overall performance. An intermediate memory level L0 is
+provided between the activation L1 and the PE array. This
+L0 memory is a FIFO buffer of size 16×8 bits, used to
+improve data locality when doing shifting window operation
+in convolution. Furthermore, a separate non-linear function
+generator (NLFG) and a max pooling unit are provided. The
+for(y=0 to Y-1); for each output row
+ for(x=0 to X/8-1); for each output column
+  for(k=0 to K/8-1); for each output channel
+   for(c=0 to C-1); for each input channel
+    for(fy=0 to Fy-1); for each filter row
+     for(fx=0 to Fx-1); for each filter column
+     parfor(k=0 to 8-1); spatial unrolled output channel
+     parfor(x=0 to 8-1); spatial unrolled output column
+      o[k][x][y] += i[c][x+fx][y+fy]*w[k][c][fx][fy]
+for(k=0 to K/8-1); for each output channel
+ for(c=0 to C/8-1); for each input channel
+ parfor(k=0 to 8-1); spatial unrolled output channel
+ parfor(c=0 to 8-1); spatial unrolled input channel
+  o[k] += i[c]*w[k][c]
+FIFO
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+C
+Weight Memory
+FIFO
+MMM
+MVM
+Weight Memory 
+OX
+K
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+K
+Bank 0
+Bank 1
+Bank 3
+Bank 0
+Bank 1
+Bank 3
+Bank 7
+Bank 7
+Fig. 7. Diagram showing the dataflow reconfiguration used to switch from
+OX|K dataflow (left) for MMM to C|K dataflow for MVM. The nested for
+loops below show the addition of parfor loops for the spatial unrolling used.
+NLFG uses LUT-based linear approximation to generate the
+various activation functions (other than ReLU) used in NN
+models such as tanh, sigmoid, etc. To control the dataflow
+and control flow inside the accelerator, a control unit with
+FSMs fetches ucode instructions from the instruction memory,
+decodes the instruction and deploys the relevant layer on the
+PE array by updating the control signals and counters that
+track the workload. The ucode instructions are generated by a
+pseudo-compiler built in python (Section V), and consists of
+CISC-like layerwise long instructions with hyperparameters
+and shown in Fig. 5. The control unit is also extended to
+enable support for efficient zero-skipping of activations in the
+case of deconvolution and zero-skipping of pruned weights in
+conjunction with the sparsity index memories (Section IV-C).
+B. Dataflow Reconfiguration
+In order to efficiently map the diverse set of ML workloads,
+runtime dataflow reconfiguration is supported in the FlexML
+accelerator at no latency overhead. The configurability enables
+efficient mapping of both: 1.) MMMs used for CNN, decon-
+volution and TCN, exploiting both input and weight spatial
+data reuse under an OX|K dataflow with output stationarity,
+and 2.) MVMs used for FC, RNNs and norm calculation
+of SVMs with batch size 1, exploiting the available input
+spatial data reuse under a C|K dataflow with partial output
+stationarity. Multiple previous works have proposed dataflow
+reconfiguration in hardware to optimally map different work-
+loads [39]–[41]. However, these works suffer from large hard-
+ware overhead and latency for diverse dataflow support and
+are not suitable for extreme edge devices. This work limits the
+dataflows to two optimal mapping schemes, thereby, keeping
+hardware and power overhead low. Moreover, none of the prior
+works have looked into mapping of TCN, AE, and SVM on
+the same hardware accelerator. Fig. 7 shows the OX|K (left)
+and C|K dataflow (right) and their hardware implementation,
+resp.. In the OX|K dataflow, the spatial unrolling is applied to
+the OX and the K dimension of the nested for loop, allocating
+the unrolled OX dimension along the columns and the unrolled
+K along the rows of the SIMD PE array of dimension 8×8.
+
+6
+Input FIFO
+Input FIFO
+Input data from activation L1
+Normal operation
+Deconvolution operation
+Input data from activation L1
+0
+0
+0
+0
+0
+0
+0
+0
+Ctrl
+Control Unit
+Fetch instruction and enable 
+deconvolution
+Control the demux and muxes 
+for deconvolution
+Skip rows and columns with all zeros
+Cycle #0: Enable demux to push data to input FIFO
+Cycle #1: Set Ctrl to 0, data sent to PEs -> a 0 b 0 c 0 d 0
+Cycle #2: Set Ctrl to 1, data sent to PEs -> 0 b 0 c 0 d 0 e, 
+Shift 0 into input FIFO
+Cycle #3: Set Ctrl to 0, data sent to PEs -> b 0 c 0 d 0 e 0
+IX
+Input Activation
+Filter
+Deconvolution layer in software
+*Orange represents pruned pixels
+IY
+Fig. 8. Representation of deconvolution layer in software (top left), control
+unit running the zero-skip operation (bottom left), the architectural change
+required on the L0 FIFO to support deconvolution (top right), and cycle by
+cycle operation of the FIFO and PEs (bottom right).
+The rest of the for loops are temporally unrolled as shown
+in the nested for loops in Fig. 7, resulting in an output
+stationary dataflow. Under this dataflow regime, the activation
+L1 memory multicasts input activation data in the vertical
+dimension to the L0 FIFO memory, which fetches 8 words
+in the first cycle followed by single word during the shifting
+window operation, thereby, reducing the memory bandwidth
+and number of memory fetches by utilizing the reuse oppor-
+tunity. The weight L1 memory provides data in the horizontal
+dimension, providing 8 words using 2 internal banks, where
+each word is multi-cast along the row. Due to the output
+stationarity, accumulation continues till the final output is
+generated which are then systolically shifted out vertically to
+the activation L1, requiring 8 cycles to complete the output
+write-back. The input data shifting inside the L0 FIFO is made
+programmable to support the variable dilation used in TCNs
+or variable strides in general.
+The alternative C|K dataflow is used for MVM, as this
+workload cannot utilize the OX|K dataflow efficiently due
+to lack of re-usability of weights. Under this dataflow, the
+C dimension is spatially unrolled along the vertical column
+dimension and the K dimension along the horizontal row
+dimension. The activation L1 memory multicasts 8 words of
+input activation along the vertical dimension, bypassing the
+L0 FIFO memory. With a batch size of 1, no weight reuse
+is available and, thus, each PE needs a new weight every
+cycle. In order to meet this requirement, the weight memory
+utilizes all of its 8 banks to unicast 64 different weight words
+to the PEs. PE rows operate on different input channels (C) of
+the same output channel (K). Hence, once the required MAC
+operations per PE are done, the outputs of PEs of the same
+row are accumulated using an adder tree and one final output
+per row is shifted out to the activation memory.
+C. Efficient Zero-skipping for Deconvolution and Blockwise
+Structured Sparsity
+The FlexML accelerator supports efficient zero-skipping of
+deconvolution workloads. As shown in Fig. 8, the input FIFO
+Sparsity Index Mem 
+storing the sparse nature 
+Sparse 
+block
+1
+0101
+0000
+0000
+3
+C
+0
+1010
+2
+C
+C
+K
+*
+K
+x8
+K
+8
+8
+Blockwise Str. Sparsity for CNN
+Input
+Weight Matrix
+Weights
+Blockwise Str. Sparsity for FC/RNN
+Output
+Control Unit
+Fetch sparsity index for block 1 to 8 from sparsity memory
+Check bit wise, if one present then update counters of 
+control FSM to skip current C
+Fetch next blockwise index and repeat
+*Orange represents pruned pixels
+Fig. 9. Blockwise structured sparsity applied to CNN and dense layers (top),
+control unit operation in tandem with sparsity index memory to support zero-
+skipping (bottom).
+is designed such that when in deconvolution mode, it only
+fetches one set of words and shuffles it with zero padding. The
+control unit skips the rows and columns with zeros that would
+result in redundant computation, resulting in a performance
+gain of up to 2× compared to running deconvolution in
+convolution mode with upsampling.
+TinyVers also supports structured sparsity, more specifically,
+blockwise kernel-level sparsity (2D) for both convolutional
+and dense layers [29], [31]. In this scheme, shown in Fig. 9,
+complete input channels of the filter kernels are pruned with a
+constraint that a block size of 8 filter kernels (K = 8) should
+share the same pruning. The block size is decided by the
+dimension of the PE array and the spatial unrolling of K along
+the horizontal dimension of the 2D PE array. In our case, the
+selected block size makes controlling the dataflow and control
+flow easier. Applying the same channel pruning to all the 8
+filter kernels mapped in parallel on the PE array makes the
+mapping efficiency higher as all the rows can still operate with
+a common control logic, and enables not only energy savings,
+but also throughput benefits. For this, the FlexML accelerator
+consists of specialized sparsity index memories which store the
+bit encoded indices of the pruned channel groups. Fig. 9 shows
+the sparsity index memory and the control flow logic used in
+the control unit. Before every filter kernel block increment, the
+control unit fetches an index memory word and checks the data
+bit-by-bit for sparsity state, as the input channels increment.
+If a sparse channel is detected, the complete computation of
+the channel is skipped, thus, avoiding any zero computation.
+D. Support Vector Machine
+The L1 and L2 norm of OC-SVM requires modification
+of the PEs in order to use the same hardware for mapping
+the workload. As shown in Fig, 6, each PE is extended with
+a subtraction block, absolute unit, rounding unit, and the
+modification of the multiplier to also enable squaring for the
+norm calculation within the PE array. The input data vector x
+and the support vector svi are of dimension D and the number
+of support vectors is N. When used in the C|K dataflow, the
+
+7
+1.00E-01
+L1 MEMORY
+2.5 mm
+2.5 mm
+L2 
+MEM
+RISC-V
+&
+ACCEL
+uDMA
+eMRAM
+WuC
+Fig. 10. Measurement setup and chip microphotograph.
+D dimension of the input data vector of x is unrolled and
+multicasted vertically (C) along the PE array, while the N
+dimension of the support vector svi are unrolled and unicasted
+horizontally (K). The results of the N norm calculations,
+computed in the PEs, are then sent to the shared L2 memory
+where it is then post-processed by the RISC-V core with the
+GNU C in-built exponential function, multiplication with α
+and summation over N to generate the final output shown in
+equation (1).
+V. DEPLOYMENT OF NEURAL NETWORKS ON TINYVERS
+Hardware used for ML applications also requires a user
+programmable full stack that can translate ML algorithms
+directly from existing ML training and inference frameworks
+like Tensorflow, Keras, PyTorch, etc. This makes the quick and
+easy deployment of various ML workloads onto an existing
+hardware possible. A python based pseudo-compiler frame-
+work created for TinyVers taking into account its heterogeneity
+is created. An ML algorithm is first quantized to selected
+precision using the QKeras framework [42] for quantized-
+aware training. The quantization-aware training framework
+takes into consideration the hardware constraints such as
+symmetric quantization and the shift based scaling of output
+in the PEs of the accelerator. The quantized model is then
+passed to a python-based NN compilation which takes in the
+hardware description and provides a set of C-based header
+files for the RISC-V core, consisting of ucode instructions for
+the accelerator, NN parameters and also a golden model for
+verification of the mapped workload.
+VI. CHIP IMPLEMENTATION AND MEASUREMENT
+The TinyVers chip microphotograph shown in Fig. 10 was
+implemented and fabricated in GlobalFoundries 22FDXTM.
+The figure shows the different sub-modules used in the SoC
+and detailed in previous sections. Fig. 10 also shows the lab
+setup used for measurements and benchmarking. The follow-
+ing subsections details the measurements and benchmarking
+done on the SoC for power, energy efficiency and performance.
+A. Peak Performance Analysis
+First, a peak performance analysis is undertaken using a
+single CNN layer with 32 input channels, 32 output channels
+and a 3×3 filter kernel. Selection of the used layer for peak
+@Vdd Mem, Vdd Logic
+0.5
+1.0
+1.5
+2.0
+3.0
+2.5
+3
+6
+9
+12
+18
+15
+5
+Clock Frequency (MHz)
+Peak energy eff. (TOPS/W)
+Throughput (GOPS)
+10
+20
+30
+40
+50
+100
+120
+150
+@0.5, 0.4V
+@0.55, 0.5V
+@0.65, 0.5V
+@0.65, 0.6V
+@0.65, 0.6V
+@0.65, 0.6V
+@0.8, 0.8V
+@0.8, 0.8V
+@0.8, 0.8V
+0.586
+2.47
+2.0
+1.9
+1.85
+1.43
+1.44
+0.833
+0.838
+0.863
+1.17
+2.35
+4.69
+3.52
+5.86
+11.7
+14.1
+17.6
+Fig. 11. Peak performance analysis of CNN3×3 layer.
+5 MHz
+10 MHz
+20 MHz
+30 MHz
+40 MHz
+50 MHz
+100 MHz
+120 MHz
+150 MHz
+0
+5,000
+10,000
+15,000
+20,000
+Power(µW)
+WuC
+L2
+L2uDMA
+L1
+Logic
+DMA
+Mram(P)
+Mram(A)
+Fig. 12. Power breakdown of the peak perf. analysis with CNN3×3. MRAM
+power consumption is negligible as it is OFF in active mode. MRAM(A) and
+MRAM(P) represents MRAM array and MRAM periphery resp..
+performance is driven by the fact that convolutional layers
+with a 3×3 filter kernel are the most commonly used layer
+in modern DNN models. The hyperparameter selection of the
+CNN layer is driven by the constraint of maximum utilization
+of the PE array and the size of the private L1 memories
+of the accelerator. The 8 bit quantized activation and non-
+sparse (structured) weights of the CNN are generated using
+the compiler framework using the Google speech dataset for
+keyword spotting [43] and verified against the golden model
+for functional correctness.
+Fig. 11 plots the peak energy efficiency and the throughput
+with respect to the clock frequency while sweeping the voltage
+supply of the logic and memories for the benchmarked CNN
+layer. For fair comparison with other SotA chips, no body
+biasing is applied. Fig. 12 shows the power breakdown of
+individual modules when running the benchmarking layer. The
+SoC shows a large flexibility in delivered performance ranging
+from high energy efficiency/low throughput of 2.5 TOPS/W,
+586 MOPS when operating at a clock frequency of 5 MHz
+with 0.4 V logic, 0.5 V memories, to low energy efficiency
+/ high throughput of 0.8 TOPS/W, 17.6 GOPS operating at
+150 MHz with 0.8 V logic and memories. This provides a
+large range for extreme edge tinyML applications to operate,
+trading-off between speed and energy efficiency.
+B. Workload Benchmarks
+Using the peak energy efficiency operating point (5 MHz,
+0.4 V logic and 0.5 V memory) from Section VI-A, further
+performance analysis of different synthetic and actual real-
+time benchmarks are evaluated. Table I shows the SoCs
+flexibility through mapping of different ML layers and full
+
+MICAS
+naikraveraels
+ADIGILENT
+二
+店
+K2_VDD_L1K2_0D_L
+ZedBoardLens: E20:X80
+2022/02/038
+TABLE I
+WORKLOAD BENCHMARKS
+Workload
+Acc.
+Power
+(µW)
+Peak
+perf.
+(GOPS)
+Peak
+(effective NZ)
+energy eff.
+(TOPS/W)
+Synthetic
+CNN@8b
+-
+237
+0.586
+2.47(2.47)
+CNN@4b
+-
+197
+1.17
+5.94(5.94)
+CNN@2b
+-
+197
+2.35
+11.9(11.9)
+CNN@8b,
+-
+239
+1.03
+4.31(2.46)
+50% sparse
+CNN@8b,
+-
+212
+3.64
+17.1(2.76)
+87.5% sparse
+FC/RNN/SVM,
+-
+140
+0.116
+0.829(0.829)
+batch=16
+Deconv@8b
+-
+235
+1.36
+5.78(2.49)
+Real-time
+TCN (KWS)
+93.3%∗
+193
+0.204
+1.05(1.05)
+CAE
+-
+209
+0.442
+2.11(1.27)
+ResNet-8
+82%+
+228
+0.267
+1.17(1.17)
+OC-SVM
+-
+129
+0.126
+0.972(0.972)
+∗ 12-class task, baseline=93.46%, + baseline=85%
+2%
+ResNet8
+OC-SVM
+CAE
+TCN
+1%
+31%
+2%
+21%
+41%
+4%
+0%
+29%
+2%
+19%
+44%
+6%
+WuC
+L2
+L2uDMA
+L1
+Logic
+DMA
+Mram(P)
+Mram(A)
+0%
+31%
+18%
+44%
+5%
+1%
+47%
+3%
+15%
+25%
+9%
+Power: 129 μW, 
+Latency: 4.3ms
+Power: 228 μW, 
+Latency: 76ms
+Power: 193 μW, 
+Latency: 11ms
+Power: 209 μW, 
+Latency: 30ms
+Fig. 13. Energy breakdown showing the distribution of measured energy of the
+chip modules for running a single inference of the four real-time workloads
+on FlexML and RISC-V with input data already available in L2 memory.
+The power and latency measurements starts from setting up of accelerator
+parameters by RISC-V, data movement from L2 to L1, inference computations,
+and ends with post processing by RISC-V core. MRAM power consumption
+is negligible as it is OFF in active mode.
+workloads. The CNN layer from Section VI-A is extended and
+measured with different precision and blockwise structured
+sparsity (BSS) levels. When moving to lower precision of INT-
+4 and INT-2, the peak throughput improves by 2× and 4×
+while the peak energy efficiency improves by 2.4× and 4.8×
+resp., achieving a maximum of 11.9 TOPS/W at INT-2. As
+shown in Table I, at 8 bit precision with 50% BSS (16/32
+input channels pruned) the performance improves by around
+1.7× while at 87.5% BSS (28/32 input channels pruned)
+the performance increases by approximately 6.9×. Further
+performance improvement can be gained when moving to
+lower precision, however, low precision combined with high
+BSS levels can cause a large drop in accuracy and, thus, is not
+explored in this benchmarking. Other synthetic benchmarks
+such as FC, RNN, SVM and a deconvolutional layer similar
+to the CNN layer in terms of hyperparameters are explored
+and the results are shown in the table. For the dense layers,
+batching of 16 is used.
+Finally, 4 real-time application benchmarks are used to
+TABLE II
+MEASUREMENT RESULTS OF DIFFERENT LOW POWER MODES.
+Power Mode
+AON
+Freq.
+(kHz)
+Core
+Freq.
+(MHz)
+Power
+(µW)
+Wakeup
+Latency
+(µs)
+Deep Sleep
+33
+-
+1.7
+788
+LP Data acq.∗
+33
+5
+23.6
+788
+Data acq.∗
+33
+5
+67
+788
+∗ @Fs=44.1 kHz
+Wake-up latency (s)
+4
+8
+12
+16
+24
+20
+0.033 
+Clock Frequency (MHz)
+Power (μW)
+1 
+5 
+10 
+788 μs  
+26 μs  
+5.2 μs  
+2.6 μs  
+20 
+40 
+1.3 μs  650 ns  
+1.7 μW  2.1 μW  
+5.8 μW  
+7.8 μW  
+12.7 μW  
+22.8 μW  
+Fig. 14. Deep sleep power-latency-frequency tradeoff.
+show the capabilities of the SoC: 1.) keyword spotting (KWS)
+using TCN model [21], [44] on google speech dataset, 2.)
+continuous machine monitoring with a convolutional auto-
+encoder (CAE) [24] on MIMII dataset [45], 3.) ResNet-8
+image classification on CIFAR-10 used in MLPerfTM tiny
+benchmark [46], and 4.) Novelty detection with OC-SVM [47].
+Table I shows the peak performance characteristics of these
+benchmarks on the SoC, more specifically the RISC-V core
+and FlexML, with 8-bit precision, a single inference, and
+assuming all input data is available in the shared L2 memory.
+For TCN and ResNet-8, hardware-aware quantization was used
+and the energy and performance metrics were measured, while
+for the CAE and OC-SVM workloads, random inputs and
+weights were used. All the 4 workloads can be deployed with
+less than 230 µW of continuous real-time power at peak energy
+efficiency between 1-2 TOPS/W. This means that the SoC can
+provide high level of flexibility in workload mapping at sub-
+mW power to enable truly power efficient tinyML application
+on extreme edge devices. Fig. 13 shows the power breakdown
+of the 4 real-time workloads. For OC-SVM (dense operation),
+the power consumption of memory dominates, due to the lack
+of re-usability of weights leading to more data fetches. On the
+other hand, power breakdown of CNN based workloads (TCN,
+ResNet8 and CAE) shows equal distribution between memory
+and logic as the dataflow exploits maximum re-usability.
+C. Power Management
+Table II shows the measured real-time power of the different
+low power modes of the SoC detailed in Section III-B. In deep
+sleep mode the SoC operates with an AON clock frequency
+of 33 kHz. In this mode, only the AON domain consisting of
+the WuC and the logic controlling the IO pads stays powered
+
+9
+1.00E-06
+1.00E-05
+1.00E-04
+1.00E-03
+1.00E-02
+1.00E-01
+Power (W)
+Time (ms)
+0
+280
+140
+Erase followed by write output 
+to MRAM
+TCN processing (16 batch)
+Boot from MRAM
+I2S LP data acq. (2s window)
+Deep sleep
+Vdd SCL, Mem: 0.55 V
+Vdd AON: 0.7 V
+Core Freq. : 5 MHz
+2000 2140 2280
+Fig. 15.
+Instantaneous power trace showing the KWS application scenario
+with one full period of smart sensing and TCN processing followed by idling.
+ON. The resulting deep sleep power measured is 1.7 µW when
+operating at 0.7 V voltage supply. When compared to the peak
+power measured for the CNN layer, the deep sleep power is
+12,000× lower. The measure latency of waking up the SoC
+from deep sleep mode to active mode is 788 µs. This wake-
+up latency can be traded off to deep sleep power by sweeping
+the AON clock frequency. Fig. 14 plots the this trade-off for
+the measured power and wake-up latency when sweeping the
+AON clock frequency. Applications that need low latency can
+operate the AON clock at 40 MHz to attain a wake-up latency
+of 650 ns at a real-time power of 22.8 µW.
+Table II also shows the measured power for the two tinyML
+optimized power modes of data acq. and LP data acq. These
+power modes are measured with an I2S protocol based win-
+dowed test vector collection with the AON clock frequency at
+33 kHz and the core and peripheral clock frequency at 5 MHz.
+The SoC is programmed to collect I2S audio data through its
+uDMA at a sampling frequency of 44.1 kHz and a sampling
+window of 2 second. The sampling clock is generated by the
+SoC using the 5 MHz clock and lasts for the duration of
+sampling window. The data acq. or LP data acq. mode is then
+initiated and power is measured. The measured power for LP
+data acq. and data acq. is 23.6 µW and 67 µW resp. which is
+850× and 300× reduced power consumption compared to the
+peak power, when the core and peripheral frequencies can be
+dynamically lowered to 5 MHz.
+D. Instantaneous Power Trace
+In order to show the complete end-to-end application de-
+ployable on the SoC and to show the SoC’s full ML func-
+tionality, duty cycling and features of power management,
+two applications are mapped onto the heterogeneous SoC
+with windowed data collection done in the LP data acq.
+mode: keyword spotting with a TCN model operating in
+continuous mode [21]; and a machine monitoring use case with
+a Mel Frequency Energy Coefficient (MFEC) based feature
+extraction with a CAE in duty cycled mode [24].
+1) Keyword-spotting Application: The first application sce-
+nario is the keyword-spotting with TCN model. In this ap-
+plication scenario, audio data from a microphone of window
+1.00E-06
+1.00E-05
+1.00E-04
+1.00E-03
+1.00E-02
+1.00E-01
+Power (W)
+MFEC processing on RISC-V
+Autoencooder processing on FlexML
+Boot from MRAM
+I2S LP data acq. (1s window)
+Deep sleep
+Time (ms)
+Vdd SCL, Mem: 0.55 V
+Vdd AON: 0.7 V
+Core Freq. : 5 MHz
+Fig. 16. Instantaneous power trace showing the machine monitoring appli-
+cation scenario with one period of smart sensing, FE, and CAE processing
+followed by idling.
+size 2 seconds (16 batches) at a sampling frequency of 44.1
+kHz is collected using the I2S peripheral interface protocol,
+the collected data is simultaneously stored in the special L2
+uDMA memory using the SoC’s uDMA with the SoC being
+in the LP data acq. mode. After 2 seconds the SoC wake’s
+up into active mode and the collected data is processed using
+the TCN model from Section VI-B. The output of the TCN
+processing is then stored into the MRAM for future processing
+or transmission while the SoC can either go into deep sleep
+mode or collect new windowed sampling data. Fig. 15 shows
+the complete instantaneous power consumption trace of the
+KWS application scenario. When operating in this duty-cycled
+mode, the average power of the complete application is 173
+µW. The power can be further reduced to 10-20 µW by using
+the deep sleep power mode of the SoC during periods of no
+sensing or computation.
+2) Machine Monitoring Application: Machine monitoring
+used for predictive maintenance is the second application
+scenario selected. In this scenario, I2S peripheral interface
+protocol is used to collect audio data from a microphone with
+window size 1 second at a sampling frequency of 16 kHz.
+The collection of I2S audio data is operated in the LP data
+acq. mode of the SoC. Once the complete windowed data is
+collected, the SoC switches to the active mode in which the
+RISC-V core is used for the MFEC based feature extraction
+followed by running the CAE on the accelerator. Fig. 16
+plots the instantaneous power trace of running the machine
+monitoring application. Unlike the previous application which
+works on raw audio data, the CAE model need pre-processing
+MFEC data. As the MFEC algorithm is not supported on
+the accelerator, it is executed on the RISC-V core with
+INT16 precision instead of INT32 or FP32 to reduce power
+consumption [52]. The power trace plots show that running
+large feature extraction on RISC-V is not energy efficient
+taking large time to complete owing to single core operation.
+The average power for continuous operation remains below
+164 µW, but for this use case, 9.5 µW is consumed with a
+duty cycling of 0.05. The MFEC execution on the RISC-V
+
+10
+TABLE III
+PERFORMANCE COMPARISON WITH STATE-OF-THE-ART.
+[48]
+[17]
+[49]
+TinyVers
+[50]
+[51]
+Extreme Edge SoCs
+edgeML Accelerators
+Technology
+28nm FDSOI
+22FDX
+55nm
+22FDX
+28nm
+65nm
+Die Area (mm2)
+4.5
+12
+10
+6.25
+0.55
+16
+Applications
+IoT GP, DNN,
+IoT GP, DNN,
+IoT GP, DNN,
+IoT GP, DNN+,
+Always-on KWS
+DNN
+NSA
+NSA
+NSA
+Trad. ML, NSA
+Supported
+CNN,
+CNN,
+CNN,
+CNN,
+DSCNN
+CNN,
+ML layers
+FC/RNN
+FC/RNN
+FC/RNN
+FC/RNN, GAN,
+FC/RNN
+AE, TCN, SVM
+Architecture
+1×RI5CY+
+10×RI5CY+
+9×RI5CY
+1×RI5CY+
+DSCNN
+DNN
+ML accel.
+ML accel.
+FlexML accel.
+accel.
+accel.
+SRAM
+464 kB (40 kB)
+128 kB(L1)
+64 kB (L1)
+132 kB (L1)
+2 kB
+256 kB
+(State retentive)
+(16-1600 kB (L2))
+(512 kB (L2))
+(64/512 kB (L2))
+eNVM
+-
+4 MB MRAM
+-
+512 kB MRAM
+-
+-
+Deep sleep power (µW)
+-
+1.7
+3.6
+1.7
+-
+-
+SRAM ret.
+6.4
+2.8-123.7
+30
+23.6-67
+-
+-
+sleep power (µW)
+Int precision (bits)
+8, 16, 32
+8, 16, 32
+8, 16, 32
+2, 4, 8
+8
+1-16
+Supply voltage (V)
+0.45-0.9
+0.5-0.8
+1-1.2
+0.4-0.9
+0.41
+0.63-1.1
+Max frequency (MHz)
+350
+450
+250
+150
+0.04
+200
+Power range
+6.4µW-96mW
+1.7µW-49.4mW
+3.6µW-75mW
+1.7µW-20mW
+0.51µW
+3.2-297mW
+Best ML perf.
+36 GOPS
+32.2 GOPS
+12 GOPS
+17.6 GOPS
+2.3 MOPS+
+691.2 GOPS
+@8b∗
+@8b∗
+@8b∗
+@8b∗∗
+@8b∗∗
+@8b ∗∗
+Best ML eff.
+1.3 TOPS/W@
+1.3 TOPS/W@
+200 GOPS/W@
+2.47 TOPS/W@
+4.5 TOPS/W@
+5.57 TOPS/W,
+@Perf
+2.8 GOPS, 8b∗
+15.6 GOPS, 8b∗
+7 GOPS, 8b∗
+0.58 GOPS, 8b∗∗
+2.3 MOPS, 8b∗∗
+8b∗∗
+11.9 TOPS/W@
+11.6 TOPS/W,
+2.4 GOPS, 2b∗∗
+4b∗∗
++ estimated at 90% utilization of MACs, ∗ Matmul, ∗∗ CNN, 1 MAC = 2 Ops
+can be optimized using special DSP extensions available with
+the PULP libraries, which is left for future work.
+VII. COMPARISON WITH SOTA
+Table III shows the comparison of our SoC with SoTA on
+two fronts: on one hand, comparing with existing extreme edge
+SoCs (left), and on the other hand, with edge ML accelera-
+tors (right). Our SoC has similar or increased flexibility in
+application mapping compared to the extreme edge SoCs on
+the left, with much improved energy efficiency and power.
+TinyVers supports not only the IoT general processing (GP),
+DNNs and near-sensor analytics (NSA) like [17], [48], [49],
+but also DNN+ such as TCN and AE and traditional ML
+like SVM, all at better energy efficiency because of efficient
+mapping. This is evident from the best energy efficiency of
+2.47 TOPS/W for running a CNN layer on TinyVers. The
+energy efficiency is further enhanced to 11.9 TOPS/W when
+the CNN workload is quantized to INT2. Compared to the
+extreme edge SoCs, TinyVers provides support for precision
+scalability and, thus, can take advantage of improved perfor-
+mance using quantization. Furthermore, by utilizing support
+for block structured sparsity, TinyVers can reach a peak
+performance of 17 TOPS/W for an 8-bit CNN layer. This is
+much higher than the efficiencies reported by [17], [48], [49].
+Compared to the edge ML accelerators on the right,
+TinyVers shows much more flexibility at comparable per-
+formance metrics in terms of energy efficiency and power
+consumption. The edgeML accelerators only support a single
+or few models extremely efficiently, but this approach has
+drawbacks in deployment for extreme edge devices. For ex-
+ample, [50] can only perform KWS with depthwise separable
+CNN and its performance is much lower than TinyVers with
+comparable energy efficiency. UNPU [51], can only support
+CNN and FC/RNN layers and also does not have a complete
+standalone SoC, which effects efficiency at the system level.
+Moreover, these edgeML accelerators cannot support any kind
+of duty cycling as they lack power management and retention
+memory support. TinyVers supports the multi-modal require-
+ments of extreme edge devices at relatively similar energy
+efficiencies of the order of TOPS/W. Moreover, it adds the
+possibility of extreme low power idle states for duty-cycling
+use cases to enable < 10µW operation, shown empirically in
+Section VI-D. To summarize, TinyVers brings the best of both
+worlds of extreme edge processors and edgeML accelerators.
+VIII. CONCLUSION
+TinyML applications at the extreme edge needs not only
+heterogeneous SoCs with flexible accelerators to support di-
+verse workloads, but also adaptive power management for
+different duty-cycling operations. Moreover, to enable such
+adaptive power management, the need for embedded non-
+volatile memories arises. TinyVers extends a RISC-V core
+with a flexible ML accelerator supporting a diverse set of
+ML workload mapping in terms of diverse compute kernels,
+different precision and structured sparsity conditions. Further-
+more, the inclusion of a WuC and an eMRAM enables the
+adaptive power management required in many duty-cycling
+use cases. Measurement result shows that the chip can achieve
+an energy efficiency range of 0.8-17 TOPS/W at 0.58 GOPS
+to 17.6 GOPS of throughput. The different low power modes
+enable the chip to achieve power range from 1.7µW-20 mW.
+The application of machine monitoring takes advantage of the
+
+11
+deep sleep mode to consume only 9.5µW of power at a duty
+cycle of 0.05. Thus, TinyVers takes a step towards creating a
+new class of ultra-low power extreme edge SoCs.
+ACKNOWLEDGMENTS
+The authors would like to thank ETHZ for their support
+on PULP platform and GlobalFoundries for 22FDX tapeout
+support. The work has been supported under ISAAC project
+(FOD Economie Belgium Energietransitiefonds (oproep II)) in
+collaboration with Magics Technologies and received funding
+from the Flemish Government (AI Research Program).
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf,len=1206
+page_content='1  TinyVers: A Tiny Versatile System-on-chip with  State-Retentive eMRAM for ML Inference at the  Extreme Edge  Vikram Jain, Sebastian Giraldo, Jaro De Roose, Linyan Mei, Bert Boons, and Marian Verhelst  Abstract—Extreme edge devices or Internet-of-thing nodes  require both ultra-low power always-on processing as well as  the ability to do on-demand sampling and processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Moreover,  support for IoT applications like voice recognition, machine  monitoring, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=', requires the ability to execute a wide range  of ML workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' This brings challenges in hardware design to  build flexible processors operating in ultra-low power regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  This paper presents TinyVers, a tiny versatile ultra-low power  ML system-on-chip to enable enhanced intelligence at the Ex-  treme Edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' TinyVers exploits dataflow reconfiguration to enable  multi-modal support and aggressive on-chip power management  for duty-cycling to enable smart sensing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The SoC  combines a RISC-V host processor, a 17 TOPS/W dataflow  reconfigurable ML accelerator, a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='7 µW deep sleep wake-up  controller, and an eMRAM for boot code and ML parameter  retention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The SoC can perform up to 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='6 GOPS while achieving  a power consumption range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='7 µW-20 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Multiple ML  workloads aimed for diverse applications are mapped on the SoC  to showcase its flexibility and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' All the models achieve  1-2 TOPS/W of energy efficiency with power consumption below  230 µW in continuous operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' In a duty-cycling use case for  machine monitoring, this power is reduced to below 10 µW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Index Terms—Extreme edge, tinyML, machine learning accel-  erators, ultra-low power, system-on-chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' INTRODUCTION  E  Xtreme edge devices [1] or Internet-of-Things (IoT)  nodes mostly perform non-vision tasks and can achieve  good accuracy, even with small and lightweight neural network  (NN) models [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' This is in contrast to more traditional tasks  designed for processing image data and contain millions to  billions of parameters and operations with high hardware re-  source demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Consider the Google voice assistant as an ex-  ample, which needs only 14 kilo bytes (kB) of NN parameters  to run a keyword-spotting application on edge devices [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The  insight that not all applications require maximum accuracy,  large and complex NN models, has resulted in a new paradigm  of ML application development, called tinyML or ML at the  extreme edge [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' This trend, at its core, has been driven by the  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Jain, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Mei, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Verhelst are with the Department of Electrical  Engineering - MICAS, KU Leuven, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Giraldo was with the Department of Electrical Engineering - MICAS,  KU Leuven, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' He is now with B12 Consulting, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' De Roose and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Boons were with the Department of Electrical  Engineering - MICAS, KU Leuven, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' They are now with Magics  Technologies, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  © 2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Permission from  IEEE must be obtained for all other uses, in any current or future media,  including reprinting/republishing this material for advertising or promotional  purposes, creating new collective works, for resale or redistribution to servers  or lists, or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  requirements imposed by battery-operated, performance- and  power-constrained IoT nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Most IoT sensor nodes consist  of a microcontroller unit (MCU) with a subset of sensors, a  memory for storing acquired data, a CPU and a wireless data  transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The presence of these MCUs for data collection  provides opportunities to process data very close to the sensor  when the NN model is small, and avoids the high penalty of  raw data transmission to more powerful edge or cloud units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Yet, this local ML processing, brings several new chal-  lenges: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=') As these nodes are battery-operated, the system is  typically severely power or energy constrained requiring ultra-  low power operation, with the ability to idle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=') the MCU,  moreover, has limited compute power and memory space,  resulting in a critical trade-off between model size, execution  performance and hardware complexity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=') despite the need  for efficiency, the system should also be flexible enough  to support different classes of NN models across different  applications, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=') it should have a small footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Several  hardware for ML have been proposed in the recent literature  and can be divided into three main categories: 1) extremely  specialized edgeML accelerators designed for ultra-low power  operation with little to no flexibility at low performance [5]–  [8],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 2) multi-modal edgeML accelerators providing medium  level of flexibility with high performance at medium to high  power consumption [9]–[13],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 3) commercial-off-the-shelf  (COTS) MCUs delivering higher flexibility but at low perfor-  mance and medium power consumption [14]–[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Most of  these hardware designs do not meet all the requirements of an  extreme edge device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' An exception is Vega [17] which presents  a complete SoC, however, the specialized accelerator of Vega  does not have the flexibility to handle all DNN workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Thus, a new class of flexible ultra-low power (ULP) platforms  towards extreme edge deployment is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  In this context, this work presents TinyVers [18], a highly  adaptive SoC platform which significantly enhances the trade-  off between energy efficiency and flexibility needed in extreme  edge devices, through the use of: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=') a RISC-V proces-  sor extended with a flexible ML accelerator (FlexML) with  dataflow reconfiguration supporting diverse ML workloads and  support for efficient zero-skipping in block structured sparsity  and deconvolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=') an embedded magnetoresistive random  access memory (eMRAM) for non-volatile storage enabling  standalone operation with efficient power-down (or idling);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=') a programmable wake-up controller (WuC) supporting  different power-on and idle modes to enable both always-on  inference as well as on-demand and duty-cycled smart sensing  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='03537v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='AR]  9 Jan 2023  2  and computation used in typical tinyML IoT applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The  SoC provides users flexibility not only in mapping diverse  ML workloads for diverse tinyML applications, but also in  supporting various use cases such as duty-cycling and smart  sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' We demonstrate TinyVers’ capabilities and improve-  ments over state-of-the-art (SotA) on diverse applications in  machine monitoring, anomaly detection, audio signal analysis,  and image classification through the use of both deep learning  as well as traditional ML workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The basics of  ML compute kernels is introduced in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Section III  discusses the architecture overview of TinyVers, followed by  Section IV providing further details of the FlexML accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Section V provides details on how the software stack for  ML deployment on TinyVers is undertaken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Subsequently,  Section VI presents the experimental results of mapping  different workloads and application use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Finally, Sec-  tion VII compares TinyVers’ performance with related works  and Section VIII concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' ALGORITHMIC BACKGROUND  ML applications heavily exploit deep neural networks  (DNN) with traditional convolutional (CNN) and fully con-  nected (FC) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' However, a plethora of new NN layer  topologies are emerging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Some examples of these are the use  of temporal convolutional networks (TCN) used in audio tasks  like keyword spotting [19]–[21], or auto-encoders (AE) using  convolution and deconvolution pairs in machine monitoring  and anomaly detection tasks [22]–[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Morever, also machine  learning models not relying on neural network layers are still  used in extreme edge IoT nodes, such as support vector ma-  chines (SVM) [25] used in novelty and anomaly detection ap-  plications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The execution efficiency of all these workloads can  can be improved with orders of magnitude when deployed on  specialized accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Yet, the wide variety in the compute  kernels of interest complicates their efficient mapping on a  single hardware platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The following subsections deal with  the different ML operation characteristics, their categorization  into mathematical operations, and their hardware implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Convolution and Dense Operation  Convolutional and dense layers are the most common com-  pute kernels used in DNNs and they can be decomposed  into matrix-matrix multiplication (MMM) and matrix-vector  multiplication (MVM) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='. These two matrix operations can  be represented mathematically as nested for loops as shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Most ML compute kernels can be categorized into one  of these two mathematical operations, with some special layers  requiring extra hardware changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' One such kernel is the TCN  layer which can be represented as a 1D CNN and requires extra  support for programmable dilation which is similar to strides  in a convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Recurrent neural networks (RNN) like long  short-term memory (LSTM) and gated recurrent unit (GRU)  can be decomposed to MVM with need for extra hardware  for activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' These hardware changes would be  discussed further in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  C  C  IY  FY  OY  OX  IX  FX  K K  Convolution Operation = MMM  Input FMAP  Weights  Output FMAP  C  C K  Dense Operation = MVM  Input FMAP  Weights  Output FMAP  TCN  CNN  GAN  AE  LSTM  FC  SVM  K  for(y=0 to Y-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' for each output row  for(x=0 to X/N-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' for each output column  for(k=0 to K/N-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' for each output channel  for(c=0 to C-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' for each input channel  for(fy=0 to Fy-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' for each filter row  for(fx=0 to Fx-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' for each filter column  o[k][x][y] += i[c][x+fx][y+fy]*w[k][c][fx][fy]  for(k=0 to K/N-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' for each output channel  for(c=0 to C/N-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' for each input channel  o[k] += i[c]*w[k][c]  PE  Spatial Unrolling X  Temporal  Unrolling  Spatial Unrolling Y  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Different ML models and their mathematical representation in terms  of MMM and MVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The nested for loop representation can be mapped onto  specialized accelerators through spatial and temporal unrolling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  When mapping MMMs and MVMs on specialized hardware  accelerators, the nested for loops can be unrolled spatially  and temporally, which is called dataflow in literature [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  On a 2D processing element (PE) array, two for loops can  be spatially unrolled, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=', the loops can be parallelized along  the X and Y dimensions, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' In the rest  of the paper, this spatial unrolling is represented as (Spatial  Unrolling X)|(Spatial Unrolling Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The remaining for loops  are temporally unrolled, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=', sequential execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Depending  on the available parallelism and available re-usability, the  spatial unrolling (X and Y) needs to be configurable, to be  able to efficiently map all workloads, detailed in Section IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Deconvolution  Autoencoders used in many machine monitoring applica-  tions consist of an encoder and a decoder pair, which tries  to reconstruct the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' After training on normal data,  a reconstruction error signals an anomaly in the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Deconvolution or transposed convolution are used in these  autoencoders and are built by combining the convolution and  upsampling into a single operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Deconvolution can be  mapped as a convolution (MMM) but needs extra hardware  to support zero-skipping of input for efficient mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Hard-  ware modification can improve the mapping efficiency of this  operation, and better exploit its inherent sparsity, as will be  discussed in Section IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Support Vector Machines (SVMs)  SVMs are ML algorithms used for classification and re-  gression tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' When classification of input data between  normal behavior and an anomaly is required, a binary classifier  called a one-class support vector machine (OC-SVM) can be  3  used [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The decision function of a OC-SVM using the  radial basis function (RBF) kernel is given by the equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  For the Laplacian kernel, the L2 norm is replaced by L1 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  f(x) =  N  �  i=0  αi · exp  −∥x−svi∥2  2σ2  − b  (1)  where x is the input vector with length D, sv are the support  vectors with length D, N is the number of support vectors,  σ the standard deviation, α the Lagrange multiplier, and b the  bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The number of support vectors N, in combination with  the vector length D, can become large in these workloads,  making the L1 and L2 norm calculation complex, and their  deployment can gain orders of magnitude in performance  when deployed on specialized accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The D and N  dimensions of the norm operations can be treated similar to  C and K dimensions of a dense layer (MVM) and can be  spatially unrolled on the PE array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' In addition to unrolling  the norms, extra hardware to support squaring, subtraction,  rounding and absolute operation needs to be added to each  PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The result of the norm calculation can then be used by a  CPU core to compute the overall kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Structured Sparsity  Exploiting sparsity in DNNs can help to reduce the com-  putational complexity and memory requirements, by skipping  zeros and compressing the NN parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' However, random  pruning or unstructured sparsity tends to be hard to efficiently  map on hardware and requires special logic for zero-skipping  and load balancing [29]–[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The structure of sparsity (gran-  ularity of pruning) has high impact on hardware efficiency and  prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Some works have found that unstructured  sparsity achieves better prediction accuracy than structured  sparsity but structured sparsity tends to be more hardware  amenable and improves computational efficiency [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Thus, a  structured sparse model could be trained with more iterations  to revert back closer to the same prediction accuracy achieving  similar overall efficiency/cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Moreover, more coarse-grained  sparsity can reduce the additional memory requirements im-  posed for storing indices of non-sparse data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  With all of these diverse ML workloads and their charac-  teristics in mind, a platform which can efficiently map all of  the above, needs to be designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' TINYVERS HARDWARE ARCHITECTURE  TinyVers, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 2, is a heterogeneous SoC  consisting of a single core RISC-V processor, a flexible ML  accelerator called FlexML, a 512 kB shared level-2 (L2)  SRAM memory, a micro-DMA (uDMA) for data movement  between peripherals/memory, a 512 kB eMRAM for non-  volatile storage, and a WuC for power management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The SoC  development is rooted in the PULPissimo platform [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' It  embeds a 2 kB read-only memory (ROM), which acts as the  first stage boot loader (FSBL) and also controls boot from  JTAG, external SPI flash or the eMRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Two communication  busses are used: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=') a logarithmic interconnect, which enables  a tightly-coupled data memory (TCDM) providing single cycle  eMRAM  (512 KB)  ROM  Shared Memory L2 (512 kB)  GPIO UART  SPI  I2C  I2S  CPI  JTAG  SCAN  CHAINS  eMRAM  CNTL  LP Data acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Memory L2  (64 kB)  TCDM interconnect  uDMA  DMA  Source  Source  Sink  RISC-V  APB  WuC (RTC  &  Power FSM)  2D SIMD  Array  8x8  Weight L1  Memory  Instruction  Memory  Activation L1  Memory  DMA  Control  Registers  Logic PD  LP Data  Acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Mem  Data Acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Mem PD  L1 PD  UDMA  PD  AON PD  MRAM  PD  Power  Modes  PD= Power Domain  Boot  OFF  OFF  OFF  OFF  OFF  OFF  OFF  OFF  OFF  OFF  OFF  OFF  ON/OFF  OFF  ON  ON  ON  ON  ON  ON  ON  ON  ON  ON  ON  ON  ON  ON  ON  ON  ON  ON  ON  ON  ON  Active  Data Acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  LP Data Acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Deep Sleep  % VDD WAKE  VDD SCL  ** VDD SRAM  ^^ VDD MRAM  # VCS MRAM  V+ bias  V- bias  ^^  #  %  %  Data  Mover  FSM  FlexML Accelerator **  **  **  **  **  FlexML  Control  Unit  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Overview of the complete TinyVers SoC showing the different power  domains (PD) with their constituting modules and the power modes supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  access to the shared L2, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=') the APB standard bus, which  is used for controlling different memory mapped modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  The interface between the SoC and FlexML accelerator is  based on the HWPE framework presented in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Using the  streamers from [33], data is moved to-and-from the shared L2  memory with the help of FlexML’s DMA engine which is a  FSM controlling the data (un)loading of its private memories  and double buffering operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Several peripheral interface  protocols are supported by the SoC including UART, SPI, I2C,  I2S, and CPI, in addition to having 32 general purpose IOs  (GPIO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Separate clocks are used for the main core logic,  the peripheral interfaces, and the always-on domain which  includes the WuC and the IO pads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Smart Sensing Modes for TinyML  IoT tinyML applications typically operate by collecting data  across a specified time window through an array of sensors,  after which the collected data can be processed to make  decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' In many applications, the time window across which  the data needs to be collected before processing can start,  can vary from a few ms to sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Moreover, during the sensor  data collection, many modules of the MCU are not used since  no heavy processing is done yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' This brings opportunities in  improving power saving in many tinyML applications: during  data collection, only the modules necessary for moving the  windowed data from the sensor peripheral interfaces to the  memory need to remain active, while e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' the CPU can be  put to sleep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Furthermore, in applications which work on  time series data like audio, the memory requirement for the  windowed data is small (< 64 kB), such that also a large part  of the main memory of the MCU can be powered-down to  avoid leakage power of the unused memory section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  To this end, TinyVers introduces two tinyML optimized  data acquisition power modes: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=') ‘Data acq.’ and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=') ‘LP  data acq.’, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The data acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' mode, targeted  towards applications with large sample data like vision, keeps  the uDMA module and the complete shared L2 memory (512  4  Full Active  Data Acq  LP Data Acq  0  100  200  300  31  20  8  325  77  10  356  97  18  Power(µW)  Dynamic  Leakage  Total  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Power simulation of post-synthesis netlist undertaken in Cadence  Genus tool for the three power modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' In all the three modes, I2S data is  collected at a sampling frequency of 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='1 kHz for a window of 2 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Full active power reported includes configuration of uDMA by RISC-V core  and interrupt handling procedure, in addition to data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Power  uDMA  Power  OFF  Power  ON  Switch  Power 1  Switch  Power 2  Power  OFF  Reset  Isolate  Clk  enable  Power  ON  Top level FSM  Bottom level FSM  Power  Logic &  L1  Power  MRAM  Power  L2 &  L2 udma  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Flow diagram showing the hierarchical FSM used in the WuC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  kB) powered up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' In contrast to that, the LP data acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' mode  only keeps part of the shared L2 memory (64 kB) powered up,  in addition to the uDMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' This mode is specifically targeted  towards applications which needs time series and audio data  like keyword spotting, machine monitoring, biosignal analysis,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 3 shows an estimation of the power saving that can  be achieved when moving from a full active mode to the  two tinyML sensing modes, with almost 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='5× improvement  between the full active and data acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' modes and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='5× between  data acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' and LP data acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Power Management  Aggressive power management is pursued in TinyVers on  top of standard low power design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The SoC is divided into 6  switchable power domains and 1 always-on domain (AON),  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Each switchable power domain consists  of multiple power gating switches, which isolate the VDD  of the power domain from the global VDD supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' These  power gating switches are controlled by control signals driven  from the WuC of the AON domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' All interconnect crossings  between the power domains are equipped with bidirectional  level shifters and isolation cells, such that the individual supply  voltages of the domains can be controlled independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  The smart WuC is in charge of this power management  control, relying on a real-time counter (RTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The counter  can be programmed by the RISC-V core with millisecond  granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The RISC-V core can instruct the WuC to bring  the SoC into one of the five supported power modes shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' To this end, the WuC encompasses hierarchical finite-  state machines (FSM) driven by the RTC, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 4,  controlling the power-up and power-down of the complete SoC  and the different power domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The top level FSM controls  the sequence of power-up/down of the different power domains  and the bottom level FSMs control the fine-grain sequence to  (de)activate the isolation cells and the power gating switches  of the individual power domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Emerging memories like ReRAM, MRAM, FeRAM, PCM,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' [34], [35], have shown promise in building cost-effective  embedded non-volatile memories (NVM) targeting applica-  tions in edge computing for automotive or industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' NVM  memories can be used as the storage space for boot code  and other parameters that need to be stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' This enables  two things: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=') Duty-cycling can be used as a means of  reducing power consumption in applications which do not  require always-on operation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=') the SoC does not need  to go to a central cloud server in order to fetch its boot codes  and NN parameters when it is power-cycled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Moreover, the  availability of the NVM embedded on-chip, avoids the high  energy cost of fetching data from off-chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  MRAM promoted as a universal memory, uses magnetic  polarity to store data in its bitcells [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Being non-volatile  and almost as dense as traditional SRAM, they are a good fit  for tinyML applications using extreme edge SoCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' With this  in mind, TinyVers integrates a 512 kB embedded MRAM on-  chip, enabling extreme power management strategies for smart  sensing and on-demand computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' In the SoC, the eMRAM  acts as a non-volatile storage for the boot code that the RISC-  V needs to wake-up and start processing, and can also store the  NN parameters of the mapped ML workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The eMRAM  can, finally, also be used as a non-volatile scratchpad space  for storing windowed data in smart sensing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The  interface between eMRAM and the shared L2 memory uses  the uDMA unit and the design is based on the work of [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' FLEXML ACCELERATOR  This section firstly describes the architecture overview of the  FlexML accelerator, followed by the dataflow reconfiguration  used for flexible mapping, efficient zero-skipping used for  deconvolution and structured sparsity, and finally the hardware  for supporting SVM, as briefly discussed in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' FlexML Architecture Overview  The FlexML accelerator is TinyVers’ specialized, versatile  hardware accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' FlexML is designed to efficiently support  the large diversity in ML workloads for tinyML applications,  while exploiting the data reuse present in individual layer  characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' This is achieved through a zero-latency runtime  dataflow reconfiguration, discussed in Section IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' As shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 5, FlexML encompasses an 8×8 single instruction  multiple data (SIMD) array of processing elements (PE),  wherein each processing element consists of a precision-  scalable multiply-accumulate (MAC) unit with support for INT  8/4/2 [37], shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' As a result of the precision-  scalability, the SIMD array can be reconfigured to be a  8×8/16/32 array of INT8/4/2 MAC units, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='. Each PE per-  forms 1/2/4 MAC operations per cycle based on the selected  precision (INT8/4/2) and the results are accumulated in a 32-  bit register with full/partial output stationarity, reducing the  movement cost of the large bit-width partial sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The final  output is passed through a ReLU function (if enabled), fol-  lowed by re-quantization to the selected precision and written  5  DMA Engine  Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Mem  Cntl  FSM  Sparsity  Mem  2x2 kB  Weight  Mem (L1)  2x32 kB  Adder Trees  Act  Mem (L1)  2x32 kB  NLFG  &  Max  Pool  Input FIFO (L0)  SIMD PE Array 8x8  IX  Layer Type  ucode  Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  K  Fx  Fy  Input  pointer  Weight  Pointer  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  IY  C  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' FlexML accelerator architecture overview with ucode instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' >>  ABS  REG  REG  Round  Sub  ReLU  Overflow  Control  Input  Activation  Output  Activation  1  0  0  0  Input  Weight  From Neighbor  PE +  8b  20b  8b  16b  …  … …  …  … …  unused  4b  4b  Gated  4b  4b  Gated  12b  9b  2b 2b 2b  …  … …  …  … …  Gated  2b  Gated  2b  2b  2b  2b  8b  6b  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Block diagram of the processing elements used in the flexML  accelerator, showing the precision-scalable MAC unit and the additional  hardware to support SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  back to the activation L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Mixed precision quantization can  help in improving performance of DNN models when moving  below 8 bits precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' However, the hardware overhead of  mixed precision can reduce the overall efficiency of PEs  due to varying bandwidth and serialized dataflow [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Thus,  FlexML only supports symmetric precision for its weights and  activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' In addition, a simple shift and ReLU is used for  normalization of output, which also keeps hardware overhead  low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' In order to maintain accuracy of the models, a hardware  aware training framework, mentioned in Section V, is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Supporting the SIMD PE array, are private level-1 (L1)  SRAM based memories for storing both weights (64 kB)  and activations (64 kB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Both the weight L1 and activation  L1 are composed of two 32 kB banks operating in a ping-  pong manner to overlap data writing and reading, improving  the overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' An intermediate memory level L0 is  provided between the activation L1 and the PE array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' This  L0 memory is a FIFO buffer of size 16×8 bits, used to  improve data locality when doing shifting window operation  in convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Furthermore, a separate non-linear function  generator (NLFG) and a max pooling unit are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The  for(y=0 to Y-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' for each output row  for(x=0 to X/8-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' for each output column  for(k=0 to K/8-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' for each output channel  for(c=0 to C-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' for each input channel  for(fy=0 to Fy-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' for each filter row  for(fx=0 to Fx-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' for each filter column  parfor(k=0 to 8-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' spatial unrolled output channel  parfor(x=0 to 8-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' spatial unrolled output column  o[k][x][y] += i[c][x+fx][y+fy]*w[k][c][fx][fy]  for(k=0 to K/8-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' for each output channel  for(c=0 to C/8-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' for each input channel  parfor(k=0 to 8-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' spatial unrolled output channel  parfor(c=0 to 8-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' spatial unrolled input channel  o[k] += i[c]*w[k][c]  FIFO  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  C  Weight Memory  FIFO  MMM  MVM  Weight Memory  OX  K  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  PE  K  Bank 0  Bank 1  Bank 3  Bank 0  Bank 1  Bank 3  Bank 7  Bank 7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Diagram showing the dataflow reconfiguration used to switch from  OX|K dataflow (left) for MMM to C|K dataflow for MVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The nested for  loops below show the addition of parfor loops for the spatial unrolling used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  NLFG uses LUT-based linear approximation to generate the  various activation functions (other than ReLU) used in NN  models such as tanh, sigmoid, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' To control the dataflow  and control flow inside the accelerator, a control unit with  FSMs fetches ucode instructions from the instruction memory,  decodes the instruction and deploys the relevant layer on the  PE array by updating the control signals and counters that  track the workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The ucode instructions are generated by a  pseudo-compiler built in python (Section V), and consists of  CISC-like layerwise long instructions with hyperparameters  and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The control unit is also extended to  enable support for efficient zero-skipping of activations in the  case of deconvolution and zero-skipping of pruned weights in  conjunction with the sparsity index memories (Section IV-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Dataflow Reconfiguration  In order to efficiently map the diverse set of ML workloads,  runtime dataflow reconfiguration is supported in the FlexML  accelerator at no latency overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The configurability enables  efficient mapping of both: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=') MMMs used for CNN, decon-  volution and TCN, exploiting both input and weight spatial  data reuse under an OX|K dataflow with output stationarity,  and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=') MVMs used for FC, RNNs and norm calculation  of SVMs with batch size 1, exploiting the available input  spatial data reuse under a C|K dataflow with partial output  stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Multiple previous works have proposed dataflow  reconfiguration in hardware to optimally map different work-  loads [39]–[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' However, these works suffer from large hard-  ware overhead and latency for diverse dataflow support and  are not suitable for extreme edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' This work limits the  dataflows to two optimal mapping schemes, thereby, keeping  hardware and power overhead low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Moreover, none of the prior  works have looked into mapping of TCN, AE, and SVM on  the same hardware accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 7 shows the OX|K (left)  and C|K dataflow (right) and their hardware implementation,  resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='. In the OX|K dataflow, the spatial unrolling is applied to  the OX and the K dimension of the nested for loop, allocating  the unrolled OX dimension along the columns and the unrolled  K along the rows of the SIMD PE array of dimension 8×8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  6  Input FIFO  Input FIFO  Input data from activation L1  Normal operation  Deconvolution operation  Input data from activation L1  0  0  0  0  0  0  0  0  Ctrl  Control Unit  Fetch instruction and enable  deconvolution  Control the demux and muxes  for deconvolution  Skip rows and columns with all zeros  Cycle #0: Enable demux to push data to input FIFO  Cycle #1: Set Ctrl to 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' data sent to PEs -> a 0 b 0 c 0 d 0  Cycle #2: Set Ctrl to 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' data sent to PEs -> 0 b 0 c 0 d 0 e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Shift 0 into input FIFO  Cycle #3: Set Ctrl to 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' data sent to PEs -> b 0 c 0 d 0 e 0  IX  Input Activation  Filter  Deconvolution layer in software  Orange represents pruned pixels  IY  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Representation of deconvolution layer in software (top left), control  unit running the zero-skip operation (bottom left), the architectural change  required on the L0 FIFO to support deconvolution (top right), and cycle by  cycle operation of the FIFO and PEs (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  The rest of the for loops are temporally unrolled as shown  in the nested for loops in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 7, resulting in an output  stationary dataflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Under this dataflow regime, the activation  L1 memory multicasts input activation data in the vertical  dimension to the L0 FIFO memory, which fetches 8 words  in the first cycle followed by single word during the shifting  window operation, thereby, reducing the memory bandwidth  and number of memory fetches by utilizing the reuse oppor-  tunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The weight L1 memory provides data in the horizontal  dimension, providing 8 words using 2 internal banks, where  each word is multi-cast along the row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Due to the output  stationarity, accumulation continues till the final output is  generated which are then systolically shifted out vertically to  the activation L1, requiring 8 cycles to complete the output  write-back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The input data shifting inside the L0 FIFO is made  programmable to support the variable dilation used in TCNs  or variable strides in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  The alternative C|K dataflow is used for MVM, as this  workload cannot utilize the OX|K dataflow efficiently due  to lack of re-usability of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Under this dataflow, the  C dimension is spatially unrolled along the vertical column  dimension and the K dimension along the horizontal row  dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The activation L1 memory multicasts 8 words of  input activation along the vertical dimension, bypassing the  L0 FIFO memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' With a batch size of 1, no weight reuse  is available and, thus, each PE needs a new weight every  cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' In order to meet this requirement, the weight memory  utilizes all of its 8 banks to unicast 64 different weight words  to the PEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' PE rows operate on different input channels (C) of  the same output channel (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Hence, once the required MAC  operations per PE are done, the outputs of PEs of the same  row are accumulated using an adder tree and one final output  per row is shifted out to the activation memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Efficient Zero-skipping for Deconvolution and Blockwise  Structured Sparsity  The FlexML accelerator supports efficient zero-skipping of  deconvolution workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 8, the input FIFO  Sparsity Index Mem  storing the sparse nature  Sparse  block  1  0101  0000  0000  3  C  0  1010  2  C  C  K K  x8  K  8  8  Blockwise Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Sparsity for CNN  Input  Weight Matrix  Weights  Blockwise Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Sparsity for FC/RNN  Output  Control Unit  Fetch sparsity index for block 1 to 8 from sparsity memory  Check bit wise, if one present then update counters of  control FSM to skip current C  Fetch next blockwise index and repeat  Orange represents pruned pixels  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Blockwise structured sparsity applied to CNN and dense layers (top),  control unit operation in tandem with sparsity index memory to support zero-  skipping (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  is designed such that when in deconvolution mode, it only  fetches one set of words and shuffles it with zero padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The  control unit skips the rows and columns with zeros that would  result in redundant computation, resulting in a performance  gain of up to 2× compared to running deconvolution in  convolution mode with upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  TinyVers also supports structured sparsity, more specifically,  blockwise kernel-level sparsity (2D) for both convolutional  and dense layers [29], [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' In this scheme, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 9,  complete input channels of the filter kernels are pruned with a  constraint that a block size of 8 filter kernels (K = 8) should  share the same pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The block size is decided by the  dimension of the PE array and the spatial unrolling of K along  the horizontal dimension of the 2D PE array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' In our case, the  selected block size makes controlling the dataflow and control  flow easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Applying the same channel pruning to all the 8  filter kernels mapped in parallel on the PE array makes the  mapping efficiency higher as all the rows can still operate with  a common control logic, and enables not only energy savings,  but also throughput benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' For this, the FlexML accelerator  consists of specialized sparsity index memories which store the  bit encoded indices of the pruned channel groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 9 shows  the sparsity index memory and the control flow logic used in  the control unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Before every filter kernel block increment, the  control unit fetches an index memory word and checks the data  bit-by-bit for sparsity state, as the input channels increment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  If a sparse channel is detected, the complete computation of  the channel is skipped, thus, avoiding any zero computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Support Vector Machine  The L1 and L2 norm of OC-SVM requires modification  of the PEs in order to use the same hardware for mapping  the workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' As shown in Fig, 6, each PE is extended with  a subtraction block, absolute unit, rounding unit, and the  modification of the multiplier to also enable squaring for the  norm calculation within the PE array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The input data vector x  and the support vector svi are of dimension D and the number  of support vectors is N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' When used in the C|K dataflow, the  7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='00E-01  L1 MEMORY  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='5 mm  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='5 mm  L2  MEM  RISC-V  &  ACCEL  uDMA  eMRAM  WuC  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Measurement setup and chip microphotograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  D dimension of the input data vector of x is unrolled and  multicasted vertically (C) along the PE array, while the N  dimension of the support vector svi are unrolled and unicasted  horizontally (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The results of the N norm calculations,  computed in the PEs, are then sent to the shared L2 memory  where it is then post-processed by the RISC-V core with the  GNU C in-built exponential function, multiplication with α  and summation over N to generate the final output shown in  equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' DEPLOYMENT OF NEURAL NETWORKS ON TINYVERS  Hardware used for ML applications also requires a user  programmable full stack that can translate ML algorithms  directly from existing ML training and inference frameworks  like Tensorflow, Keras, PyTorch, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' This makes the quick and  easy deployment of various ML workloads onto an existing  hardware possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' A python based pseudo-compiler frame-  work created for TinyVers taking into account its heterogeneity  is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' An ML algorithm is first quantized to selected  precision using the QKeras framework [42] for quantized-  aware training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The quantization-aware training framework  takes into consideration the hardware constraints such as  symmetric quantization and the shift based scaling of output  in the PEs of the accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The quantized model is then  passed to a python-based NN compilation which takes in the  hardware description and provides a set of C-based header  files for the RISC-V core, consisting of ucode instructions for  the accelerator, NN parameters and also a golden model for  verification of the mapped workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' CHIP IMPLEMENTATION AND MEASUREMENT  The TinyVers chip microphotograph shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 10 was  implemented and fabricated in GlobalFoundries 22FDXTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  The figure shows the different sub-modules used in the SoC  and detailed in previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 10 also shows the lab  setup used for measurements and benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The follow-  ing subsections details the measurements and benchmarking  done on the SoC for power, energy efficiency and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Peak Performance Analysis  First, a peak performance analysis is undertaken using a  single CNN layer with 32 input channels, 32 output channels  and a 3×3 filter kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Selection of the used layer for peak  @Vdd Mem, Vdd Logic  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
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+page_content=' (TOPS/W)  Throughput (GOPS)  10  20  30  40  50  100  120  150  @0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
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+page_content='69  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='52  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='86  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='7  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='1  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='6  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Peak performance analysis of CNN3×3 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  5 MHz  10 MHz  20 MHz  30 MHz  40 MHz  50 MHz  100 MHz  120 MHz  150 MHz  0  5,000  10,000  15,000  20,000  Power(µW)  WuC  L2  L2uDMA  L1  Logic  DMA  Mram(P)  Mram(A)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Power breakdown of the peak perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' analysis with CNN3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' MRAM  power consumption is negligible as it is OFF in active mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' MRAM(A) and  MRAM(P) represents MRAM array and MRAM periphery resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='.  performance is driven by the fact that convolutional layers  with a 3×3 filter kernel are the most commonly used layer  in modern DNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The hyperparameter selection of the  CNN layer is driven by the constraint of maximum utilization  of the PE array and the size of the private L1 memories  of the accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The 8 bit quantized activation and non-  sparse (structured) weights of the CNN are generated using  the compiler framework using the Google speech dataset for  keyword spotting [43] and verified against the golden model  for functional correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 11 plots the peak energy efficiency and the throughput  with respect to the clock frequency while sweeping the voltage  supply of the logic and memories for the benchmarked CNN  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' For fair comparison with other SotA chips, no body  biasing is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 12 shows the power breakdown of  individual modules when running the benchmarking layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The  SoC shows a large flexibility in delivered performance ranging  from high energy efficiency/low throughput of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='5 TOPS/W,  586 MOPS when operating at a clock frequency of 5 MHz  with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='4 V logic, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='5 V memories, to low energy efficiency  / high throughput of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='8 TOPS/W, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='6 GOPS operating at  150 MHz with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='8 V logic and memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' This provides a  large range for extreme edge tinyML applications to operate,  trading-off between speed and energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Workload Benchmarks  Using the peak energy efficiency operating point (5 MHz,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='4 V logic and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='5 V memory) from Section VI-A, further  performance analysis of different synthetic and actual real-  time benchmarks are evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Table I shows the SoCs  flexibility through mapping of different ML layers and full  MICAS  naikraveraels  ADIGILENT  二  店  K2_VDD_L1K2_0D_L  ZedBoardLens: E20:X80  2022/02/038  TABLE I  WORKLOAD BENCHMARKS  Workload  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Power  (µW)  Peak  perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  (GOPS)  Peak  (effective NZ)  energy eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  (TOPS/W)  Synthetic  CNN@8b 237  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='586  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='47(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='47)  CNN@4b 197  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='17  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='94(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='94)  CNN@2b 197  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='35  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='9(11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='9)  CNN@8b, 239  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='03  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='31(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='46)  50% sparse  CNN@8b, 212  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='64  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='1(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='76)  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='5% sparse  FC/RNN/SVM, 140  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='116  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='829(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='829)  batch=16  Deconv@8b 235  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='36  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='78(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='49)  Real-time  TCN (KWS)  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='3%∗  193  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='204  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='05(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='05)  CAE 209  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='442  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='11(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='27)  ResNet-8  82%+  228  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='267  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='17(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='17)  OC-SVM 129  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='126  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='972(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='972)  ∗ 12-class task, baseline=93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='46%, + baseline=85%  2%  ResNet8  OC-SVM  CAE  TCN  1%  31%  2%  21%  41%  4%  0%  29%  2%  19%  44%  6%  WuC  L2  L2uDMA  L1  Logic  DMA  Mram(P)  Mram(A)  0%  31%  18%  44%  5%  1%  47%  3%  15%  25%  9%  Power: 129 μW,  Latency: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='3ms  Power: 228 μW,  Latency: 76ms  Power: 193 μW,  Latency: 11ms  Power: 209 μW,  Latency: 30ms  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Energy breakdown showing the distribution of measured energy of the  chip modules for running a single inference of the four real-time workloads  on FlexML and RISC-V with input data already available in L2 memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  The power and latency measurements starts from setting up of accelerator  parameters by RISC-V, data movement from L2 to L1, inference computations,  and ends with post processing by RISC-V core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' MRAM power consumption  is negligible as it is OFF in active mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The CNN layer from Section VI-A is extended and  measured with different precision and blockwise structured  sparsity (BSS) levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' When moving to lower precision of INT-  4 and INT-2, the peak throughput improves by 2× and 4×  while the peak energy efficiency improves by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='4× and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='8×  resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=', achieving a maximum of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='9 TOPS/W at INT-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' As  shown in Table I, at 8 bit precision with 50% BSS (16/32  input channels pruned) the performance improves by around  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='7× while at 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='5% BSS (28/32 input channels pruned)  the performance increases by approximately 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='9×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Further  performance improvement can be gained when moving to  lower precision, however, low precision combined with high  BSS levels can cause a large drop in accuracy and, thus, is not  explored in this benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Other synthetic benchmarks  such as FC, RNN, SVM and a deconvolutional layer similar  to the CNN layer in terms of hyperparameters are explored  and the results are shown in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' For the dense layers,  batching of 16 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Finally, 4 real-time application benchmarks are used to  TABLE II  MEASUREMENT RESULTS OF DIFFERENT LOW POWER MODES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Power Mode  AON  Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  (kHz)  Core  Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  (MHz)  Power  (µW)  Wakeup  Latency  (µs)  Deep Sleep  33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='7  788  LP Data acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='∗  33  5  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='6  788  Data acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='∗  33  5  67  788  ∗ @Fs=44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='1 kHz  Wake-up latency (s)  4  8  12  16  24  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='033  Clock Frequency (MHz)  Power (μW)  1  5  10  788 μs  26 μs  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='2 μs  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='6 μs  20  40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='3 μs  650 ns  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='7 μW  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='1 μW  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='8 μW  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='8 μW  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='7 μW  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='8 μW  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Deep sleep power-latency-frequency tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  show the capabilities of the SoC: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=') keyword spotting (KWS)  using TCN model [21], [44] on google speech dataset, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=')  continuous machine monitoring with a convolutional auto-  encoder (CAE) [24] on MIMII dataset [45], 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=') ResNet-8  image classification on CIFAR-10 used in MLPerfTM tiny  benchmark [46], and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=') Novelty detection with OC-SVM [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Table I shows the peak performance characteristics of these  benchmarks on the SoC, more specifically the RISC-V core  and FlexML, with 8-bit precision, a single inference, and  assuming all input data is available in the shared L2 memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  For TCN and ResNet-8, hardware-aware quantization was used  and the energy and performance metrics were measured, while  for the CAE and OC-SVM workloads, random inputs and  weights were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' All the 4 workloads can be deployed with  less than 230 µW of continuous real-time power at peak energy  efficiency between 1-2 TOPS/W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' This means that the SoC can  provide high level of flexibility in workload mapping at sub-  mW power to enable truly power efficient tinyML application  on extreme edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 13 shows the power breakdown  of the 4 real-time workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' For OC-SVM (dense operation),  the power consumption of memory dominates, due to the lack  of re-usability of weights leading to more data fetches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' On the  other hand, power breakdown of CNN based workloads (TCN,  ResNet8 and CAE) shows equal distribution between memory  and logic as the dataflow exploits maximum re-usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Power Management  Table II shows the measured real-time power of the different  low power modes of the SoC detailed in Section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' In deep  sleep mode the SoC operates with an AON clock frequency  of 33 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' In this mode, only the AON domain consisting of  the WuC and the logic controlling the IO pads stays powered  9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='00E-06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='00E-05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='00E-04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='00E-03  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='00E-02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='00E-01  Power (W)  Time (ms)  0  280  140  Erase followed by write output  to MRAM  TCN processing (16 batch)  Boot from MRAM  I2S LP data acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' (2s window)  Deep sleep  Vdd SCL, Mem: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='55 V  Vdd AON: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='7 V  Core Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' : 5 MHz  2000 2140 2280  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Instantaneous power trace showing the KWS application scenario  with one full period of smart sensing and TCN processing followed by idling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  ON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The resulting deep sleep power measured is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='7 µW when  operating at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='7 V voltage supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' When compared to the peak  power measured for the CNN layer, the deep sleep power is  12,000× lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The measure latency of waking up the SoC  from deep sleep mode to active mode is 788 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' This wake-  up latency can be traded off to deep sleep power by sweeping  the AON clock frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 14 plots the this trade-off for  the measured power and wake-up latency when sweeping the  AON clock frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Applications that need low latency can  operate the AON clock at 40 MHz to attain a wake-up latency  of 650 ns at a real-time power of 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='8 µW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Table II also shows the measured power for the two tinyML  optimized power modes of data acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' and LP data acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' These  power modes are measured with an I2S protocol based win-  dowed test vector collection with the AON clock frequency at  33 kHz and the core and peripheral clock frequency at 5 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  The SoC is programmed to collect I2S audio data through its  uDMA at a sampling frequency of 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='1 kHz and a sampling  window of 2 second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The sampling clock is generated by the  SoC using the 5 MHz clock and lasts for the duration of  sampling window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The data acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' or LP data acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' mode is then  initiated and power is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The measured power for LP  data acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' and data acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' is 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='6 µW and 67 µW resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' which is  850× and 300× reduced power consumption compared to the  peak power, when the core and peripheral frequencies can be  dynamically lowered to 5 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Instantaneous Power Trace  In order to show the complete end-to-end application de-  ployable on the SoC and to show the SoC’s full ML func-  tionality, duty cycling and features of power management,  two applications are mapped onto the heterogeneous SoC  with windowed data collection done in the LP data acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  mode: keyword spotting with a TCN model operating in  continuous mode [21];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' and a machine monitoring use case with  a Mel Frequency Energy Coefficient (MFEC) based feature  extraction with a CAE in duty cycled mode [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  1) Keyword-spotting Application: The first application sce-  nario is the keyword-spotting with TCN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' In this ap-  plication scenario, audio data from a microphone of window  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='00E-06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='00E-05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='00E-04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='00E-03  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='00E-02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='00E-01  Power (W)  MFEC processing on RISC-V  Autoencooder processing on FlexML  Boot from MRAM  I2S LP data acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' (1s window)  Deep sleep  Time (ms)  Vdd SCL, Mem: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='55 V  Vdd AON: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='7 V  Core Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' : 5 MHz  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Instantaneous power trace showing the machine monitoring appli-  cation scenario with one period of smart sensing, FE, and CAE processing  followed by idling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  size 2 seconds (16 batches) at a sampling frequency of 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='1  kHz is collected using the I2S peripheral interface protocol,  the collected data is simultaneously stored in the special L2  uDMA memory using the SoC’s uDMA with the SoC being  in the LP data acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' After 2 seconds the SoC wake’s  up into active mode and the collected data is processed using  the TCN model from Section VI-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The output of the TCN  processing is then stored into the MRAM for future processing  or transmission while the SoC can either go into deep sleep  mode or collect new windowed sampling data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 15 shows  the complete instantaneous power consumption trace of the  KWS application scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' When operating in this duty-cycled  mode, the average power of the complete application is 173  µW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The power can be further reduced to 10-20 µW by using  the deep sleep power mode of the SoC during periods of no  sensing or computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  2) Machine Monitoring Application: Machine monitoring  used for predictive maintenance is the second application  scenario selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' In this scenario, I2S peripheral interface  protocol is used to collect audio data from a microphone with  window size 1 second at a sampling frequency of 16 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  The collection of I2S audio data is operated in the LP data  acq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' mode of the SoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Once the complete windowed data is  collected, the SoC switches to the active mode in which the  RISC-V core is used for the MFEC based feature extraction  followed by running the CAE on the accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' 16  plots the instantaneous power trace of running the machine  monitoring application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Unlike the previous application which  works on raw audio data, the CAE model need pre-processing  MFEC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' As the MFEC algorithm is not supported on  the accelerator, it is executed on the RISC-V core with  INT16 precision instead of INT32 or FP32 to reduce power  consumption [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The power trace plots show that running  large feature extraction on RISC-V is not energy efficient  taking large time to complete owing to single core operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  The average power for continuous operation remains below  164 µW, but for this use case, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='5 µW is consumed with a  duty cycling of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The MFEC execution on the RISC-V  10  TABLE III  PERFORMANCE COMPARISON WITH STATE-OF-THE-ART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  [48]  [17]  [49]  TinyVers  [50]  [51]  Extreme Edge SoCs  edgeML Accelerators  Technology  28nm FDSOI  22FDX  55nm  22FDX  28nm  65nm  Die Area (mm2)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='5  12  10  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='55  16  Applications  IoT GP, DNN,  IoT GP, DNN,  IoT GP, DNN,  IoT GP, DNN+,  Always-on KWS  DNN  NSA  NSA  NSA  Trad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' ML, NSA  Supported  CNN,  CNN,  CNN,  CNN,  DSCNN  CNN,  ML layers  FC/RNN  FC/RNN  FC/RNN  FC/RNN, GAN,  FC/RNN  AE, TCN, SVM  Architecture  1×RI5CY+  10×RI5CY+  9×RI5CY  1×RI5CY+  DSCNN  DNN  ML accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  ML accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  FlexML accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  accel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  SRAM  464 kB (40 kB)  128 kB(L1)  64 kB (L1)  132 kB (L1)  2 kB  256 kB  (State retentive)  (16-1600 kB (L2))  (512 kB (L2))  (64/512 kB (L2))  eNVM 4 MB MRAM 512 kB MRAM Deep sleep power (µW) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='7 SRAM ret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='8-123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='7  30  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='6-67 sleep power (µW)  Int precision (bits)  8, 16, 32  8, 16, 32  8, 16, 32  2, 4, 8  8  1-16  Supply voltage (V)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='45-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='5-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='8  1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='63-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='1  Max frequency (MHz)  350  450  250  150  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='04  200  Power range  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='4µW-96mW  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='7µW-49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='4mW  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='6µW-75mW  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='7µW-20mW  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='51µW  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='2-297mW  Best ML perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  36 GOPS  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='2 GOPS  12 GOPS  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='6 GOPS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='3 MOPS+  691.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='2 GOPS  @8b∗  @8b∗  @8b∗  @8b∗∗  @8b∗∗  @8b ∗∗  Best ML eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='3 TOPS/W@  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='3 TOPS/W@  200 GOPS/W@  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='47 TOPS/W@  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='5 TOPS/W@  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='57 TOPS/W,  @Perf  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='8 GOPS, 8b∗  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='6 GOPS, 8b∗  7 GOPS, 8b∗  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='58 GOPS, 8b∗∗  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='3 MOPS, 8b∗∗  8b∗∗  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='9 TOPS/W@  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='6 TOPS/W,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='4 GOPS, 2b∗∗  4b∗∗  + estimated at 90% utilization of MACs, ∗ Matmul, ∗∗ CNN, 1 MAC = 2 Ops  can be optimized using special DSP extensions available with  the PULP libraries, which is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' COMPARISON WITH SOTA  Table III shows the comparison of our SoC with SoTA on  two fronts: on one hand, comparing with existing extreme edge  SoCs (left), and on the other hand, with edge ML accelera-  tors (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Our SoC has similar or increased flexibility in  application mapping compared to the extreme edge SoCs on  the left, with much improved energy efficiency and power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  TinyVers supports not only the IoT general processing (GP),  DNNs and near-sensor analytics (NSA) like [17], [48], [49],  but also DNN+ such as TCN and AE and traditional ML  like SVM, all at better energy efficiency because of efficient  mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' This is evident from the best energy efficiency of  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='47 TOPS/W for running a CNN layer on TinyVers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The  energy efficiency is further enhanced to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='9 TOPS/W when  the CNN workload is quantized to INT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Compared to the  extreme edge SoCs, TinyVers provides support for precision  scalability and, thus, can take advantage of improved perfor-  mance using quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Furthermore, by utilizing support  for block structured sparsity, TinyVers can reach a peak  performance of 17 TOPS/W for an 8-bit CNN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' This is  much higher than the efficiencies reported by [17], [48], [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Compared to the edge ML accelerators on the right,  TinyVers shows much more flexibility at comparable per-  formance metrics in terms of energy efficiency and power  consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The edgeML accelerators only support a single  or few models extremely efficiently, but this approach has  drawbacks in deployment for extreme edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' For ex-  ample, [50] can only perform KWS with depthwise separable  CNN and its performance is much lower than TinyVers with  comparable energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' UNPU [51], can only support  CNN and FC/RNN layers and also does not have a complete  standalone SoC, which effects efficiency at the system level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  Moreover, these edgeML accelerators cannot support any kind  of duty cycling as they lack power management and retention  memory support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' TinyVers supports the multi-modal require-  ments of extreme edge devices at relatively similar energy  efficiencies of the order of TOPS/W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Moreover, it adds the  possibility of extreme low power idle states for duty-cycling  use cases to enable < 10µW operation, shown empirically in  Section VI-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' To summarize, TinyVers brings the best of both  worlds of extreme edge processors and edgeML accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' CONCLUSION  TinyML applications at the extreme edge needs not only  heterogeneous SoCs with flexible accelerators to support di-  verse workloads, but also adaptive power management for  different duty-cycling operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Moreover, to enable such  adaptive power management, the need for embedded non-  volatile memories arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' TinyVers extends a RISC-V core  with a flexible ML accelerator supporting a diverse set of  ML workload mapping in terms of diverse compute kernels,  different precision and structured sparsity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Further-  more, the inclusion of a WuC and an eMRAM enables the  adaptive power management required in many duty-cycling  use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Measurement result shows that the chip can achieve  an energy efficiency range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='8-17 TOPS/W at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='58 GOPS  to 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='6 GOPS of throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The different low power modes  enable the chip to achieve power range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='7µW-20 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  The application of machine monitoring takes advantage of the  11  deep sleep mode to consume only 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='5µW of power at a duty  cycle of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' Thus, TinyVers takes a step towards creating a  new class of ultra-low power extreme edge SoCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The authors would like to thank ETHZ for their support  on PULP platform and GlobalFoundries for 22FDX tapeout  support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content=' The work has been supported under ISAAC project  (FOD Economie Belgium Energietransitiefonds (oproep II)) in  collaboration with Magics Technologies and received funding  from the Flemish Government (AI Research Program).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
+page_content='  REFERENCES  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
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+page_content='org/abs/2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
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+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQf7wU5/content/2301.03537v1.pdf'}
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diff --git a/AdE3T4oBgHgl3EQfsgtu/content/tmp_files/2301.04668v1.pdf.txt b/AdE3T4oBgHgl3EQfsgtu/content/tmp_files/2301.04668v1.pdf.txt
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@@ -0,0 +1,867 @@
+Trapped Ion Quantum Computing using Optical Tweezers and the Magnus Effect
+M. Mazzanti,1 R. Gerritsma,1, 2 R. J. C. Spreeuw,1, 2 and A. Safavi-Naini2, 3
+1Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, 1098 XH Amsterdam, Netherlands
+2QuSoft, Science Park 123, 1098 XG Amsterdam, the Netherlands
+3Institute for Theoretical Physics, Institute of Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands
+(Dated: January 13, 2023)
+We consider the implementation of quantum logic gates in trapped ions using tightly focused optical tweezers.
+Strong polarization gradients near the tweezer focus lead to qubit-state dependent forces on the ion. We show
+that these may be used to implement quantum logic gates on pairs of ion qubits in a crystal. The qubit-state
+dependent forces generated by this effect live on the plane perpendicular to the direction of propagation of
+the laser beams opening new ways of coupling to motional modes of an ion crystal. The proposed gate does
+not require ground state cooling of the ions and does not rely on the Lamb-Dicke approximation, although the
+waist of the tightly focused beam needs to be comparable with its wavelength in order to achieve the needed
+field curvature. Furthermore, the gate can be performed on both ground state and magnetic field insensitive
+clock state qubits without the need for counter-propagating laser fields. This simplifies the setup and eliminates
+errors due to phase instabilities between the gate laser beams. Finally, we show that imperfections in the gate
+execution, in particular pointing errors < 30 nm in the tweezers reduce the gate fidelity from F ≳ 0.99998 to
+≳ 0.999.
+Trapped ions are one of the most mature platforms for the
+implementation of quantum computing and quantum logic
+gates have been implemented with very high fidelity in these
+systems [1, 2]. Usually, the quantum logic gates in trapped
+ions rely on state-dependent forces applied to the ions by
+laser fields or magnetic fields.
+The exchange of motional
+quanta between the ions then leads to effective qubit-qubit in-
+teractions. Several recent works have explored how the use
+of state-of-the-art optical tweezer technology can benefit the
+trapped ion quantum computer. Optical tweezers can be used
+to confine atoms very strongly by inducing a dipole in them
+and find application in neutral atomic quantum simulators, in
+which tweezers are used to levitate individual atoms [3–7].
+In trapped ions, tweezers may be used to tune the soundwave
+spectrum in the ion crystal and thereby to program the inter-
+actions between the qubits [8–10]. Furthermore, in a recent
+work [11] we have proposed combining state-dependent opti-
+cal tweezers with oscillating electric fields to build a universal
+trapped ion quantum computer with extremely long-ranged in-
+teractions between the qubits.
+In this work, we consider another scenario, in which we
+make use of the strong polarization gradients that occur in op-
+tical tweezers. We note that strong gradients in optical po-
+tentials have been previously investigated to implement two-
+qubit gates without the need for ground-state cooling [12–
+14]. However, our approach utilizes the state-dependent dis-
+placement of the tweezer potential due to polarization gradi-
+ents [15–17]. We propose to use this optical analogue of the
+Magnus effect to implement quantum logic gates in trapped
+ions.
+Setup – We consider linearly x-polarized, Gaussian tweez-
+ers, pointing in the −y direction and tightly focused at two
+qubits between which we wish to implement a quantum logic
+gate. The quantum computing platform here considered is a
+linear crystal of N alkali-like trapped ions of mass m. In the
+focal plane the ions experience a strong polarization gradient
+along the x direction, such that the polarization is linear (x)
+in the center and circular (σ±)z in the wings of the Gaussian.
+b)
+a)
+⃗y
+⃗x
+⃗z
+mj
+P1/2
+−1/2
++1/2
+|0⟩
+|1⟩
+Ω−
+S1/2
+Ω+
+nX+
++
++
++
++
++
++
++
+−6 −4 −2
+0
+2
+4
+6
+x/λ
+−4
+−2
+0
+2
+4
+z/λ
+(σ−)z
+−6 −4 −2
+0
+2
+4
+6
+x/λ
+−4
+−2
+0
+2
+4 (σ+)z
+−6 −4 −2
+0
+2
+4
+6
+x/λ
+−4
+−2
+0
+2
+4
+z/λ
+(σ−)z
+−6 −4 −2
+0
+2
+4
+6
+x/λ
+−4
+−2
+0
+2
+4 (σ+)z
+Laguerre-Gaussian
+Gaussian
+FIG. 1. Schematic representation of the two-qubit gate. a) We apply
+tweezers propagating along the −y direction on the two ions forming
+the gate. The tweezer intensity can be decomposed into three polar-
+ization components. b) Simplified level scheme of an alkaline-earth
+like ion without nuclear spin showing the encoding of the qubit in
+its Zeeman ground states. The two polarization components of the
+tweezer couple to different states in the P1/2 manifold with detuning
+∆. This causes the minima of the tweezer potentials to be shifted by
+an amount ±λ depending on the qubit state. Bottom : main polar-
+ization components for a Gaussian and Laguerre-Gaussian (l = 1,
+n = 0) tightly focused tweezer.
+A direct calculation [18] decomposing the field in the focal
+plane into its circular components (σ±)z (and πz) shows that,
+to a good approximation, the circular components are near-
+Gaussian distributions, displaced in opposite directions along
+the x axis. We depict this setup in Fig. 1. Note that the circu-
+lar components rotate in the xy plane, i.e. a plane containing
+the k vector of the light. As shown in Fig. 1, the (σ±)z com-
+ponent is displaced by an amount ±λ ≡ ±λ/2π, with λ the
+arXiv:2301.04668v1  [quant-ph]  11 Jan 2023
+
+2
+tweezer wavelength. As the total field is the superposition
+of two displaced Gaussians, its intensity is slightly elongated
+along x. Hollow tweezers (Gaussian-Laguerre) can be used
+instead of Gaussian ones. This will provide the needed field
+curvature while keeping near-zero intensity at the center of
+the beam, drastically reducing the probability of off-resonant
+scattering that might limit the gate fidelity.
+For simplicity, we first consider ions without nuclear spin,
+such as 40Ca+, 88Sr+, 138Ba+ and 174Yb+. The qubits are
+encoded in the electronic ground states 2S1/2 and |0⟩ = |j =
+1/2, mj = 1/2⟩ and |1⟩ = |j = 1/2, mj = −1/2⟩ with
+j the total electronic angular momentum and mj its projec-
+tion on the quantization axis. The magnetic field lies along
+the z-direction and the tweezers are polarized along the x-
+direction, such that the ions experience linearly polarized laser
+light. The direction along the x-axis is the long direction of
+the ion trap, with trap frequency ωx. The motion of the ions
+along the x-direction can be described by collective modes of
+harmonic motion with frequencies ωm and mode vectors bi,m,
+with m labeling the mode and i the ion [19].
+We choose the detuning between the tweezers and the D1
+transition to be large enough to avoid photon scattering, but
+much smaller than the spin-orbit coupling splitting of the 2P
+state. In this way, we can neglect coupling to the P3/2 state.
+In what follows we will show that this requirement can be
+satisfied experimentally. Close to the center of the tweezer,
+strong polarization gradients appear and as a result, the two
+qubit states experience slightly different tweezer potentials. In
+particular, as we show in Fig. 1(a), the optical Magnus effect
+causes each qubit state to experience a tweezer potential that
+is offset from the apparent center of the tweezer by ∼ λ [16].
+Hence, we may approximate the tweezer potential as :
+ˆU(x) = −U0 exp
+�
+−2(ˆx + ˆσzλ)2/w2
+0
+�
+(1)
+≈ − ˜U0 + 1
+2mω2
+twˆx2 + gx ˆσz
+(2)
+with ωtw =
+�
+4 ˜U0(w2
+0 − 4λ2)/(mw4
+0), g = 4 ˜U0λ/w2
+0, and
+˜U0 = U0 exp(−2λ2/w2
+0) ≈ U0. Here U0 is the tweezer po-
+tential in the center and the beam waist is w0. Our approxima-
+tion replaces the tweezer potential with a harmonic potential
+and is valid for w0 ≫ lm, with lm =
+�
+ℏ/2mωm. The last
+term in U(x) is the result of the spin-dependent force g cou-
+pling the internal state of the qubit, ˆσz, to its motion ˆx. Thus,
+the optical Magnus effect allows us to straightforwardly im-
+plement a quantum gate.
+Tweezer Hamiltonian – In the interaction picture with re-
+spect to ˆH0 = ℏωmˆa†
+mˆam the tweezer Hamiltonian on ions i
+and j is:
+ˆH1 = A(t)
+�1
+2mω2
+tw
+�
+ˆx2
+i + ˆx2
+j
+�
++ g
+�
+ˆσ(i)
+z ˆxi + ˆσ(j)
+z ˆxj
+��
+.
+(3)
+Here, ˆxi = �
+m lmbim
+�
+ˆame−iωmt + ˆa†
+meiωmt�
+is the posi-
+tion operator of ion i in the interaction picture, with ˆa†
+m the
+creation operator for the mode m, and 0 ≤ A(t) ≤ 1 speci-
+fies the time-dependence of the tweezer intensity. The qubit-
+state independent terms in ˆH1 do not alter the dynamics of the
+quantum gate. We ignore these terms and arrive at:
+ˆH2 = A(t)g
+�
+ˆxiˆσ(i)
+z
++ ˆxjˆσ(j)
+z
+�
+,
+(4)
+which takes the form of a spin-phonon coupling Hamilto-
+nian reminiscent of the Mølmer-Sørenson scheme for phonon-
+mediated quantum gates in trapped ions [20]. However, at this
+stage we still have various choices available for A(t), depend-
+ing on which type of quantum gate we would like to imple-
+ment. For instance, pulsed A(t) could be used to perform fast
+gates. Here, we choose A(t) to obtain a geometric phase gate.
+For this, we set 2A(t) = 1−cos(νt+φ) where φ = 0 assures
+a smooth ramp of the tweezer intensity and ν = ωc + δ with
+the subscript c denoting the center-of-mass (c.o.m.) mode for
+which ωc = ωx and bi,c = 1/
+√
+N. We write the operators ˆxi
+and ˆxj in terms of ˆac and ˆa†
+c and perform the rotating wave
+approximation to arrive at:
+ˆH3 =
+glc
+4
+√
+N
+�
+ˆaceiδt + ˆa†
+ce−iδt� �
+ˆσ(i)
+z
++ ˆσ(j)
+z
+�
+.
+(5)
+To derive the qubit-qubit interactions forming the geomet-
+ric phase gate, we perform a unitary transformation ˆU1 =
+e−iδˆa†
+c ˆact to eliminate the time dependence, followed by a
+Lang-Firsov [21] transformation, ˆU2 = exp
+�
+ˆα
+�
+ˆa†
+c − ˆac
+��
+with ˆα = − ˜g
+δ
+�
+ˆσ(i)
+z
++ ˆσ(j)
+z
+�
+. Disregarding qubit-independent
+terms, we obtain
+Heff = 2˜g2
+ℏδ ˆσ(i)
+z ˆσ(j)
+z ,
+(6)
+with ˜g = glc/(4
+√
+N) = ˜η ˜U0, with the proportionality factor
+˜η = λlc/(
+√
+Nw2
+0). This Hamiltonian generates qubit-qubit
+interactions that can be used to implement a geometric phase
+gate by setting the gate time τ = 2π/δ and ˜g2τ
+ℏ2δ = π/4.
+Characterization of the gate – We analyse the gate dynam-
+ics by performing numerical simulation of the full dynamics
+generated by the Hamiltonian ˆHsim = ˆH0 + ˆU (xi) + ˆU (xj)
+for a two dimensional ion crystal where the tweezers po-
+tentials ˆU (xi;j) on ions i and j have been expanded up to
+fourth-order including spin-independent terms. We use real-
+istic experimental parameters: ∼ 156 µW of tweezer laser
+power focused to a waist of w0 ∼ 0.5 µm and tuned 15 THz
+to the red from the 2S1/2 →
+2P1/2 transition in 174Yb+
+(λ = 369.5 nm). This results in ˜U0/h ∼ 1.6 MHz, ˜g/h =
+2.1 kHz/
+√
+N, and setting δ = 2π × 12.2 kHz/
+√
+N the gate
+time for the geometric phase gate is τ = 170
+√
+Nµs. With a
+calculated qubit-state independent tweezer potential of ωtw ∼
+2π × 37 kHz, the center-of-mass mode frequency (ωc/2π ∼
+1 MHz) is shifted by ∼ 2ω2
+tw/ωcN ∼ 2π × 710/N Hz.
+This shift can easily be taken into account by correcting δ
+accordingly. In these estimates, we neglected the contribu-
+tion from other dipole allowed transitions, that are detuned by
+∼ 66 THz (the relatively weak 2S1/2 → 3[3/2]3/2 transition)
+and 115 THz (the strong D2 line) or more.
+We consider the gate unitary with a spin-echo sequence
+given by U(0, τ) = X⊗2U(τ/2, τ)X⊗2U(0, τ/2), where
+
+3
+FIG. 2. We calculate the gate fidelity for a ground state cooled ion
+¯nc, ¯ns = 0 (blue), sub-Doppler cooled thermal state with ¯nc =
+0.62, ¯ns = 0.23 (orange) and ¯nc = 15, ¯ns = 0.23 (red, using in
+this case a Fock cutoff nc ≤ 120, ns ≤ 10). (a) Process fidelity
+of the two-qubit Magnus gate for different gate times. (b) Effects of
+misalignment ϵ (orange) and intensity noise Λ1/τ (blue) on the gate
+fidelity. The size of each intensity noise data point represents the
+standard deviation of 20 simulation where we generated a random
+Gaussian noise with σ = Λ1/τ on each of the two pulses. This
+implies a noise on the laser intensity at frequency 1/τ that can not
+be removed by the spin-echo sequence.
+X⊗2 is a qubit flip on both qubits. This spin echo sequence is
+needed in order to remove local rotations on the qubits states
+and possible timing errors. We calculate the unitary time evo-
+lution operator U(0, τ) for a system of two ions with their
+motional c.o.m. and stretch modes and truncate their respec-
+tive Hilbert spaces to nc ≤ 18 and ns ≤ 10. In figure 2 we
+show the process fidelity of the gate assuming the ions are in
+their motional ground state (¯n = 0) as a function of gate time.
+The gate fidelity of F = 0.999988 with nc = ns = 0 rivals
+the current standard approaches. Moreover, the performance
+of our gate is robust to the thermal occupation of the motional
+modes. We characterize the gate performance in presence of
+thermal phonons using the average gate fidelity [18, 22] and
+find that it depends weakly on the motional state of the two
+ions. In fact, using ¯nc = 0.62, ¯ns = 0.23, the fidelity is
+almost unaltered at Fth = 0.999989.
+One of main experimental challenges is perfect tweezer
+alignment. We have studied the resilience of the gate against
+misalignment of the tweezer in the x-direction, which we
+denote by ϵ.
+In the presence of misalignment,
+˜U0
+→
+|0⟩
+|1⟩
+∆
+mF
+−1
+0
++1
+171Yb+
+FIG. 3. Relevant energy levels of 171Yb+ for implementing the gate
+on hyperfine qubit splitted by ωq. The coupling can be achieved
+using a pair of Raman beams detuned from the upper state 2P1/2 by
+∆. In the brackets are the angular contributions to the various dipole
+transition elements.
+TABLE I. Main sources of gate errors. We estimate γph as the prob-
+ability of a off-resonant scattering in for 174Yb+ during the gate time
+(τ = 240 µs) for a Gaussian and Laguerre-Gaussian beams. Other
+typical sources of errors are misalignment (ϵ), tweezer intensity noise
+(Λ1/τ) and timing (∆τ). The values here reported are for laser pa-
+rameters used in our numerical simulations.
+Error source
+γph
+Gaussian
+γph
+Laguerre-Gaussian
+ϵ
+30 nm
+Λ1/τ
+0.5%
+∆τ
+±5 µs
+1 − F
+2 × 10−3
+10−6
+1.3 × 10−3 9.3 × 10−5 2.7 × 10−4
+U0 exp−2(ϵ+ˆσzλ)2/ω2
+0.
+Thus, the misalignment has two ef-
+fects: (i) it changes the tweezer potential at the center of the
+tweezer and therefore the phase accumulation in the phase
+gate, and (ii) it shifts the potential in a qubit-state-dependent
+way. The second contribution is corrected to lowest order by
+a spin-echo sequence. Figure 2(b) shows the infidelity as
+a function of ϵ. Here we assume that the tweezers are mis-
+aligned on both ions in the same way which seems the experi-
+mentally most likely case. The unitary U(0, τ) leads to phase
+space trajectories for ⟨x(t)⟩ and ⟨px(t)⟩ associated with the
+c.o.m. motion[18]. As expected, we find approximately cir-
+cular phase-space orbits for the even parity states |00⟩, |11⟩,
+and very little motion for the odd parity ones. We see that ev-
+ery state combination leads to ion motion, but the difference
+in motion still leads to a high fidelity of ≳ 0.999 as shown in
+Figure 2(b).
+Clock state case – While the calculation was performed
+for the electron spin qubit states in 174Yb+, it should also
+be possible to use the hyperfine clock states |F = mF = 0⟩
+and |F = 1, mF = 0⟩ in 171Yb+. This qubit is insensitive to
+magnetic field noise and coherence times of up to an hour have
+been measured [23]. In this case, the tweezers are formed by a
+bichromatic co-propagating laser field detuned by ∆ from the
+D1 transition at 369.5 nm with overall detuning ∆ ≪ ωFS, the
+fine structure splitting. We set the frequency difference in the
+
+4
+bichromatic tweezer to 12.6 GHz, corresponding to the tran-
+sition between the qubit states [24]. The tweezer laser then
+causes Raman coupling between the qubit states via two dis-
+tinct paths. In the first path, the qubits are coupled via the state
+|P1/2, F = 1, mF = −1⟩ due to the σ− polarization compo-
+nent in the tweezer. In the other, the qubits are coupled via
+the state |P1/2, F = 1, mF = +1⟩ due to the σ+ component
+in the tweezer. We denote the Rabi frequencies of each path
+as Ω±
+1,2(x). The corresponding Raman couplings of each path
+interfere destructively in the center of the tweezer due to a rel-
+ative minus sign between Ω+
+1 (x) and Ω+
+2 (x) in their Clebsch-
+Gordan coefficient, ∝ (Ω−
+1 (0)Ω−
+2 (0)+Ω+
+1 (0)Ω+
+2 (0))/∆ = 0.
+However, the Magnus effect causes a strong position depen-
+dence of the relative strength of both paths of magnitude
+Ωeff(x) = Ω−
+1 (x)Ω−
+2 (x)
+∆
++ Ω+
+1 (x)Ω+
+2 (x)
+∆
+≈ Ω2
+∆
+4λx
+w2
+0
+,
+(7)
+where we assumed x ≪ λ ≪ w0 and |Ω±
+i (0)| = Ω/
+√
+2 with
+i = 1, 2, such that both laser frequencies have the same power.
+As a result, a qubit state-dependent force appears as in Eq.
+(4), except that we must now replace ˆσ(i,j)
+z
+→ ˆσ(i,j)
+x
+and the
+gate takes the form of the usual Mølmer-Sørensen interaction
+∝ ˆσ(i)
+x ˆσ(j)
+x
+[20]. Amplitude modulation via A(t) allows again
+for resonant enhancement of the gate.
+In addition to the Raman coupling, we obtain a tweezer po-
+tential (AC Stark shift) for each qubit state of magnitude
+δ|k⟩
+AC(x) =
+�
+i=1,2
+�
+j=+,−
+|Ωj
+i(x)|2
+∆i,|k⟩
+(8)
+with ∆1,|0⟩ = ∆ − ωq, ∆2,|0⟩ = ∆, ∆1,|1⟩ = ∆ and ∆2,|1⟩ =
+∆ + ωq. This causes an additional trapping potential Φ(x) ≈
+1
+2mω2
+twx2 that is independent of the qubit state as before, as
+well as a position-dependent differential Stark shift δAC(x) =
+δ|1⟩
+AC(x) − δ|0⟩
+AC(x). In the limit ωq ≪ |∆|,
+δAC(x) ≈ − ωq
+∆2
+�
+i=1,2
+�
+j=+,−
+|Ωj
+i(x)|2
+(9)
+= −ωq
+∆
+˜U0(x)
+(10)
+This differential Stark shift is estimated to be small, δAC/2π ≈
+2.7 kHz for the numbers used in the simulations, and can be
+compensated by a corresponding Raman detuning.
+Photon scattering on the D1 transition can be estimated as
+γph ∼ ˜U0Γ/(ℏ∆) ∼ 13 s−1 with Γ = 1.23 × 108 s−1 in
+Yb+. This adverse effect may be reduced significantly by em-
+ploying hollow tweezers [11, 25, 26] at the expense of added
+complexity. For a hollow beam with a waist w0 = 0.5 µm
+and ∼ 160 µW we obtain a reduction in scattering rate of
+∼ 10−6 s−1. As long as ωtw ≪ Ωrf, the drive frequency of the
+Paul trap, no parametric excitations can occur and micromo-
+tion of the ions is not a problem. Other errors, such as due to
+intensity noise of the laser, heating of the ions due to electric
+field noise and decoherence due to magnetic field noise have
+the same effect as in other gate implementations. Finally, we
+note that because the tweezers are far detuned from the closest
+transitions, the exact overall frequency of the tweezer laser is
+irrelevant.
+Conclusions In conclusion, we have described a novel
+type of quantum phase gate based on the optical Magnus ef-
+fect using optical tweezers in a linear chain of trapped ions.
+The main benefit is that the gate does not require counter-
+propagating laser fields, greatly simplifying the setup and
+eliminating errors due to phase instabilities between the gate
+laser beams. Furthermore, the state-dependent force gener-
+ated by the Magnus effect allows to perform the gate by cou-
+pling to motional modes on the plane perpendicular to the di-
+rection of propagation of the tweezers allowing novel experi-
+mental implementations. The proposed gate does not require
+ground state cooling and can perform a quantum logic gate
+on any pair of ion qubits by spatial addressing. The expected
+gate fidelity rivals the state of the art also for ions that are not
+cooled to the ground-state of motion.
+ACKNOWLEDGEMENTS
+This work was supported by the Netherlands Organiza-
+tion for Scientific Research (Grant Nos.
+680.91.120 and
+680.92.18.05, (R.G.). A.S.N is supported by the Dutch Re-
+search Council (NWO/OCW), as part of the Quantum Soft-
+ware Consortium programme (project number 024.003.037).
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+Supplementary material for :
+Trapped Ion Quantum Computing using Optical Tweezers and the Magnus Effect
+M. Mazzanti,1 R. Gerritsma,1, 2 R. J. C. Spreeuw,1, 2 and A. Safavi-Naini2, 3
+1Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, 1098 XH Amsterdam, Netherlands
+2QuSoft, Science Park 123, 1098 XG Amsterdam, the Netherlands
+3Institute for Theoretical Physics, Institute of Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands
+(Dated: January 13, 2023)
+APPENDIX I : OPTICAL MAGNUS EFFECT
+A key characteristic of a tightly focused beam is the strong
+field curvature near the focus. This not only affects the local
+intensity but also its polarization structure. To calculate this,
+we take a superposition of plane waves labeled by their wave
+vector in spherical coordinates, k = (k, θ, φ). Taking k =
+ω/c as fixed we write
+E(r) ∝
+� 2π
+0
+dφ
+� π
+0
+dθ sin θ ux(θ, φ) a(θ, φ) eik·r
+with ux(θ, φ) a polarization vector obtained by co-rotating
+the x unit vector when k is rotated from z to (θ, φ), such
+that ux(θ, φ) · k = 0, see also Ref. [S1]. In the calcula-
+tion we center the beam around θ = 0, and the focal plane
+is given by r = (x, y, 0).
+The shape of the beam is de-
+termined by the amplitude function a(θ, φ). For a Gaussian
+beam we set a(θ, φ) = exp(−θ2/w2
+θ); for the lowest or-
+der (l = 1) Laguerre-Gaussian (LG) beam we set a(θ, φ) =
+θ exp(iφ − θ2/w2
+θ). After performing the above integral we
+rotate the results for tweezers propagating along the −y di-
+rection. Finally, the circular field components σ± shown in
+Fig. 1 of the main text are obtained as the projection onto unit
+vectors (x ± iy)/
+√
+2. In Figure S-1, all three polarization
+components for a Laguerre-Gaussian beam are shown. Note
+that the σ− and σ+ components have similar intensity while
+the π-polarization is suppressed by a factor ∼ 100.
+−6 −4 −2 0
+2
+4
+6
+x/λ
+−4
+−2
+0
+2
+4
+z/λ
+(σ−)z
+−6 −4 −2 0
+2
+4
+6
+x/λ
+−4
+−2
+0
+2
+4 (π)z
+−6 −4 −2 0
+2
+4
+6
+x/λ
+−4
+−2
+0
+2
+4 (σ+)z
+FIG. S-1. Intensity of the polarization components for a LG beam
+calculated at the focus. The π-polarization component has been en-
+hanced by a factor 100 to make it visible. Here we set wθ = 0.6
+APPENDIX II : PHASE-SPACE DYNAMICS
+We study the phase-space dynamics of the ions by simulat-
+ing the time dependent Hamiltonian using trotterization with
+time-steps of 10−4 τ. At each time-step we evaluate the ex-
+pectation value of the ⟨ˆx⟩ and ⟨ˆp⟩ for the center of mass mode.
+As expected, we find approximately circular phase-space or-
+bits for the even parity states |00⟩, |11⟩, and very little motion
+for the odd parity ones. In Fig. S-2 it is possible to see the evo-
+lution in phase-space for all the four spin states in case of per-
+fectly aligned and slightly misaligned tweezers. As described
+in the main text we simulate numerically the full Hamiltonian
+defined as ˆHsim = ˆH0 + ˆU (xi) + ˆU (xj) where in case of
+misalignment ϵ, ˆU (x) reads as :
+U(x) ≈ −U0 e−2((ˆx−ˆϵ)+ˆσzλ)2/w2
+0
+≈ − ˜U0 + 4 ˜U0
+ˆσzλ − ˆϵ
+w2
+0
+ˆx
++ 1
+2
+˜U0
+�
+4
+�
+w2
+0 − 4λ2�
+w4
+0
+�
+ˆx2 − 1
+2
+˜U0
+�
+16
+�
+ˆϵ2 − 2ˆσzˆϵλ
+�
+w4
+0
+�
+ˆx2
+−
+�
+8 ˜U0ˆσzλ3w2
+0 − 4
+�
+3ˆϵ2 + λ2�
+3w6
+0
+�
+ˆx3
++
+�
+8 ˜U0ˆϵ3w2
+0 − 4
+�
+ˆϵ2 + 3λ2�
+3w6
+0
+�
+ˆx3
+−
+�
+2 ˜U0ˆσzλˆϵ−48w2
+0 + 64
+�
+ˆϵ2 + λ2�
+3w8
+0
+�
+ˆx4
++
+�
+2 ˜U0
+3w4
+0 − 24w2
+0
+�
+ˆϵ2 + λ2�
++ 16
+�
+ˆϵ4 + 6ˆϵ2λ2 + λ4�
+3w8
+0
+�
+ˆx4.
+with
+˜U0 = U0e−2(ˆϵ+ˆσzλ)2/w2
+0
+A small tweezer misalignment ϵ gives rise to new spin-
+dependent terms in the Hamiltonian that shift the trapping po-
+tential in a state dependent way. In Fig.S-2 is shown how the
+dynamics is affected in the case where the tweezers are mis-
+aligned by 30 nm.
+APPENDIX III : GATE FIDELITY
+We characterize the gate by calculating the average process
+fidelity as follows : [S2]:
+¯F( ˆUid, ˆU ˆ
+Hsim) =
+�
+j tr
+�
+ˆUidˆσ†
+j ˆU †
+idˆσj( ˆU ˆ
+Hsim)
+�
++ d2
+d2 (d + 1)
+,
+
+2
+−1.4
+−0.7
+0.0
+0.7
+1.4
+⟨ˆx⟩
+−1.4
+−0.7
+0.0
+0.7
+1.4
+⟨ˆpx⟩
+ϵ = 0
+−2.1 −1.4 −0.7 0.0
+0.7
+1.4
+⟨ˆx⟩
+−1.4
+−0.7
+0.0
+0.7
+1.4
+⟨ˆpx⟩
+ϵ = 30 nm
+ψ↑↑
+ψ↓↑
+ψ↑↓
+ψ↓↓
+FIG. S-2. Center of mass mode phase-space dynamics for perfectly
+aligned tweezer (left) and for 30 nm misaligned ones (right). For the
+simulation we used the same parameters as for τ/2 = 120 µs point
+in Figure 1(a) of the main text.
+where ˆUid is the unitary of an ideal geometric phase gate and
+ˆσj( ˆU ˆ
+Hsim) ≡ trFS( ˆU ˆ
+Hsim [|n⟩⟨n| � ˆσj] ˆU †
+ˆ
+Hsim) projects the
+unitary matrix generated by the time evolution of the Hamil-
+tonian used for the simulations ˆU ˆ
+Hsim on the Fock state |n⟩
+and on a d-dimensional representation Pauli matrices.
+[S1] R. J. Spreeuw, Phys. Rev. Lett. 125, 233201 (2020).
+[S2] M. A. Nielsen, Physics Letters A 303, 249 (2002).
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf,len=487
+page_content='Trapped Ion Quantum Computing using Optical Tweezers and the Magnus Effect  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Mazzanti,1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Gerritsma,1, 2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Spreeuw,1, 2 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Safavi-Naini2, 3  1Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, 1098 XH Amsterdam, Netherlands  2QuSoft, Science Park 123, 1098 XG Amsterdam, the Netherlands  3Institute for Theoretical Physics, Institute of Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands  (Dated: January 13, 2023)  We consider the implementation of quantum logic gates in trapped ions using tightly focused optical tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Strong polarization gradients near the tweezer focus lead to qubit-state dependent forces on the ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We show  that these may be used to implement quantum logic gates on pairs of ion qubits in a crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The qubit-state  dependent forces generated by this effect live on the plane perpendicular to the direction of propagation of  the laser beams opening new ways of coupling to motional modes of an ion crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The proposed gate does  not require ground state cooling of the ions and does not rely on the Lamb-Dicke approximation, although the  waist of the tightly focused beam needs to be comparable with its wavelength in order to achieve the needed  field curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Furthermore, the gate can be performed on both ground state and magnetic field insensitive  clock state qubits without the need for counter-propagating laser fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This simplifies the setup and eliminates  errors due to phase instabilities between the gate laser beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Finally, we show that imperfections in the gate  execution, in particular pointing errors < 30 nm in the tweezers reduce the gate fidelity from F ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='99998 to  ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Trapped ions are one of the most mature platforms for the  implementation of quantum computing and quantum logic  gates have been implemented with very high fidelity in these  systems [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Usually, the quantum logic gates in trapped  ions rely on state-dependent forces applied to the ions by  laser fields or magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  The exchange of motional  quanta between the ions then leads to effective qubit-qubit in-  teractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Several recent works have explored how the use  of state-of-the-art optical tweezer technology can benefit the  trapped ion quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Optical tweezers can be used  to confine atoms very strongly by inducing a dipole in them  and find application in neutral atomic quantum simulators, in  which tweezers are used to levitate individual atoms [3–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  In trapped ions, tweezers may be used to tune the soundwave  spectrum in the ion crystal and thereby to program the inter-  actions between the qubits [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Furthermore, in a recent  work [11] we have proposed combining state-dependent opti-  cal tweezers with oscillating electric fields to build a universal  trapped ion quantum computer with extremely long-ranged in-  teractions between the qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  In this work, we consider another scenario, in which we  make use of the strong polarization gradients that occur in op-  tical tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We note that strong gradients in optical po-  tentials have been previously investigated to implement two-  qubit gates without the need for ground-state cooling [12–  14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' However, our approach utilizes the state-dependent dis-  placement of the tweezer potential due to polarization gradi-  ents [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We propose to use this optical analogue of the  Magnus effect to implement quantum logic gates in trapped  ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Setup – We consider linearly x-polarized, Gaussian tweez-  ers, pointing in the −y direction and tightly focused at two  qubits between which we wish to implement a quantum logic  gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The quantum computing platform here considered is a  linear crystal of N alkali-like trapped ions of mass m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In the  focal plane the ions experience a strong polarization gradient  along the x direction, such that the polarization is linear (x)  in the center and circular (σ±)z in the wings of the Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  b)  a)  ⃗y  ⃗x  ⃗z  mj  P1/2  −1/2  +1/2  |0⟩  |1⟩  Ω−  S1/2  Ω+  nX+  +  +  +  +  +  +  +  −6 −4 −2  0  2  4  6  x/λ  −4  −2  0  2  4  z/λ  (σ−)z  −6 −4 −2  0  2  4  6  x/λ  −4  −2  0  2  4 (σ+)z  −6 −4 −2  0  2  4  6  x/λ  −4  −2  0  2  4  z/λ  (σ−)z  −6 −4 −2  0  2  4  6  x/λ  −4  −2  0  2  4 (σ+)z  Laguerre-Gaussian  Gaussian  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Schematic representation of the two-qubit gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' a) We apply  tweezers propagating along the −y direction on the two ions forming  the gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The tweezer intensity can be decomposed into three polar-  ization components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' b) Simplified level scheme of an alkaline-earth  like ion without nuclear spin showing the encoding of the qubit in  its Zeeman ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The two polarization components of the  tweezer couple to different states in the P1/2 manifold with detuning  ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This causes the minima of the tweezer potentials to be shifted by  an amount ±λ depending on the qubit state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Bottom : main polar-  ization components for a Gaussian and Laguerre-Gaussian (l = 1,  n = 0) tightly focused tweezer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  A direct calculation [18] decomposing the field in the focal  plane into its circular components (σ±)z (and πz) shows that,  to a good approximation, the circular components are near-  Gaussian distributions, displaced in opposite directions along  the x axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We depict this setup in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Note that the circu-  lar components rotate in the xy plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' a plane containing  the k vector of the light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 1, the (σ±)z com-  ponent is displaced by an amount ±λ ≡ ±λ/2π, with λ the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='04668v1  [quant-ph]  11 Jan 2023  2  tweezer wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' As the total field is the superposition  of two displaced Gaussians, its intensity is slightly elongated  along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Hollow tweezers (Gaussian-Laguerre) can be used  instead of Gaussian ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This will provide the needed field  curvature while keeping near-zero intensity at the center of  the beam, drastically reducing the probability of off-resonant  scattering that might limit the gate fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  For simplicity, we first consider ions without nuclear spin,  such as 40Ca+, 88Sr+, 138Ba+ and 174Yb+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The qubits are  encoded in the electronic ground states 2S1/2 and |0⟩ = |j =  1/2, mj = 1/2⟩ and |1⟩ = |j = 1/2, mj = −1/2⟩ with  j the total electronic angular momentum and mj its projec-  tion on the quantization axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The magnetic field lies along  the z-direction and the tweezers are polarized along the x-  direction, such that the ions experience linearly polarized laser  light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The direction along the x-axis is the long direction of  the ion trap, with trap frequency ωx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The motion of the ions  along the x-direction can be described by collective modes of  harmonic motion with frequencies ωm and mode vectors bi,m,  with m labeling the mode and i the ion [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  We choose the detuning between the tweezers and the D1  transition to be large enough to avoid photon scattering, but  much smaller than the spin-orbit coupling splitting of the 2P  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In this way, we can neglect coupling to the P3/2 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  In what follows we will show that this requirement can be  satisfied experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Close to the center of the tweezer,  strong polarization gradients appear and as a result, the two  qubit states experience slightly different tweezer potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In  particular, as we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 1(a), the optical Magnus effect  causes each qubit state to experience a tweezer potential that  is offset from the apparent center of the tweezer by ∼ λ [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Hence, we may approximate the tweezer potential as :  ˆU(x) = −U0 exp  �  −2(ˆx + ˆσzλ)2/w2  0  �  (1)  ≈ − ˜U0 + 1  2mω2  twˆx2 + gx ˆσz  (2)  with ωtw =  �  4 ˜U0(w2  0 − 4λ2)/(mw4  0), g = 4 ˜U0λ/w2  0, and  ˜U0 = U0 exp(−2λ2/w2  0) ≈ U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Here U0 is the tweezer po-  tential in the center and the beam waist is w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Our approxima-  tion replaces the tweezer potential with a harmonic potential  and is valid for w0 ≫ lm, with lm =  �  ℏ/2mωm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The last  term in U(x) is the result of the spin-dependent force g cou-  pling the internal state of the qubit, ˆσz, to its motion ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Thus,  the optical Magnus effect allows us to straightforwardly im-  plement a quantum gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Tweezer Hamiltonian – In the interaction picture with re-  spect to ˆH0 = ℏωmˆa†  mˆam the tweezer Hamiltonian on ions i  and j is:  ˆH1 = A(t)  �1  2mω2  tw  �  ˆx2  i + ˆx2  j  �  + g  �  ˆσ(i)  z ˆxi + ˆσ(j)  z ˆxj  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  (3)  Here, ˆxi = �  m lmbim  �  ˆame−iωmt + ˆa†  meiωmt�  is the posi-  tion operator of ion i in the interaction picture, with ˆa†  m the  creation operator for the mode m, and 0 ≤ A(t) ≤ 1 speci-  fies the time-dependence of the tweezer intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The qubit-  state independent terms in ˆH1 do not alter the dynamics of the  quantum gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We ignore these terms and arrive at:  ˆH2 = A(t)g  �  ˆxiˆσ(i)  z  + ˆxjˆσ(j)  z  �  ,  (4)  which takes the form of a spin-phonon coupling Hamilto-  nian reminiscent of the Mølmer-Sørenson scheme for phonon-  mediated quantum gates in trapped ions [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' However, at this  stage we still have various choices available for A(t), depend-  ing on which type of quantum gate we would like to imple-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' For instance, pulsed A(t) could be used to perform fast  gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Here, we choose A(t) to obtain a geometric phase gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  For this, we set 2A(t) = 1−cos(νt+φ) where φ = 0 assures  a smooth ramp of the tweezer intensity and ν = ωc + δ with  the subscript c denoting the center-of-mass (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=') mode for  which ωc = ωx and bi,c = 1/  √  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We write the operators ˆxi  and ˆxj in terms of ˆac and ˆa†  c and perform the rotating wave  approximation to arrive at:  ˆH3 =  glc  4  √  N  �  ˆaceiδt + ˆa†  ce−iδt� �  ˆσ(i)  z  + ˆσ(j)  z  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  (5)  To derive the qubit-qubit interactions forming the geomet-  ric phase gate, we perform a unitary transformation ˆU1 =  e−iδˆa†  c ˆact to eliminate the time dependence, followed by a  Lang-Firsov [21] transformation, ˆU2 = exp  �  ˆα  �  ˆa†  c − ˆac  ��  with ˆα = − ˜g  δ  �  ˆσ(i)  z  + ˆσ(j)  z  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Disregarding qubit-independent  terms, we obtain  Heff = 2˜g2  ℏδ ˆσ(i)  z ˆσ(j)  z ,  (6)  with ˜g = glc/(4  √  N) = ˜η ˜U0, with the proportionality factor  ˜η = λlc/(  √  Nw2  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This Hamiltonian generates qubit-qubit  interactions that can be used to implement a geometric phase  gate by setting the gate time τ = 2π/δ and ˜g2τ  ℏ2δ = π/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Characterization of the gate – We analyse the gate dynam-  ics by performing numerical simulation of the full dynamics  generated by the Hamiltonian ˆHsim = ˆH0 + ˆU (xi) + ˆU (xj)  for a two dimensional ion crystal where the tweezers po-  tentials ˆU (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='j) on ions i and j have been expanded up to  fourth-order including spin-independent terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We use real-  istic experimental parameters: ∼ 156 µW of tweezer laser  power focused to a waist of w0 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='5 µm and tuned 15 THz  to the red from the 2S1/2 →  2P1/2 transition in 174Yb+  (λ = 369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='5 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This results in ˜U0/h ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='6 MHz, ˜g/h =  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='1 kHz/  √  N, and setting δ = 2π × 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='2 kHz/  √  N the gate  time for the geometric phase gate is τ = 170  √  Nµs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' With a  calculated qubit-state independent tweezer potential of ωtw ∼  2π × 37 kHz, the center-of-mass mode frequency (ωc/2π ∼  1 MHz) is shifted by ∼ 2ω2  tw/ωcN ∼ 2π × 710/N Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  This shift can easily be taken into account by correcting δ  accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In these estimates, we neglected the contribu-  tion from other dipole allowed transitions, that are detuned by  ∼ 66 THz (the relatively weak 2S1/2 → 3[3/2]3/2 transition)  and 115 THz (the strong D2 line) or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  We consider the gate unitary with a spin-echo sequence  given by U(0, τ) = X⊗2U(τ/2, τ)X⊗2U(0, τ/2), where  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We calculate the gate fidelity for a ground state cooled ion  ¯nc, ¯ns = 0 (blue), sub-Doppler cooled thermal state with ¯nc =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='62, ¯ns = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='23 (orange) and ¯nc = 15, ¯ns = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='23 (red, using in  this case a Fock cutoff nc ≤ 120, ns ≤ 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' (a) Process fidelity  of the two-qubit Magnus gate for different gate times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' (b) Effects of  misalignment ϵ (orange) and intensity noise Λ1/τ (blue) on the gate  fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The size of each intensity noise data point represents the  standard deviation of 20 simulation where we generated a random  Gaussian noise with σ = Λ1/τ on each of the two pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This  implies a noise on the laser intensity at frequency 1/τ that can not  be removed by the spin-echo sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  X⊗2 is a qubit flip on both qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This spin echo sequence is  needed in order to remove local rotations on the qubits states  and possible timing errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We calculate the unitary time evo-  lution operator U(0, τ) for a system of two ions with their  motional c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' and stretch modes and truncate their respec-  tive Hilbert spaces to nc ≤ 18 and ns ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In figure 2 we  show the process fidelity of the gate assuming the ions are in  their motional ground state (¯n = 0) as a function of gate time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  The gate fidelity of F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='999988 with nc = ns = 0 rivals  the current standard approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Moreover, the performance  of our gate is robust to the thermal occupation of the motional  modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We characterize the gate performance in presence of  thermal phonons using the average gate fidelity [18, 22] and  find that it depends weakly on the motional state of the two  ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In fact, using ¯nc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='62, ¯ns = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='23, the fidelity is  almost unaltered at Fth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='999989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  One of main experimental challenges is perfect tweezer  alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We have studied the resilience of the gate against  misalignment of the tweezer in the x-direction, which we  denote by ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  In the presence of misalignment,  ˜U0  →  |0⟩  |1⟩  ∆  mF  −1  0  +1  171Yb+  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Relevant energy levels of 171Yb+ for implementing the gate  on hyperfine qubit splitted by ωq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The coupling can be achieved  using a pair of Raman beams detuned from the upper state 2P1/2 by  ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In the brackets are the angular contributions to the various dipole  transition elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Main sources of gate errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We estimate γph as the prob-  ability of a off-resonant scattering in for 174Yb+ during the gate time  (τ = 240 µs) for a Gaussian and Laguerre-Gaussian beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Other  typical sources of errors are misalignment (ϵ), tweezer intensity noise  (Λ1/τ) and timing (∆τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The values here reported are for laser pa-  rameters used in our numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Error source  γph  Gaussian  γph  Laguerre-Gaussian  ϵ  30 nm  Λ1/τ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='5%  ∆τ  ±5 µs  1 − F  2 × 10−3  10−6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='3 × 10−3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='3 × 10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7 × 10−4  U0 exp−2(ϵ+ˆσzλ)2/ω2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Thus, the misalignment has two ef-  fects: (i) it changes the tweezer potential at the center of the  tweezer and therefore the phase accumulation in the phase  gate, and (ii) it shifts the potential in a qubit-state-dependent  way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The second contribution is corrected to lowest order by  a spin-echo sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Figure 2(b) shows the infidelity as  a function of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Here we assume that the tweezers are mis-  aligned on both ions in the same way which seems the experi-  mentally most likely case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The unitary U(0, τ) leads to phase  space trajectories for ⟨x(t)⟩ and ⟨px(t)⟩ associated with the  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' motion[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' As expected, we find approximately cir-  cular phase-space orbits for the even parity states |00⟩, |11⟩,  and very little motion for the odd parity ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We see that ev-  ery state combination leads to ion motion, but the difference  in motion still leads to a high fidelity of ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='999 as shown in  Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Clock state case – While the calculation was performed  for the electron spin qubit states in 174Yb+, it should also  be possible to use the hyperfine clock states |F = mF = 0⟩  and |F = 1, mF = 0⟩ in 171Yb+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This qubit is insensitive to  magnetic field noise and coherence times of up to an hour have  been measured [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In this case, the tweezers are formed by a  bichromatic co-propagating laser field detuned by ∆ from the  D1 transition at 369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='5 nm with overall detuning ∆ ≪ ωFS, the  fine structure splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We set the frequency difference in the  4  bichromatic tweezer to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='6 GHz, corresponding to the tran-  sition between the qubit states [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The tweezer laser then  causes Raman coupling between the qubit states via two dis-  tinct paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In the first path, the qubits are coupled via the state  |P1/2, F = 1, mF = −1⟩ due to the σ− polarization compo-  nent in the tweezer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In the other, the qubits are coupled via  the state |P1/2, F = 1, mF = +1⟩ due to the σ+ component  in the tweezer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We denote the Rabi frequencies of each path  as Ω±  1,2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The corresponding Raman couplings of each path  interfere destructively in the center of the tweezer due to a rel-  ative minus sign between Ω+  1 (x) and Ω+  2 (x) in their Clebsch-  Gordan coefficient, ∝ (Ω−  1 (0)Ω−  2 (0)+Ω+  1 (0)Ω+  2 (0))/∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  However, the Magnus effect causes a strong position depen-  dence of the relative strength of both paths of magnitude  Ωeff(x) = Ω−  1 (x)Ω−  2 (x)  ∆  + Ω+  1 (x)Ω+  2 (x)  ∆  ≈ Ω2  ∆  4λx  w2  0  ,  (7)  where we assumed x ≪ λ ≪ w0 and |Ω±  i (0)| = Ω/  √  2 with  i = 1, 2, such that both laser frequencies have the same power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  As a result, a qubit state-dependent force appears as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  (4), except that we must now replace ˆσ(i,j)  z  → ˆσ(i,j)  x  and the  gate takes the form of the usual Mølmer-Sørensen interaction  ∝ ˆσ(i)  x ˆσ(j)  x  [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Amplitude modulation via A(t) allows again  for resonant enhancement of the gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  In addition to the Raman coupling, we obtain a tweezer po-  tential (AC Stark shift) for each qubit state of magnitude  δ|k⟩  AC(x) =  �  i=1,2  �  j=+,−  |Ωj  i(x)|2  ∆i,|k⟩  (8)  with ∆1,|0⟩ = ∆ − ωq, ∆2,|0⟩ = ∆, ∆1,|1⟩ = ∆ and ∆2,|1⟩ =  ∆ + ωq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This causes an additional trapping potential Φ(x) ≈  1  2mω2  twx2 that is independent of the qubit state as before, as  well as a position-dependent differential Stark shift δAC(x) =  δ|1⟩  AC(x) − δ|0⟩  AC(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In the limit ωq ≪ |∆|,  δAC(x) ≈ − ωq  ∆2  �  i=1,2  �  j=+,−  |Ωj  i(x)|2  (9)  = −ωq  ∆  ˜U0(x)  (10)  This differential Stark shift is estimated to be small, δAC/2π ≈  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7 kHz for the numbers used in the simulations, and can be  compensated by a corresponding Raman detuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Photon scattering on the D1 transition can be estimated as  γph ∼ ˜U0Γ/(ℏ∆) ∼ 13 s−1 with Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='23 × 108 s−1 in  Yb+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This adverse effect may be reduced significantly by em-  ploying hollow tweezers [11, 25, 26] at the expense of added  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' For a hollow beam with a waist w0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='5 µm  and ∼ 160 µW we obtain a reduction in scattering rate of  ∼ 10−6 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' As long as ωtw ≪ Ωrf, the drive frequency of the  Paul trap, no parametric excitations can occur and micromo-  tion of the ions is not a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Other errors, such as due to  intensity noise of the laser, heating of the ions due to electric  field noise and decoherence due to magnetic field noise have  the same effect as in other gate implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Finally, we  note that because the tweezers are far detuned from the closest  transitions, the exact overall frequency of the tweezer laser is  irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Conclusions In conclusion, we have described a novel  type of quantum phase gate based on the optical Magnus ef-  fect using optical tweezers in a linear chain of trapped ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  The main benefit is that the gate does not require counter-  propagating laser fields, greatly simplifying the setup and  eliminating errors due to phase instabilities between the gate  laser beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Furthermore, the state-dependent force gener-  ated by the Magnus effect allows to perform the gate by cou-  pling to motional modes on the plane perpendicular to the di-  rection of propagation of the tweezers allowing novel experi-  mental implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The proposed gate does not require  ground state cooling and can perform a quantum logic gate  on any pair of ion qubits by spatial addressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The expected  gate fidelity rivals the state of the art also for ions that are not  cooled to the ground-state of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  This work was supported by the Netherlands Organiza-  tion for Scientific Research (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  680.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='120 and  680.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='05, (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='N is supported by the Dutch Re-  search Council (NWO/OCW), as part of the Quantum Soft-  ware Consortium programme (project number 024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='037).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Ballance, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Harty, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Linke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Sepiol, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lucas,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 117, 060504 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Gaebler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Tan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Wan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Bowler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Keith,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Glancy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Coakley, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Knill, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Leibfried, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Wineland,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 117, 060505 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Léséleuc, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lienhard, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lahaye, and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' Bernien, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' Levine, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' Senko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Vuletic, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' Young, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Kaufman, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' Keesling, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Semeghini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' Bernien, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Greiner, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' Rajabi, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Islam,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Schüssler, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Wu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Corboz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Gerritsma, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Safavi-Naini,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Schüssler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Arias Espinoza, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Wu,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Gerritsma, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Safavi-Naini, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Cirac and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Cirac, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Zoller, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' Zhuang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' He, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' Sheng, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Xu,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+page_content=' Wolf, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Schmiegelow, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Schmidt-  Kaler, arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='07095 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Supplementary material for :  Trapped Ion Quantum Computing using Optical Tweezers and the Magnus Effect  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Mazzanti,1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Gerritsma,1, 2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Spreeuw,1, 2 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Safavi-Naini2, 3  1Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, 1098 XH Amsterdam, Netherlands  2QuSoft, Science Park 123, 1098 XG Amsterdam, the Netherlands  3Institute for Theoretical Physics, Institute of Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands  (Dated: January 13, 2023)  APPENDIX I : OPTICAL MAGNUS EFFECT  A key characteristic of a tightly focused beam is the strong  field curvature near the focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This not only affects the local  intensity but also its polarization structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' To calculate this,  we take a superposition of plane waves labeled by their wave  vector in spherical coordinates, k = (k, θ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Taking k =  ω/c as fixed we write  E(r) ∝  � 2π  0  dφ  � π  0  dθ sin θ ux(θ, φ) a(θ, φ) eik·r  with ux(θ, φ) a polarization vector obtained by co-rotating  the x unit vector when k is rotated from z to (θ, φ), such  that ux(θ, φ) · k = 0, see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' [S1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In the calcula-  tion we center the beam around θ = 0, and the focal plane  is given by r = (x, y, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  The shape of the beam is de-  termined by the amplitude function a(θ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' For a Gaussian  beam we set a(θ, φ) = exp(−θ2/w2  θ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' for the lowest or-  der (l = 1) Laguerre-Gaussian (LG) beam we set a(θ, φ) =  θ exp(iφ − θ2/w2  θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' After performing the above integral we  rotate the results for tweezers propagating along the −y di-  rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Finally, the circular field components σ± shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 1 of the main text are obtained as the projection onto unit  vectors (x ± iy)/  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In Figure S-1, all three polarization  components for a Laguerre-Gaussian beam are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Note  that the σ− and σ+ components have similar intensity while  the π-polarization is suppressed by a factor ∼ 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  −6 −4 −2 0  2  4  6  x/λ  −4  −2  0  2  4  z/λ  (σ−)z  −6 −4 −2 0  2  4  6  x/λ  −4  −2  0  2  4 (π)z  −6 −4 −2 0  2  4  6  x/λ  −4  −2  0  2  4 (σ+)z  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' S-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Intensity of the polarization components for a LG beam  calculated at the focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The π-polarization component has been en-  hanced by a factor 100 to make it visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Here we set wθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='6  APPENDIX II : PHASE-SPACE DYNAMICS  We study the phase-space dynamics of the ions by simulat-  ing the time dependent Hamiltonian using trotterization with  time-steps of 10−4 τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' At each time-step we evaluate the ex-  pectation value of the ⟨ˆx⟩ and ⟨ˆp⟩ for the center of mass mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  As expected, we find approximately circular phase-space or-  bits for the even parity states |00⟩, |11⟩, and very little motion  for the odd parity ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' S-2 it is possible to see the evo-  lution in phase-space for all the four spin states in case of per-  fectly aligned and slightly misaligned tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' As described  in the main text we simulate numerically the full Hamiltonian  defined as ˆHsim = ˆH0 + ˆU (xi) + ˆU (xj) where in case of  misalignment ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' ˆU (x) reads as :  U(x) ≈ −U0 e−2((ˆx−ˆϵ)+ˆσzλ)2/w2  0  ≈ − ˜U0 + 4 ˜U0  ˆσzλ − ˆϵ  w2  0  ˆx  + 1  2  ˜U0  �  4  �  w2  0 − 4λ2�  w4  0  �  ˆx2 − 1  2  ˜U0  �  16  �  ˆϵ2 − 2ˆσzˆϵλ  �  w4  0  �  ˆx2  −  �  8 ˜U0ˆσzλ3w2  0 − 4  �  3ˆϵ2 + λ2�  3w6  0  �  ˆx3  +  �  8 ˜U0ˆϵ3w2  0 − 4  �  ˆϵ2 + 3λ2�  3w6  0  �  ˆx3  −  �  2 ˜U0ˆσzλˆϵ−48w2  0 + 64  �  ˆϵ2 + λ2�  3w8  0  �  ˆx4  +  �  2 ˜U0  3w4  0 − 24w2  0  �  ˆϵ2 + λ2�  + 16  �  ˆϵ4 + 6ˆϵ2λ2 + λ4�  3w8  0  �  ˆx4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  with  ˜U0 = U0e−2(ˆϵ+ˆσzλ)2/w2  0  A small tweezer misalignment ϵ gives rise to new spin-  dependent terms in the Hamiltonian that shift the trapping po-  tential in a state dependent way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='S-2 is shown how the  dynamics is affected in the case where the tweezers are mis-  aligned by 30 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  APPENDIX III : GATE FIDELITY  We characterize the gate by calculating the average process  fidelity as follows : [S2]:  ¯F( ˆUid, ˆU ˆ  Hsim) =  �  j tr  �  ˆUidˆσ†  j ˆU †  idˆσj( ˆU ˆ  Hsim)  �  + d2  d2 (d + 1)  ,  2  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='4  ⟨ˆx⟩  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='4  ⟨ˆpx⟩  ϵ = 0  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='4  ⟨ˆx⟩  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='4  ⟨ˆpx⟩  ϵ = 30 nm  ψ↑↑  ψ↓↑  ψ↑↓  ψ↓↓  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' S-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Center of mass mode phase-space dynamics for perfectly  aligned tweezer (left) and for 30 nm misaligned ones (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' For the  simulation we used the same parameters as for τ/2 = 120 µs point  in Figure 1(a) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  where ˆUid is the unitary of an ideal geometric phase gate and  ˆσj( ˆU ˆ  Hsim) ≡ trFS( ˆU ˆ  Hsim [|n⟩⟨n| � ˆσj] ˆU †  ˆ  Hsim) projects the  unitary matrix generated by the time evolution of the Hamil-  tonian used for the simulations ˆU ˆ  Hsim on the Fock state |n⟩  and on a d-dimensional representation Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [S1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Spreeuw, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 125, 233201 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [S2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Nielsen, Physics Letters A 303, 249 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
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+Cinematic Techniques in Narrative Visualization
+Matthew Conlen
+Our World in Data
+matt.conlen@ourworldindata.org
+Jeffrey Heer
+University of Washington
+jheer@uw.edu
+Hillary Mushkin
+California Institute of Technology
+hmushkin@caltech.edu
+Scott Davidoff
+Jet Propulsion Laboratory
+California Institute of Technology
+scott.davidoff@jpl.nasa.gov
+ABSTRACT
+The many genres of narrative visualization (e.g. data comics, data
+videos) each offer a unique set of affordances and constraints. To
+better understand a genre that we call cinematic visualizations—3D
+visualizations that make highly deliberate use of a camera to convey
+a narrative—we gathered 50 examples and analyzed their traditional
+cinematic aspects to identify the benefits and limitations of the form.
+While the cinematic visualization approach can violate traditional
+rules of visualization, we find that through careful control of the
+camera, cinematic visualizations enable immersion in data-driven,
+anthropocentric environments, and can naturally incorporate in-
+situ narrators, concrete scales, and visual analogies. Our analysis
+guides our design of a series of cinematic visualizations, created for
+NASA’s Earth Science Communications team. We present one as a
+case study to convey design guidelines covering cinematography,
+lighting, set design, and sound, and discuss challenges in creating
+cinematic visualizations.
+1
+INTRODUCTION
+Within narrative visualization [57], researchers have identified gen-
+res (such as data comics [3] and data videos [1]) that help better
+unpack and situate their specific application and the features that
+they employ. cinematic visualizations embed data into a three-
+dimensional, time varying scene, utilizing one or more cameras to
+direct the relationship between a viewer and the scene to tell a dra-
+matic data-driven story. This cinematic approach is different from
+the one typically used in information visualization, where graph-
+ics are reduced to a minimal form, incorporating only essential
+elements like axes and data-driven marks [64]. Cinematic visual-
+izations are more maximal: non-data marks are not compressed or
+reduced, instead entire digital worlds are built up around data points
+and included in the visible frame. This technique allows viewers
+to feel present in locations augmented with data-bound objects,
+known as data visceralizations [41]. Narrative documentary visual-
+izations [10] can be produced through the careful editorial direction
+of the cinematography, editing, mise-en-scène, and sound [5].
+Through an analysis of 50 existing cinematic visualizations, we
+identified four salient techniques (in-situ narrators, resolution of
+scale, anthropocentric perspective, and story-driven cameras) that
+cinematic visualizations employ to dramatically engage their au-
+dience through emotionally resonant data-stories. We show how
+these techniques are used throughout the examples analyzed, dis-
+cuss constraints associated with them, and reason about why cine-
+matic visualizations may be effective despite the known pitfalls of
+3D visualization.
+Using the lessons learned from this formal analysis, we produced
+a web-based article containing a series of cinematic visualizations
+relating to climate change, which was published by NASA’s Earth
+Science Communications team 1. We contribute the design process
+for one of these visualizations as a case study, presenting design
+artifacts that were created during our process (both successful
+and unsuccessful), and provide concrete guidelines for designers
+of cinematic visualizations. Our analysis and design artifacts are
+available at https://cinematic-visualization.github.io/.
+2
+RELATED WORK
+Narrative visualizations are used to improve memorability [7, 8],
+to instill empathy or emotion [9], to frame a message [33], and to
+improve engagement [19, 28]. Segel & Heer [57] provided an ini-
+tial characterization of the design space of narrative visualizations,
+which was later elaborated to include additional techniques [60].
+Hullman et al. [34] focused on the role of sequence in narrative
+visualization, characterizing a set of transition types and other high
+level strategies for sequencing visualizations. Tools have been cre-
+ated to support narrative visualization authoring [2, 11, 18, 56], and
+a small number of empirical evaluations of narrative visualizations
+have been conducted [9, 19, 46, 69]. Further work has investigated
+specific genres of narrative visualization such as data comics [3],
+and new genres have emerged beyond Segel & Heer’s initial set,
+such as “scrollytelling.” Here we add to the ongoing conversation
+around narrative visualization by identifying another such genre:
+cinematic visualization. Kosara & McKinley [39] identified the op-
+portunity for narrative visualization researchers to learn from other
+disciplines that engaged heavily with storytelling and multimedia,
+this paper draws on film art scholarship, incorporates a formal
+system of cinematic style into our analysis, discussion, and design
+of cinematic visualizations.
+2.1
+Data Videos & VR
+Data videos were included in the initial set of genres put forth by
+Segel & Heer [57] and first studied closely by Amini et al. [1]. Not
+all data videos are cinematic visualizations (for example, we do
+not consider a video consisting of a sequence of two-dimensional
+infographics to be cinematic), and not all cinematic visualizations
+1https://climate.nasa.gov/news/2933/visualizing-the-quantities-of-climate-change/
+arXiv:2301.03109v1  [cs.HC]  8 Jan 2023
+
+, ,
+Conlen, et al.
+Figure 1: The Dangers of Storm Surge (CV42) is a mixed reality video produced by the Weather Channel. The video opens with
+a close up shot of a news anchor wearing a rain jacket, standing in front of a house (1A). There are audible sounds of rain
+under the anchor’s voice and water dripping down the windows of the house. The camera pulls back revealing that the live
+anchor is being composited into a 3D scene of a suburban neighborhood during a storm surge (1B). There are very few data
+points actually encoded as visual elements. The piece simply shows water rising from zero, to three, to six, to nine feet (1C-D)
+as the anchor narrates with details in reference to the danger of storm surge associated with hurricanes.
+are data videos (for example one in which the visualization is deeply
+tied to the text of an interactive news article). While Amini et al.
+were primarily concerned with the narrative structure and attention
+cues of data videos, we additionally consider the visual and auditory
+style of cinematic visualizations in detail. Under our formal style
+system, our analysis of editing is most closely related to Amini’s
+work, however that is only one of four dimensions we consider.
+Bradbury & Guadagno [10] studied viewer preferences in docu-
+mentary narrative visualization (a subgenre of data videos in which
+data is presented using the techniques of documentary film), and
+found that audiences may prefer when documentary data videos
+include voice-over narration and on-screen narrators. We build on
+their analysis of the use of narrators and narration, in particular
+during our discussions of in-situ narrators that interact with data-
+bound objects digitally rendered into the space around them, and
+of the use of sound in cinematic visualizations. Video producers
+have extended the traditional documentary visualization format to
+enable interactivity such as user selected paths through the content
+and manipulable graphics [29, 63].
+Immersive data stories [35] have been discussed within the
+emerging field of immersive analytics [44] and have been shown
+to allow viewers to examine data at multiple scales, support im-
+mersive exploration, and create affective personal experiences with
+data [36]. Lee et al. [41] introduced data visceralizations, where
+physical quantities are visualized in 3D virtual reality scenes. This
+paper helps to bridge the gap between data visceralization and nar-
+rative visualization by showing how cinematic techniques can be
+used to create author-guided narrative visualizations using data vis-
+ceralizations. Cinematic visualizations similarly attempt to immerse
+viewers and create emotionally resonant experiences, although in
+contrast to immersive visualizations they are typically viewed on a
+standard 2D screen with limited (or no) user control of the camera.
+There are several toolkits for creating immersive data visualizations
+and data stories on augmented reality devices [21, 55, 59].
+2.2
+3D Computer Graphics
+Animation [62] has been a partner discipline with visualization for
+some time. Classic principles of animation [61] have been adapted
+for digital usage [40] and subsequently for information visualiza-
+tion [30]. With realistic camera models [38] and improving render-
+ing capabilities [20] digital animation became a tool to create Holly-
+wood films [31]. While 3D graphics have been used in visualization
+to limited success, e.g., to display hierarchical information [17], the
+use of 3D graphics in information visualization is often avoided.
+A broad body of research documents potential pitfalls, including
+that volume is not a perceptually effective encoding channel [16],
+and that 3D projections introduce distortion and occlusion [67].
+We find that designers of cinematic visualizations may intention-
+ally use suboptimal encodings in support of more visceral [41] and
+emotionally resonant [28] graphics.
+The use of 3D does find more regular application in scientific visu-
+alization [4, 65], including its use in storytelling [22, 43]. Borkiewicz
+used the term cinematic scientific visualization [6] to refer to a class
+of narrative data videos that focus on scientific data. Here we use
+cinematic visualization in a similar way but do not restrict the data
+to be strictly scientific or inherently spatial. Unlike Borkiewicz,
+our description encapsulates visualizations which are not embed-
+ded in films, but may be, for example, displayed as an animation
+accompanying a news article.
+2.3
+Film Art
+Bordwell and Thompson [5] define narrative and style as the two
+major formal systems of film. While prior work has examined se-
+quence [34] and narrative structure & attention cues [1] in data
+videos, we observe that cinematic style has far less visibility in the
+critical vocabulary of data visualization. Style plays a crucial roll
+in filmmaking, enabling directors to “confirm our expectations, or
+modify them, or cheat, or challenge them. [...] A director directs
+not only the cast and crew. A director also directs us, directs our
+attention, shapes our reaction.” [5] This paper brings Bordwell and
+Thompson’s formal system of cinematic style into the world of data
+visualization, and uses it to examine how narrative visualizations
+borrow techniques from cinema while departing from many of the
+traditional practices advocated by visualization research.
+Style consists of four features, which together make up a film’s
+style, each now briefly described. Mise-en-scène refers to every-
+thing that is seen in the frame, including lighting, actors, objects,
+backdrops, and so on [27]. Cinematography refers to the use of the
+
+Cinematic Techniques in Narrative Visualization
+, ,
+Figure 2: (A) In [REALISTIC] Elephant rocket fuel - Saturn V (CV29), a model Saturn V rocket takes off, however, instead of
+flames exiting the bottom of the spacecraft, elephants are expelled, the number of elephants represents the corresponding
+mass of fuel. This video may not make for a particularly effective visualization in terms of conveying precise quantities, but
+the style successfully uses humor in order to call attention to the fact that rocket launches use a quantity of fuel so great
+it is appropriate to measure it in terms of dozens of elephants. (B) In Here are 120 million Monopoly pieces, roughly one for
+every household in the United States (CV6) by the New York Times the pile of Monopoly pieces is first seen from a far, before
+the reader scrolls down the page to trigger the camera zooming in to the very top of the pile, dramatically revealing what a
+disproportionately small portion of families provide most political funding.
+camera, how shots are composed and framed [26]. By placing ele-
+ments at specific locations within the frame, they can be perceived
+either as the subject or the background of the image [25]. Editing
+is the composition of multiple pieces of footage in time or space,
+creating transitions between perspectives and scenes [54]. Sound
+is the audio used, whether it be music, voice over, or sounds from
+characters or objects on screen [32]. Our analysis of cinematic visu-
+alization identified techniques along these dimensions of style that
+designers can use to enhance their presentation of data narratives.
+3
+CINEMATIC VISUALIZATION SURVEY
+We collected cinematic visualization to analyze by surveying liter-
+ature on narrative visualization [1, 6, 34, 35, 43, 57, 60], browsing
+the information visualization awards website Information is Beau-
+tiful [45] and the PacificVIS storytelling contest [50], and searching
+for news articles, blog posts, conference talks, and videos which
+were described using combinations of the keywords cinematic, data,
+data video, dataviz, datavis, visualization, news, newsgames, immer-
+sive, mixed reality, 3d, and video. We searched the portfolios of the
+creators of the visualizations found initially and their collaborators.
+A full list of the cinematic visualizations can be seen in Figure 7 in
+the appendix of this paper; we refer to these studies by identifiers
+throughout the paper (e.g., CV4 refers to the fourth example in the
+table). Our analysis considered 50 cinematic visualizations. While
+the corpus is not exhaustive, the examples expose the variety of
+media (interactive news articles, YouTube videos, and TV segments)
+which cinematic visualizations occupy and the messages that they
+deliver. The examples visualized a broad range of data types, in-
+cluding datasets both with and without physical and geographic
+dimensions.
+Rather than empirically evaluate specific design patterns utilized
+in the visualizations, we turn to the means of understanding plot
+devices [57], sequencing [1], and film style [5]. We analyzed the
+style of each example along the dimensions of mise-en-scène, cine-
+matography, editing, and sound using the 4-step analysis process
+described by Bordwell and Thompson [5], a canonical method of
+film analysis. For each example we first identified the main com-
+municative goals of the visualizations, and then studied the salient
+techniques applied within the mise-en-scène, cinematography, edit-
+ing, and sound which supported these narrative goals. We then used
+iterative coding to categorize the salient techniques used across
+the examples. Usage of these techniques are shown in Figure 7, for
+example we recorded many ways in which a viewer’s attention
+is guided (through color, light, annotations, and narrators in the
+mise-en-scène) and use of cinematographic techniques like point-
+of-view perspective and user-controlled cameras. The table shows
+that the medium of the cinematic visualization has some impact on
+the techniques used, for example cinematic visualizations embed-
+ded in online articles rarely use sound, but often utilize user-paced
+segments, while those presented as videos make heavy use of sound.
+
+A
+B
+Here are 120 million
+ Monopoly pieces, roughly
+ one for every household
+in the United States.
+Just 158 families have
+provided nearly half of the
+early money for efforts to
+capture the White House., ,
+Conlen, et al.
+Figure 3: VFX Artist Reveals the True Scale of the Universe fea-
+tures a live-action narrator alongside scaled-down 3D mod-
+els of celestial bodies.
+3.1
+Design Techniques
+Through this analysis we identified salient recurring techniques that
+were frequently applied to support the communicative goals of the
+visualization, including the use of in-situ narrators, anthropocentric
+perspective, resolution of scale, and story-driven cameras.
+In-situ narrators mediate interactions with diegetic data.
+Perhaps the most novel technique that we identified in cinematic
+visualizations is the use of in-situ narrators, in which the mise-en-
+scène contains a character that interacts directly with on-screen,
+diegetic data.2 In contrast to traditional documentary visualization
+narrators who might participate from off-screen (“voice of god”) or
+refer to data visualizations rendered as two-dimensional holograms
+or composited over top the video [10], in-situ narrators are under-
+stood by the viewer to be able to see and interact with the diegetic
+data either through the use of superimposed data visceralizations
+2Something which is diegetic exists in the same universe as the characters on screen; we
+use the phrase diegetic data to refer to data-driven elements which are part of—rather
+than composited over—the scene shown in the frame.
+(CV35, 40, 42, 43) or, in one case, data physicalization [37] (CV41).
+This (typically) mixed reality environment serves an important
+role for narrative visualization, allowing the on-screen narrator to
+mediate interactions between the audience and the graphics, letting
+them provide additional context and push the storyline forward.
+These narrators, essential components of the mise-en-scène, can
+also help concretize a visualization’s anthropocentric perspective,
+reinforcing the idea that data is being displayed at a human scale.
+In The Dangers of Storm Surge (CV42), one exemplar of this
+technique (Figure 1) produced by the Weather Channel, a news
+anchor wearing a blue jacket explains the dangers associated with
+flooding due to storm surge. The graphics are coordinated with
+the narrator’s script and appear to respond to his dialogue, the
+composition of the frame inviting comparison between the man and
+the height of the water. The narrator is the primary subject from the
+start of the clip, positioned centrally in frame and maintaining focus
+due to visual cues like his bright blue coat, the circular platform
+upon which he stands, and the shot composition. To call attention
+to the water’s height at certain key moments, a brightly colored
+annotation is projected onto the crest of the surge.
+An anthropocentric perspective transports viewers and
+enables drama. One notable aspect of cinema is how the camera is
+able to transport the audience into the scene: people watching sus-
+pend disbelief [24] to allow themselves to wholeheartedly imagine,
+or “believe”, that they are in the scene, seeing things through the
+camera lens. That is, the camera’s perspective becomes the viewer’s
+point of view, they are one and the same. The height, angle, and
+distance of a camera in relation to objects in the scene all play a role
+in how a viewer will interpret and respond to the frame that they
+ultimately see [5]. When a camera is placed high above a setting,
+the viewer feels like they are also high above it. When a camera
+is placed at eye level, a viewer feels as if they are standing there
+watching the subject. For example, both CV1 and CV26 utilize unit
+visualizations and concrete scales to visualize quantities in relation
+to the size of Manhattan, but each uses perspective to impact the
+viewer’s experience in a different way. In CV1 the data being dis-
+played (plastic bottle usage) is not directly related to the locations
+being used as concrete scale referents, and an overview shot is
+used, letting the viewer absorb the scale of the data rather than
+the details and textures of the city itself. In contrast, CV26 begins
+with a shot from a camera placed at eye-level, looking at several
+of the city’s ubiquitous yellow taxis, transporting viewers to the
+city at street level, and forcing them to reckon with the data being
+displayed (New York City’s annual green house gas emissions) in a
+much more visceral way [41].
+Some cinematic visualizations place the camera perspective
+somewhere that is humanly impossible. However, if the audience
+suspends disbelief, the camera can carry the viewer through these
+otherwise inaccessible spaces, for example, CV12 shows an anima-
+tion of the Cassini spacecraft as it orbited and eventually crashed
+into Saturn. Choice and Chance (CV11), visualizes the events of the
+2016 Pulse night club shooting in Tampa Bay, positions a camera
+looking “through” the roof of a nightclub. Because the scene is
+shot using a digital model instead of a real location, the roof of the
+club can simply be removed and problems of occlusion go away.
+Changing perspectives can also shift the subject of the scene or
+add emotional content, for example, when the camera moves to
+
+A
+SUBSCRIBE
+B
+c
+Rige!Cinematic Techniques in Narrative Visualization
+, ,
+Figure 4: New York City’s greenhouse gas emissions as one-ton spheres of carbon dioxide gas, a cinematic visualization produced
+by Carbon Visuals and released online. The cinematic visualization uses a variety of different camera views, along with stark
+colors to guide viewers through an explanation of the scale of the city’s greenhouse gas emissions. The number of instances
+of the blue sphere is driven by the rate of emissions. As this number grows the city buildings serve as a concrete scale.
+reveal something that wasn’t already in the frame, the audience
+experiences seeing it for the first time. In Choice and Chance the
+camera moves to different vantage points throughout the model as
+the story progresses. The camera remains in an overview shot for
+the majority of the article, but moves to ground level at the climax,
+elevating the intensity of the shot by placing the viewer into the
+perspective of a bystander.
+Author-defined camera trajectories can be played, paused,
+and (lightly) modified by viewers. The cinematic visualizations
+that we analyzed tended to use author-driven narrative structures [57],
+with most user interactions consisting of the user clicking or scrolling
+to trigger the visualization to continue to the next stage (e.g., CV2,
+5-17, 21-22). Operationally, this requires animating the position
+and orientation of a digital camera model along a track specified
+by the author, and has been used heavily by cinematic visualiza-
+tions embedded in articles (16 out of 22). The other way in which
+(constrained) interactivity was employed was allowing the manipu-
+lation of 3D models. In most cases this means the user can position
+the camera at a particular location around the model (see CV17 for a
+stereotypical example). These models might be scientific (CV13,17)
+or cultural (CV5) objects that would be otherwise inaccessible to
+the audience viewing the visualization. It is common for orbital
+cameras to be used, constraining the camera’s focus to remain on
+a particular object of interest while allowing the user to exercise
+control over viewing angle and zoom level (Fig. 7D). Cinematic
+visualizations that support these interactions must be rendered in
+real-time, limiting the fidelity at which the models may be rendered.
+Visualization techniques are combined toward resolution
+of scale. While we traditionally think of 3D graphics as ineffective
+for encoding quantities [16], a recurring theme in our examples is
+the use of 3D graphics to visualize and communicate quantities of
+a massive scale (e.g., CV1, 6, 8, 26-28). Quantities at a scale beyond
+what we experience in daily life (i.e. hyperobjects [47]), like amount
+of carbon dioxide emitted from NYC annually (CV26), may be es-
+pecially difficult for people to picture because we rarely, if ever,
+interact with quantities of such a size. Cinematic visualizations can
+convey a quantity of scale in a concrete and affecting way by using
+cinematography to establish the viewer’s point of view from the
+ground, a position which often serves as the implicit zero point
+of a y-axis. We observed that several visualization techniques are
+naturally expressed in cinematic visualizations, including data vis-
+ceralizations [41], unit visualization [51] and concrete scales [14].
+For example, in CV27 the viewer sees a city park, including trees,
+people standing in a grassy field, and a ten meter tall blue sphere
+representing the actual size of one metric ton of CO2 (data vis-
+ceralization). As the scene progresses, many more spheres appear,
+each representing one metric ton of CO2 (unit visualization), until
+so many appear that the camera must zoom out, above the park,
+observing the growing pile of spheres in comparison to the city
+buildings (concrete scale).
+Objects which are used as backdrops—for example a city skyline
+(CV11) or parked car (CV42, Fig. 1)—may serve double duty as
+concrete scale referents and contextual elements. The use of 3D
+graphics affords designers the ability to use concrete scales (CV1,
+26) and visual analogies (CV29, 36) to (re-)contextualize the size of
+objects, and digital sets are constructed to facilitate comparisons
+that are impossible to make directly in the physical world (CV1,
+27) and use point-of-view perspective to impart a visceral sense of
+magnitude. The visual medium is rich with possibilities for analogy.
+For example, in [REALISTIC] Elephant rocket fuel - Saturn V (CV29,
+Fig. 2), designer Maxim Sachs renders the launch of the Saturn V
+rocket, except that the rocket expels elephants behind it as it travels,
+rather than exhaust. The elephants represent the mass of fuel that
+is being expended. By juxtaposing these images, Sachs is able to re-
+frame an abstract quantity of rocket fuel in terms that people may
+have more familiarity with, and do it with a sense of humor that
+may make the visualization overall more memorable or engaging
+for its audience [8]. In a more typical case, the narrator of CV40
+asks the audience to imagine if Earth were the size of a tennis ball,
+and then, using this new scale, shows the relative size of different
+planets, moons, and stars. These planets are compared against one
+another, rendered into real-world footage including a narrator who
+provides guidance and relevant facts about the celestial objects.
+
+, ,
+Conlen, et al.
+Figure 5: How Much is a Gigatonne? shows one gigatonne of ice in Central Park, New York. A digital set (A) is designed including
+multiple cameras, lighting, and data-driven and contextual elements. Footage from the various cameras is composed to create
+the final sequence (B-E). This was one of several videos that we developed for an article published on NASA’s climate website.
+View the full videos at https://cinematic-visualization.github.io/.
+They are shown embedded into several settings, for example an
+office, a Los Angeles street, and the New York City skyline.
+3.2
+Constraints
+The time-based format does not support a high data density.
+Traditional information graphics often present a data-dense display
+with minimal “non-data ink” [64] to remove possible distractions
+and optimize the display for tasks such as value look-up and com-
+parison. In some cases, designers may choose to add additional
+illustrative features to increase the memorability of the visualiza-
+tion [7]. In contrast, cinematic visualizations utilize diegetic data,
+embedded in a three dimensional scene with other elements which
+contextualize the scene (see CV35 for a striking example). In cine-
+matic visualizations (e.g. CV40,42) the elements surrounding the
+data fulfill a dual role as both data and non-data ink; they add
+spatial presence to the visualization [12], supporting a sense of
+transportation to the virtual world for viewers, while simultane-
+ously serving as guides and axes, points of reference for concrete
+scales [14]. Rather than densely packing data, we see that cinematic
+visualizations often only show one or a few data points in the frame,
+favoring to include additional contextual elements that help add
+emotional resonance to the data-story being told.
+Designers trade-off between perceptual effectiveness and
+dramatic narrative. Visualizations that employ 3D graphics are
+often ineffective perceptually. These graphics may use sub-optimal
+encoding channels like volume and can further bias judgement
+through distortion and occlusion. Cinematic visualizations are not
+appropriate when the task is centered around value judgements.
+Instead, we see cinematic visualizations effectively used when a
+rough estimate of values is sufficient and the precise value is not
+of central importance (e.g. CV29). Many of the cinematic visual-
+izations that we analyzed use a volume encoding to display data
+(CV1,6,26,27,35). Volume is a less effective encoding channel com-
+pared to position and may cause the audience to misestimate the
+true quantity. This trade-off may be acceptable depending on the
+data being presented and the precision with which the author hopes
+it will be apprehended.
+4
+CASE STUDY: HOW MUCH IS A
+GIGATONNE?
+We collected and studied the aforementioned cinematic visualiza-
+tions while exploring designs to support the communication ob-
+jectives of NASA’s Earth Science Communications Team. Climate
+change is a complex, multi-faceted issue of global importance [49]
+and the team is tasked with maintaining climate.nasa.gov, a website
+that tracks vital statistics about Earth’s climate, and delivers up-
+dates about global warming to a diverse global audience of millions
+of readers. The team uses traditional information graphics [48], as
+well as narrative visualizations (e.g., [53]), to highlight how scien-
+tists know that anthropogenic global warming is truly happening,
+what changes have taken place in Earth’s climate so far, and why it
+is an important topic for readers to understand even if it does not
+seem to be affecting them. However, the team sought data-driven
+stories that more viscerally engaged their audience and connect
+
+Digital set design
+Cam1 (God's eye view)
+D
+Cam2 (bird's eye view)
+Lighting: Global Illumination
+Data-driven element
+Geographic elements
+Cam3 (point-of-view)
+Texture from satellite images
+A
+Rendered output
+B
+Central Park
+D
+C
+ewYorkCit
+God's eye view (Establishing shot)
+Point-of-view (Establishing)
+Point-of-view (Initial action)
+Medium-long shot (Peak)Cinematic Techniques in Narrative Visualization
+, ,
+Figure 6: We explored many different designs, these were left on the cutting room floor. The designs were dropped for reasons
+including poor perceptual effectiveness (A-C), locations too small for the scale of the data (D-F), and designs too illustrative
+and not physically accurate enough (G-H). It was particularly difficult to identify locations that were broadly recognizable
+from a 3D reconstruction but also suitable to server as a concrete scale referent.
+the planetary scale data of climate change to a human scale that
+readers can readily understand.
+Within the domain of climate change communication is a range
+of research investigating how to effectively communicate the latest
+science to a broad audience. High level principles of climate change
+communication have been synthesized by the Center for Research
+on Environmental Decisions [58]. We think cinematic visualizations
+are well suited to satisfy principles “Get Your Audience’s Attention“
+and “Translate Scientific Data Into Concrete Experience.” Here
+we describe how our work creates connections between ongoing
+investigations in narrative visualization, computer graphics, and
+film art to achieve this.
+Guided by editorial priorities set by NASA’s Earth Science Com-
+munication team, we produced an article consisting of a several
+cinematic visualizations to communicate massive quantities related
+to climate change. We endeavoured to make them interpretable and
+meaningful to a broad public audience. These visualizations were
+eventually published to an audience of millions. Here we describe
+our design process to create cinematic visualizations, identifying a
+general workflow of use to practitioners who wish to create this
+type of visualization themselves, and to tool-builders who wish to
+provide better support for authoring cinematic visualizations in
+the future. As with visualization production in general, these steps
+are not necessarily linear; rather, the process is iterative and error
+prone, and may require going back to earlier steps if it becomes
+apparent that a design is not working. We experienced many failed
+attempts (see Figure 6) before arriving at our final designs.
+4.1
+Pre-Production
+Narrative. Quantities of ice loss are measured in gigatonnes, a
+unit of mass corresponding to one million metric tons. Statistics
+about ice loss are often reported using this unit, for example Earth’s
+polar ice caps are losing about 426 gigatonnes of ice per year, at
+the time of writing. The scale of the unit here hides the fact that
+426 gigatonnes is a massive amount of ice. Our goal was to provide
+a visualization that would allow our audience to better interpret
+these statistics going forward. We collected statistics on ice loss in
+Greenland and Antarctica (the two ice sheets) over the course of
+significant periods, such as the amount of ice lost between 2002-
+2017 when NASA’s Grace satellite was actively observing the polar
+ice caps, or since the start of the 20th century (5,000 and 49,000
+gigatonnes, respectively).
+We settled on cinematic visualization because it is a natural fit
+for the use of concrete scales, we wanted to draw people’s attention,
+there is a relatively small amount of data that we are showing, and
+we wanted to display the data in a context that conveyed corporeal
+urgency. Given the affordances identified in Section 3, a cinematic
+visualization was an appropriate choice for our task of visualizing
+quantities related to climate change in a way that would capture
+the attention of our audience and allow them to comprehend the
+data in a concrete way. We ultimately chose the form factor for our
+visualization to be an interactive article containing a series of short
+cinematic visualizations. The visualizations were embedded as pre-
+rendered videos, which could be loaded dynamically, allowing for
+a certain amount of interactivity. Depending on the use case, one
+must determine whether real-time rendering is needed or not. Using
+real-time rendering limits the level of photorealism [52], but enables
+another level of interactivity, letting the user control the camera
+and interact with elements in the scene (Fig. 7D). We intended
+the narrative structure of our visualization to be largely author-
+driven [57], and decided that real-time rendering was not required.
+After determining that a cinematic visualization was appropri-
+ate, we began outlining possible scripts and creating storyboards
+in which we sketched ideas for locations, cinematography, and se-
+quencing of shots. We first sought to identify locations that would
+serve as effective backdrops, allowing people to gain a concrete
+understanding of the size of data in familiar locations. We consid-
+ered natural locations like the Grand Canyon, Monument Valley,
+Mt. Everest, and Uluru, urban environments like Houston, New
+York City, San Francisco, and St. Louis, and other man-made sites
+like football stadiums and the Hoover Dam. Within each of these
+environments we created sketches to help determine the camera
+placement, mise-en-scène, data, and annotations that the visualiza-
+tions would require, and wrote rough scripts to define the narrative
+structure.
+While we wanted to place data in a variety of different envi-
+ronments so that our diverse audience would be able to connect,
+
+2000
+1979
+2009
+Carbon
+Emissions
+7021
+M, ,
+Conlen, et al.
+ultimately many of these locations were not used. See Figure 6 for
+examples of some of the locations that were not able to support
+both focus and context at an anthropocentric perspective. The final
+article consisted of videos visualizing one, then 5,000, then 49,000
+gigatonnes of ice. The videos were embedded throughout the text
+of an article which provided context. In the first and last videos the
+user could click to choose to play videos displaying the relevant
+quantity of ice in different locations. Here we look closely at the
+design process for the first video, showing one gigatonne of ice in
+Central Park, New York City.
+4.2
+Principal Photography
+With the storyboards and scripts ready, the source footage that
+would make up the final video needed to be created. We chose to
+use Blender for this process, which provides both an interactive
+GUI-based interface as well as a Python API that allowed us to
+load, transform, and bind data to objects in a 3D scene. We created
+renders for many different scenes, although ultimately ended up
+using a small number of them in our published pieces.
+Mise-en-scène. The elements that constitute the mise-en-scène
+of a cinematic visualization need to be created and arranged. Be-
+cause many of our scenes take place in real-world locations, we
+were able to utilize existing open data sets to import geographic
+data, including 3D models of buildings and terrain data. In addition
+to elements derived from real-world locations, we added elements
+which would be parameterized by data, for example the large block
+of ice placed in Central Park (Figure 5). After the models have been
+created, they need to be assigned a material, which (along with
+lighting) will determine how they appear in final renders. We chose
+to use a flat shading for the buildings and other environmental
+elements. This gave these elements less visual weight while still al-
+lowing them to be easily identifiable. We considered using a similar
+flat style for the data elements, but ultimately decided to add a more
+photorealistic ice material which would allow the data to stand out
+against the buildings and reinforce the idea that we were showing
+a concrete amount of ice. While many of the examples that we saw
+utilize a studio lighting setup to control shadows and reflection, we
+opted to use simple global illumination to emulate the sun shining
+in our outdoor scene. This meant our lighting was realistic for the
+location and the setup was quite simple, but we were limited in our
+ability to use lighting as a tool to guide attention, as we saw it used
+(for example) in CV15.
+With the scene constructed, the next step was to bind the data.
+This was the point at which we realized that many of the set lo-
+cations were not going to work with the data we were hoping to
+visualize (“data changes everything” [66]). For example, a gigatonne
+of ice placed in a football stadium (Fig. 6D) would extend over 200
+kilometers into the sky, making it difficult to view both the diegetic
+data and the stadium itself simultaneously. For our visualizations
+we were simply assigning the dimensions of a primitive 3D object
+based on calculations related to the mass of ice melt over specific
+periods, along with the density of ice, in order to create blocks of
+ice which were physically representative of the quantity lost.
+Cinematography. After we incorporated our data into the scene
+it was time to add animation and cinematography. Blender supports
+a keyframe-based animation system which made it simple to add
+basic animations to the size and locations of elements in the scene,
+as well as the position and perspective of cameras. Working off of
+the storyboards that we had created, we placed cameras (shown
+in Figure 5) that would be physically realistic and familiar: we use
+three cameras, one a human point-of-view, one a bird’s eye view
+(as if it were taken from a helicopter circling the city), and one a
+"god’s eye view" taken from the perspective of a satellite overhead.
+The satellite camera allowed us to create an initial establishing shot,
+while the other cameras provided views that supported a ground-
+level view as well as an overview. When sequenced together, these
+camera perspectives allow us to present focus plus context [13] to
+the viewer, and support our narrative goals [1].
+4.3
+Post-Production
+Once the source material was created, we needed to edit it to form
+a coherent narrative, for example by combining multiple videos in
+sequence, adding annotations on top of the video to add context,
+and adding sound to add presence, guide attention, and provide
+details. Any visual effects must be added at this stage. For example,
+in the case of embedding digital data objects into physical footage
+of a narrator, a “match moving” process to align the digital and
+physical scenes would need to be performed [23].
+Editing. We combined footage from multiple cameras, compos-
+ing shots into a narrative structure, starting with establishing shots,
+then initial action, peak, and finally release [1]. The sequence of
+images is important to advance the role of narrative, pacing, and
+mood. Narrative visualizations often include annotations to provide
+additional context and explain to viewers what it is they are seeing.
+In the case of cinematic visualizations these annotations can be
+composited over the source footage using standard video editing
+software. Some examples that we saw embed annotations directly
+into the 3D scene itself, which requires them to be embedded in the
+source footage directly. We chose to composite annotations rather
+than include them “in-situ” as it facilitated more rapid iteration dur-
+ing the editing process, allowing us to change the timing, location,
+and content of annotations, without needing to re-render any of
+the source footage — a potentially time-consuming process.
+Sound. In our work we ultimately did not use audio, instead
+opting to embed the videos in a larger text article, which would
+serve to provide viewers with context for the visualization. This is
+a limitation and something to be explored more in future work, as
+audio can be a useful tool in cinematic visualization to set tone and
+drive narrative.
+4.4
+Publication
+Once the article was completed and approved for publication, it
+was posted to NASA’s climate website. We did not collect detailed
+metrics on how readers interacted with the videos on the article
+itself, but can see how users responded to posts on the NASA
+Climate Facebook, Instagram, and Twitter pages. These posts—
+which contained a link to the article and (in some cases) directly
+embedded the video set in New York City—were collectively viewed
+tens of thousands of times, received thousands of engagements
+(likes, comments, shares), and the article was subsequently shared
+by other organizations such as the United States Department of
+
+Cinematic Techniques in Narrative Visualization
+, ,
+Agriculture and the World Meteorological Organization, as well as
+by individual scientists and meteorologists.
+Across all of the social platforms users left 94 direct comments,
+with topics ranging from positive (for example, some explicitly
+expressing that they like this type of visualization “We need more
+of these types of comparisons in the media”, “This is an amazing
+visualization. Thanks NASA!”, or asking for similar visualizations
+of different quantities “It would be very interesting to see this illus-
+tration but with the predicted sea level after all the ice in Greenland
+and Antarctica melt. Can you show that?”) to concern about the
+data being visualized (“Oh my God. Come to our aid.”, “Thanks for
+helping us comprehend the enormity of this sad news!”, a GIF of
+a cartoon rodent crying) to climate change denial (“Where’s your
+proof”, “Wow, as much as 2 millimetres. Measured by satellite too”).
+The comments were distributed roughly uniformly across the three
+types (positive attitude toward visualization, concern about climate
+change, and climate change denial), but varied heavily across plat-
+forms, with users on Facebook expressing concern or denying that
+there is a climate problem, users on Instagram leaving both positive
+and concerned comments, and users on Twitter expressing a range
+of concern, denial, and a positive attitude toward the graphic.
+5
+DISCUSSION
+Cinematic visualizations can engage viewers with dramatic and
+visceral presentations of data, highlighting particularly important
+data points, and presenting an author-guided tour through data
+embedded in a relevant context. On the other hand, they may be
+poor choices for communicating large amounts of data and are
+not optimal in terms of perceptual effectiveness. If a cinematic
+visualization is appropriate, it will require a broad range of skills —
+such as cinematography, narrative, 3D modeling, video editing, and
+possibly acting — and a time-consuming iterative design process.
+5.1
+Challenges of Creating Cinematic
+Visualizations
+While cinematic visualizations can capture the attention of their
+audience and help viewers relate to the data in a concrete way,
+they can be challenging and time-consuming to produce. Here we
+discuss some of the challenges inherent in creating an effective
+cinematic visualization.
+One of the most apparent difficulties of cinematic visualization
+is the potentially overwhelming size of the design space. Works
+in this genre typically use three visual dimensions, plus time and
+sound. The methods that allow us to analyze and critique cinematic
+visualizations (e.g., [5]) do not necessarily help us to create them.
+That is, they are difficult to use generatively. While information
+designers are familiar with the attention to detail that is required
+when placing objects in a frame in order to achieve an effective
+visual hierarchy, in cinematic visualizations there are also objects
+outside of the frame that affect the style and tone of the visualization.
+For example, the placement of the camera in relation to the subjects,
+the focal length of the camera, and the placement and strength of
+light sources are all instrumental in creating a shot which can easily
+be decoded by viewers.
+There is a diversity of tasks that need to be completed in order
+to create a cinematic visualization, each requiring a separate set of
+skills. For example, in addition to skills required for traditional visu-
+alization (data analysis, transformation, and visualization) and nar-
+rative visualization (understanding audience, storytelling, graphic
+design), cinematic visualization will often make use of animation,
+cinematography, lighting, motion graphics, 3D modeling, sound
+design, video editing, and (sometimes) acting. The skills that make
+one a good 3D modeler are not necessarily the same skills that make
+one a good storyteller, and so graphics of this type often require
+a diverse team to create. Furthermore, for ray-tracing renderers,
+there is a large gap between prototypes and final rendered output,
+challenging the iterative design process.
+5.2
+Considerations for Cinematic Visualization
+Creators
+While cinematic visualizations share many of the same design goals
+of more traditional narrative visualization (e.g., guide the viewers’
+attention), the way in which these goals are operationalized differ.
+Here we highlight ways that these design goals were operational-
+ized across the four dimensions of style, both in our own work and
+in the examples analyzed. For a full breakdown of the techniques
+used in each example, see Figure 7.
+Mise-en-scène. Objects’ sizes, colors, shapes, textures, and place-
+ment in relation to one another can all be used create an effective
+visual hierarchy. For example, to guide a user’s attention in a cin-
+ematic visualization, a designer might choose to use lighting to
+cast a glow around an object (CV11), or change the object’s color
+(CV2, CV13) so that it stands out. In How Much is a Gigatonne,
+the ice’s large size, color, and shine draw a viewers attention to
+it in contrast with the surrounding buildings, which are smaller,
+grayscale, and matte. The mise-en-scène is designed both to com-
+municate information—including using narrators (CV42), diegetic
+data (CV35), and visual analogies (CV6)—and to add dramatic affect
+(e.g. CV11, CV40).
+Cinematography. Perspective can be used both to drive narra-
+tive and to set tone, as well as to provide focus plus context. The
+position (CV26), angle (CV28), or focus (CV2) of a camera can be
+modified so that the object becomes the focal point of the frame.
+To help narrow the large space of possible cinematic visualizations,
+and make effective use of the frame, designers of cinematic visual-
+ization may study how shots are composed and sequenced in films.
+In How Much is a Gigatonne?, we rendered footage from multiple
+cameras in order to create close-up, medium, and wide shots. Some
+cinematic visualizations enable limited user-control of the camera,
+for example letting the user trigger the next stage of animation
+(CV9) or rotate their perspective (CV13). Often the camera needs
+to track a particular object in the scene (CV12). If this object is in
+motion you may need to set your camera to track it. Planning the
+path of the camera so that the object of interest is not occluded by
+other objects and so that motion is smooth and visually pleasing
+can be difficult. This may be done algorithmically [15, 68] or by
+hand.
+Editing. Putting the footage into a particular order progressively
+reveals information to convey the authors’ intended message. Edi-
+tors may use footage from one camera at one location (CV29), or
+multiple cameras at multiple locations (CV40). The editing tech-
+niques used in data videos—particularly the use of establishing,
+
+, ,
+Conlen, et al.
+initial, peak, and release shots—has been studied in more depth by
+Amini et al. [1]. Similar to movie makers, creators of cinematic visu-
+alizations may use the technique of storyboarding to prototype and
+communicate their scenes in a lo-fidelity form before endeavouring
+on the time intensive task of 3D modeling and rendering. In How
+Much is a Gigatonne we use establishing shots to situate the viewer
+before initiating action from the perspective of the ground level (an
+anthropocentric perspective), before cutting to the vantage point
+of a helicopter, using the city skyline as a concrete scale.
+Sound. Audio can set tone (CV25), cue attention (CV28), and
+impart additional details through narration on (CV40) or off-screen
+(CV45). Music (CV29) and ambient sound (CV26) can affect the tone
+of the visualization and add presence to the scene, for example
+CV29 uses combines techno music and a visual analogy of of the
+weight of rocket fuel (measured in elephants) to create a humorous
+juxtaposition which may make the visualization more approachable
+and less dry. CV26 uses diegetic sound (taxi cabs honking) to rein-
+force the anthropocentric perspective. In How Much is a Gigatonne
+we did not use sound (neither did most of the other visualizations
+that we analyzed which used an “article” format), but effective use
+of both the visual and auditory channels has been shown to lead to
+improved outcomes in multimedia learning contexts [42].
+5.3
+Implications for Authoring Tools
+As cinematic visualization is a newly emerging genre, there is rel-
+atively little tool support to facilitate authoring of this type of
+visualization. Instead, creators turn to general purpose 3D software
+that was designed to support a breadth of use cases such as architec-
+tural design, modeling, and narrative animation. These tools, while
+powerful and expressive, may overwhelm users with complexity
+that is incidental to the task of creating a cinematic visualization.
+For example, objects are assigned materials which are powered by
+low-level shader code. One can not choose, e.g., between “realistic”
+or “cartoon” aesthetics but instead must compose low level shader
+components to achieve the desired look.
+These tools do not support the basic building blocks of visualiza-
+tion, such as easily ingesting data and binding data values to objects
+in a scene. Instead, users must write custom scripts to handle any
+such task. The interfaces in general are multi-modal: most 3D mod-
+eling work is done directly through a GUI, but data-driven work
+needs to be done in code; shaders are described using a directed
+graph. Authors are forced to context switch between drastically
+different environments, arguably making it harder to iterate.
+The task of 3D rendering can be computationally intensive. De-
+pending on the output resolution, complexity of the scene, and
+computing power available, a short (30 seconds) animation could
+take several hours to render. There is a large gap between the
+fidelity of the final renders and what a designer sees while con-
+structing the scene. This setup makes it important to create test
+renders frequently, but makes it hard to have a rapid feedback loop.
+5.4
+Limitations of our Work
+Our survey was limited to 50 examples, taken from a limited set of
+sources. While not exhaustive, the examples implement a range of
+design techniques across a variety of applications. We do not pro-
+vide an empirical evaluation of the work surveyed, instead choosing
+to use techniques of film criticism in order to analyze patterns used
+and identify the communication intentions of their producers. We
+similarly did not empirically evaluate our own work, and instead
+provide an account of our design process and detail our reasoning
+for important decisions that were made along the way. Our work
+does not fully utilize the design space of cinematic visualizations
+that we identified; for example, we did not use sound at all, and all
+narration was done through written text with a few small overlays
+in the video. The experience might be improved by incorporating
+narration either on-screen or off [10].
+6
+CONCLUSION
+We presented cinematic visualization, a genre of narrative visu-
+alization that uses techniques from cinema in order to enhance the
+presentation of data-driven stories. A central contribution of this
+work is to identify a new genre of narrative visualization that we
+then analyze in depth. The importance of genre is clear in other art
+forms like literature and cinema; however, it is invoked less often
+in the context of visualization research. We believe that this type of
+work is crucial for understanding the design of narrative visualiza-
+tions, and thinking rigorously about how they can be constructed
+and deployed. While past work on narrative visualization looked
+specifically at the narrative structure, here we look at both narra-
+tive and style as formal systems that contribute to the dramatic
+experience of watching a cinematic visualization. To do this, we
+turned to theory from another form of art, film, in order to provide
+grounding in the features of style, and used analysis techniques
+established in that domain to deconstruct our case studies.
+We analyzed a variety of examples of cinematic visualization and
+the techniques that they employ towards certain narrative applica-
+tions. Many of these visualizations show a relatively small amount
+of data (e.g., focusing on a single rate or quantity) as opposed to
+being data-dense. The non-data elements of the scene play an im-
+portant role: they are used to set the location in which the shot is
+taking place and provide cues to viewers about where they are, what
+they are looking at, and why it is relevant. This approach is quite
+different from typical information visualizations, where data may
+be reduced to a minimal form, such as a line or a bar. Cinematic
+visualization instead tends to be more maximal in its approach,
+such that the non-data ink is not reduced or omitted, but rather
+used to build up entire digital worlds around data points. This style
+encourages viewers to feel present in locations augmented with
+data objects, or to viscerally experience events that happened in
+the past, or are happening far away in the universe.
+Rendering data in 3D is a fraught endeavor, as the values being
+rendered can be obscured by humans’ relatively poor ability to
+estimate and compare volume, and because the 3D projection can
+introduce distortion when trying to read values. Why would the cre-
+ators choose to follow a cinematic path over one that more clearly
+and directly communicates the underlying data with precision?
+We argue that in choosing to treat a visualization as a cinematic
+experience, its authors might be looking beyond the immediate
+data, in order to viscerally ground that data in meaningful context.
+In other words, analytic precision is only one of several objectives
+that a visualization might help accomplish. In choosing 3D, we
+might diminish precision in service of other objectives.
+
+Cinematic Techniques in Narrative Visualization
+, ,
+ACKNOWLEDGEMENTS
+We would like to thank Susan Callery, Holly Shaftel, Randal Jackson,
+Daniel Bailey, Michael Gunson, Josh Willis, Joe Witte, and the
+Earth Science Communications Team at NASA’s Jet Propulsion
+Laboratory for their support of this work. A portion of this research
+was carried out at the Jet Propulsion Laboratory, California Institute
+of Technology, under a contract with the National Aeronautics and
+Space Administration (80NM0018D0004).
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+
+Cinematic Techniques in Narrative Visualization
+, ,
+Figure 7: We analyzed the style of 50 cinematic visualizations using the features of mise-en-scène, cinematography, editing, and
+sound. An HTML version of this table including URLs for each row can be found at https://cinematic-visualization.github.io/.
+
+Cinematography
+Mise-en-scene
+Editing
+Audio
+rative Text
+Composited Annotations
+Jser-controlled camera
+Realistic Background
+ - Opacity
+mera
+er-triggered steps
+3617 - L
+n-Situ Annotations
+Composited Narra
+Point-of-view can
+nera
+Visual Analogy
+rator
+it Visualizatiol
+Scale
+punos oeba
+Reality
+n-situ Narra
+notation
+Annotation
+Annotation
+dded
+ncrete
+erview
+pax
+ISIC
+0
+Author
+Publishel
+Titte
+cV1
+Drowning in plastic
+Simon Scarr, et al.
+Reuters
+CV2
+Is This the Neighborhood New York Deserves?
+Michael Kimmelman 
+NYT
+CV3
+Krigsskipet som krasjet og sank
+B
+Stangvik, et al.
+VG
+CV4
+How China Turned a City Into a Prison
+Chris Buckley, et al.
+NYT
+CV5
+The Forbidden City's unique architecture
+Marco Hernandez
+South China Morning Post
+CV6
+Here are 120 million Monopoly pieces, roughly one.
+Confessore, et al.
+NYT
+CV7
+Building Katie's New Face
+Jason Treat
+NatGeo
+CV8
+Mass Exodus: The scale of the Rohingya crisis
+Christian Inton
+Reuters
+This 3-D Simulation Shows Why Social Distancing.
+CV9
+Parshina-Kottas, et al.
+NYT
+CV10 Tracking China's Muslim Gulag
+Simon Scarr
+Reuters 
+CV11
+Choice and Chance
+Staff
+Tampa Bay Times
+cV12 Cassini's Grand Tour
+Nadia Drake, et al.
+NatGeo
+CV13 Resurecting a Dragon
+Brian T Jacobs
+NatGeo
+CV14Want to fireproof your house? Here's where to start
+Kyle Kim, et al.
+LATimes 
+CV15Apoll 11 -As They Shot It
+Jonathan Corum, et al. NYT
+CV16 The Atlas of Moons
+NatGeo Staff
+NatGeo
+CV17 Explore a Toad's Digital Clone
+Brian T. Jacobs
+NatGeo
+CV18 THE THOMAS FIRE: 40 DAYS OF DEVASTATION
+Joe Fox
+LATimes
+D.
+CV19 Is the Nasdag in Another Bubble?
+Roger Kenny, et al.
+Wall Street Journal
+CV20 A 3-D View of a Chart That Predicts The Econom...
+Gregor Aisch, et al.
+NYT
+CV21
+Inside the Taser
+Simon Scarr
+Reuters
+CV22 Seeing Earth from Outer Space
+Matthew Conlen
+The Pudding
+C.
+CV23 Of Catastrophes and Rescues: Making the Invisib..
+Peter Mindek, et al.
+PacificVis
+CV24A Visualization of Two-stage Autoignition of n-dod..
+Yucong Ye, et al.
+PacificVis
+CV25 How Mariano Rivera Dominates Hitters
+Graham Roberts, et al. NYTimes
+Real World Visuals
+Real World Visuals
+CV27 CCS:A2 degree solution
+Real World Visuals
+Real world Visuals
+CV28CARS
+Real World Visuals
+Real world Visuals
+CV29[REALISTIC] Elephant rocket fuel - Saturn V
+Maxim Sachs
+YOUTUBE
+CV30 University of Exeter greenhouse gas emissions in R..
+Real World Visuals
+Real world Visuals
+CV31if The World Were 100 People
+Gabriel Reilich, et al.
+Good Magazine
+CV32 Up - and down - from Ground Zero
+Graham Roberts, et al. NYT
+CV33 The Birth of a Virtual Cell
+Peter Mindek, et al.
+PacificVis
+CV34
+The Nuclear Threat - The Shadow Peace
+Neil Halloran
+Youtube
+CV35What f Carbon Left Your Tailpipe as Solid Chunks?
+Sukee Bennett
+PBS Nova
+CV36Stay Home, Flatten the Curve
+keta
+Youtube
+CV37 Chart Party: We decided to erase the three-pointer
+Jon Bois 
+Youtube
+cV38 200 Countries, 200 Years, 4 Minutes
+Hans Rosling
+Youtube
+CV39 The best stats you've ever seen
+Hans Rosling 
+TED 
+CV40 VFX Artist Reveals the True Scale of the Universe
+Wren Weichman
+Corridor Crew
+CV41
+Helge Ingstad
+Hallvard Sandberg
+NRKbeta
+CV42The dangers of storm surge
+The Weather Channel
+The Weather Channel
+CV43 Survive the Tornado
+The Weather Channel
+The Weather Channel
+CV44
+Television Elections Coverage
+KING5 TV
+KING5
+CV45 Powers of Ten
+Eames & Eames
+IBM
+CV46 Strange things happen when you rotate in 4 dimen...
+Hamish Todd
+Youtube
+CV47 Discovering Gale Crater
+Armand Emamdjomeh LATimes
+A
+CV48 Taiwan earthquake: Survivors found in rubble of Ta..
+ Malachy Brown,
+SketchFab / Australian
+cV49 Four of the Best Olympians, as You've Never Seen...
+John Branch
+NYT
+CV50 How We Created a Virtual Crime Scene to Investig.
+Malachy Brown,
+NYT
+CV13
+CV40
+CV26
+CV35
+D. Author-Guided
+A. In-situ Narrator
+B. Anthropocentric Perspective
+C. Resolving Scale
+Interactive Camera
\ No newline at end of file
diff --git a/CtE1T4oBgHgl3EQfWAR_/content/tmp_files/load_file.txt b/CtE1T4oBgHgl3EQfWAR_/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf,len=873
+page_content='Cinematic Techniques in Narrative Visualization  Matthew Conlen  Our World in Data  matt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='conlen@ourworldindata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='org  Jeffrey Heer  University of Washington  jheer@uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='edu  Hillary Mushkin  California Institute of Technology  hmushkin@caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='edu  Scott Davidoff  Jet Propulsion Laboratory  California Institute of Technology  scott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='davidoff@jpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='gov  ABSTRACT  The many genres of narrative visualization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' data comics, data  videos) each offer a unique set of affordances and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' To  better understand a genre that we call cinematic visualizations—3D  visualizations that make highly deliberate use of a camera to convey  a narrative—we gathered 50 examples and analyzed their traditional  cinematic aspects to identify the benefits and limitations of the form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  While the cinematic visualization approach can violate traditional  rules of visualization, we find that through careful control of the  camera, cinematic visualizations enable immersion in data-driven,  anthropocentric environments, and can naturally incorporate in-  situ narrators, concrete scales, and visual analogies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Our analysis  guides our design of a series of cinematic visualizations, created for  NASA’s Earth Science Communications team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We present one as a  case study to convey design guidelines covering cinematography,  lighting, set design, and sound, and discuss challenges in creating  cinematic visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  1  INTRODUCTION  Within narrative visualization [57], researchers have identified gen-  res (such as data comics [3] and data videos [1]) that help better  unpack and situate their specific application and the features that  they employ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' cinematic visualizations embed data into a three-  dimensional, time varying scene, utilizing one or more cameras to  direct the relationship between a viewer and the scene to tell a dra-  matic data-driven story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' This cinematic approach is different from  the one typically used in information visualization, where graph-  ics are reduced to a minimal form, incorporating only essential  elements like axes and data-driven marks [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Cinematic visual-  izations are more maximal: non-data marks are not compressed or  reduced, instead entire digital worlds are built up around data points  and included in the visible frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' This technique allows viewers  to feel present in locations augmented with data-bound objects,  known as data visceralizations [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Narrative documentary visual-  izations [10] can be produced through the careful editorial direction  of the cinematography, editing, mise-en-scène, and sound [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Through an analysis of 50 existing cinematic visualizations, we  identified four salient techniques (in-situ narrators, resolution of  scale, anthropocentric perspective, and story-driven cameras) that  cinematic visualizations employ to dramatically engage their au-  dience through emotionally resonant data-stories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We show how  these techniques are used throughout the examples analyzed, dis-  cuss constraints associated with them, and reason about why cine-  matic visualizations may be effective despite the known pitfalls of  3D visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Using the lessons learned from this formal analysis, we produced  a web-based article containing a series of cinematic visualizations  relating to climate change, which was published by NASA’s Earth  Science Communications team 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We contribute the design process  for one of these visualizations as a case study, presenting design  artifacts that were created during our process (both successful  and unsuccessful), and provide concrete guidelines for designers  of cinematic visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Our analysis and design artifacts are  available at https://cinematic-visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='io/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  2  RELATED WORK  Narrative visualizations are used to improve memorability [7, 8],  to instill empathy or emotion [9], to frame a message [33], and to  improve engagement [19, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Segel & Heer [57] provided an ini-  tial characterization of the design space of narrative visualizations,  which was later elaborated to include additional techniques [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Hullman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' [34] focused on the role of sequence in narrative  visualization, characterizing a set of transition types and other high  level strategies for sequencing visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Tools have been cre-  ated to support narrative visualization authoring [2, 11, 18, 56], and  a small number of empirical evaluations of narrative visualizations  have been conducted [9, 19, 46, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Further work has investigated  specific genres of narrative visualization such as data comics [3],  and new genres have emerged beyond Segel & Heer’s initial set,  such as “scrollytelling.” Here we add to the ongoing conversation  around narrative visualization by identifying another such genre:  cinematic visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Kosara & McKinley [39] identified the op-  portunity for narrative visualization researchers to learn from other  disciplines that engaged heavily with storytelling and multimedia,  this paper draws on film art scholarship, incorporates a formal  system of cinematic style into our analysis, discussion, and design  of cinematic visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='1  Data Videos & VR  Data videos were included in the initial set of genres put forth by  Segel & Heer [57] and first studied closely by Amini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Not  all data videos are cinematic visualizations (for example, we do  not consider a video consisting of a sequence of two-dimensional  infographics to be cinematic), and not all cinematic visualizations  1https://climate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='gov/news/2933/visualizing-the-quantities-of-climate-change/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='03109v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='HC]  8 Jan 2023  , ,  Conlen, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Figure 1: The Dangers of Storm Surge (CV42) is a mixed reality video produced by the Weather Channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The video opens with  a close up shot of a news anchor wearing a rain jacket, standing in front of a house (1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' There are audible sounds of rain  under the anchor’s voice and water dripping down the windows of the house.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The camera pulls back revealing that the live  anchor is being composited into a 3D scene of a suburban neighborhood during a storm surge (1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' There are very few data  points actually encoded as visual elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The piece simply shows water rising from zero, to three, to six, to nine feet (1C-D)  as the anchor narrates with details in reference to the danger of storm surge associated with hurricanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  are data videos (for example one in which the visualization is deeply  tied to the text of an interactive news article).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' While Amini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  were primarily concerned with the narrative structure and attention  cues of data videos, we additionally consider the visual and auditory  style of cinematic visualizations in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Under our formal style  system, our analysis of editing is most closely related to Amini’s  work, however that is only one of four dimensions we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Bradbury & Guadagno [10] studied viewer preferences in docu-  mentary narrative visualization (a subgenre of data videos in which  data is presented using the techniques of documentary film), and  found that audiences may prefer when documentary data videos  include voice-over narration and on-screen narrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We build on  their analysis of the use of narrators and narration, in particular  during our discussions of in-situ narrators that interact with data-  bound objects digitally rendered into the space around them, and  of the use of sound in cinematic visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Video producers  have extended the traditional documentary visualization format to  enable interactivity such as user selected paths through the content  and manipulable graphics [29, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Immersive data stories [35] have been discussed within the  emerging field of immersive analytics [44] and have been shown  to allow viewers to examine data at multiple scales, support im-  mersive exploration, and create affective personal experiences with  data [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' [41] introduced data visceralizations, where  physical quantities are visualized in 3D virtual reality scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' This  paper helps to bridge the gap between data visceralization and nar-  rative visualization by showing how cinematic techniques can be  used to create author-guided narrative visualizations using data vis-  ceralizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Cinematic visualizations similarly attempt to immerse  viewers and create emotionally resonant experiences, although in  contrast to immersive visualizations they are typically viewed on a  standard 2D screen with limited (or no) user control of the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  There are several toolkits for creating immersive data visualizations  and data stories on augmented reality devices [21, 55, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='2  3D Computer Graphics  Animation [62] has been a partner discipline with visualization for  some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Classic principles of animation [61] have been adapted  for digital usage [40] and subsequently for information visualiza-  tion [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' With realistic camera models [38] and improving render-  ing capabilities [20] digital animation became a tool to create Holly-  wood films [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' While 3D graphics have been used in visualization  to limited success, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=', to display hierarchical information [17], the  use of 3D graphics in information visualization is often avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  A broad body of research documents potential pitfalls, including  that volume is not a perceptually effective encoding channel [16],  and that 3D projections introduce distortion and occlusion [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  We find that designers of cinematic visualizations may intention-  ally use suboptimal encodings in support of more visceral [41] and  emotionally resonant [28] graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  The use of 3D does find more regular application in scientific visu-  alization [4, 65], including its use in storytelling [22, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Borkiewicz  used the term cinematic scientific visualization [6] to refer to a class  of narrative data videos that focus on scientific data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Here we use  cinematic visualization in a similar way but do not restrict the data  to be strictly scientific or inherently spatial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Unlike Borkiewicz,  our description encapsulates visualizations which are not embed-  ded in films, but may be, for example, displayed as an animation  accompanying a news article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='3  Film Art  Bordwell and Thompson [5] define narrative and style as the two  major formal systems of film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' While prior work has examined se-  quence [34] and narrative structure & attention cues [1] in data  videos, we observe that cinematic style has far less visibility in the  critical vocabulary of data visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Style plays a crucial roll  in filmmaking, enabling directors to “confirm our expectations, or  modify them, or cheat, or challenge them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='] A director directs  not only the cast and crew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' A director also directs us, directs our  attention, shapes our reaction.” [5] This paper brings Bordwell and  Thompson’s formal system of cinematic style into the world of data  visualization, and uses it to examine how narrative visualizations  borrow techniques from cinema while departing from many of the  traditional practices advocated by visualization research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Style consists of four features, which together make up a film’s  style, each now briefly described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Mise-en-scène refers to every-  thing that is seen in the frame, including lighting, actors, objects,  backdrops, and so on [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Cinematography refers to the use of the  Cinematic Techniques in Narrative Visualization  , ,  Figure 2: (A) In [REALISTIC] Elephant rocket fuel - Saturn V (CV29), a model Saturn V rocket takes off, however, instead of  flames exiting the bottom of the spacecraft, elephants are expelled, the number of elephants represents the corresponding  mass of fuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' This video may not make for a particularly effective visualization in terms of conveying precise quantities, but  the style successfully uses humor in order to call attention to the fact that rocket launches use a quantity of fuel so great  it is appropriate to measure it in terms of dozens of elephants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' (B) In Here are 120 million Monopoly pieces, roughly one for  every household in the United States (CV6) by the New York Times the pile of Monopoly pieces is first seen from a far, before  the reader scrolls down the page to trigger the camera zooming in to the very top of the pile, dramatically revealing what a  disproportionately small portion of families provide most political funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  camera, how shots are composed and framed [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' By placing ele-  ments at specific locations within the frame, they can be perceived  either as the subject or the background of the image [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Editing  is the composition of multiple pieces of footage in time or space,  creating transitions between perspectives and scenes [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Sound  is the audio used, whether it be music, voice over, or sounds from  characters or objects on screen [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Our analysis of cinematic visu-  alization identified techniques along these dimensions of style that  designers can use to enhance their presentation of data narratives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  3  CINEMATIC VISUALIZATION SURVEY  We collected cinematic visualization to analyze by surveying liter-  ature on narrative visualization [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 35,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 43,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 57,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 60],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' browsing  the information visualization awards website Information is Beau-  tiful [45] and the PacificVIS storytelling contest [50],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' and searching  for news articles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' blog posts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' conference talks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' and videos which  were described using combinations of the keywords cinematic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  data video,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' dataviz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' datavis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' visualization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' news,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' newsgames,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' immer-  sive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' mixed reality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 3d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' and video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We searched the portfolios of the  creators of the visualizations found initially and their collaborators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  A full list of the cinematic visualizations can be seen in Figure 7 in  the appendix of this paper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' we refer to these studies by identifiers  throughout the paper (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=', CV4 refers to the fourth example in the  table).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Our analysis considered 50 cinematic visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' While  the corpus is not exhaustive, the examples expose the variety of  media (interactive news articles, YouTube videos, and TV segments)  which cinematic visualizations occupy and the messages that they  deliver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The examples visualized a broad range of data types, in-  cluding datasets both with and without physical and geographic  dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Rather than empirically evaluate specific design patterns utilized  in the visualizations, we turn to the means of understanding plot  devices [57], sequencing [1], and film style [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We analyzed the  style of each example along the dimensions of mise-en-scène, cine-  matography, editing, and sound using the 4-step analysis process  described by Bordwell and Thompson [5], a canonical method of  film analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' For each example we first identified the main com-  municative goals of the visualizations, and then studied the salient  techniques applied within the mise-en-scène, cinematography, edit-  ing, and sound which supported these narrative goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We then used  iterative coding to categorize the salient techniques used across  the examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Usage of these techniques are shown in Figure 7, for  example we recorded many ways in which a viewer’s attention  is guided (through color, light, annotations, and narrators in the  mise-en-scène) and use of cinematographic techniques like point-  of-view perspective and user-controlled cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The table shows  that the medium of the cinematic visualization has some impact on  the techniques used, for example cinematic visualizations embed-  ded in online articles rarely use sound, but often utilize user-paced  segments, while those presented as videos make heavy use of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  A  B  Here are 120 million  Monopoly pieces, roughly  one for every household  in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Just 158 families have  provided nearly half of the  early money for efforts to  capture the White House.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=', ,  Conlen, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Figure 3: VFX Artist Reveals the True Scale of the Universe fea-  tures a live-action narrator alongside scaled-down 3D mod-  els of celestial bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='1  Design Techniques  Through this analysis we identified salient recurring techniques that  were frequently applied to support the communicative goals of the  visualization, including the use of in-situ narrators, anthropocentric  perspective, resolution of scale, and story-driven cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  In-situ narrators mediate interactions with diegetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Perhaps the most novel technique that we identified in cinematic  visualizations is the use of in-situ narrators, in which the mise-en-  scène contains a character that interacts directly with on-screen,  diegetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='2 In contrast to traditional documentary visualization  narrators who might participate from off-screen (“voice of god”) or  refer to data visualizations rendered as two-dimensional holograms  or composited over top the video [10], in-situ narrators are under-  stood by the viewer to be able to see and interact with the diegetic  data either through the use of superimposed data visceralizations  2Something which is diegetic exists in the same universe as the characters on screen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' we  use the phrase diegetic data to refer to data-driven elements which are part of—rather  than composited over—the scene shown in the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  (CV35, 40, 42, 43) or, in one case, data physicalization [37] (CV41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  This (typically) mixed reality environment serves an important  role for narrative visualization, allowing the on-screen narrator to  mediate interactions between the audience and the graphics, letting  them provide additional context and push the storyline forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  These narrators, essential components of the mise-en-scène, can  also help concretize a visualization’s anthropocentric perspective,  reinforcing the idea that data is being displayed at a human scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  In The Dangers of Storm Surge (CV42), one exemplar of this  technique (Figure 1) produced by the Weather Channel, a news  anchor wearing a blue jacket explains the dangers associated with  flooding due to storm surge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The graphics are coordinated with  the narrator’s script and appear to respond to his dialogue, the  composition of the frame inviting comparison between the man and  the height of the water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The narrator is the primary subject from the  start of the clip, positioned centrally in frame and maintaining focus  due to visual cues like his bright blue coat, the circular platform  upon which he stands, and the shot composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' To call attention  to the water’s height at certain key moments, a brightly colored  annotation is projected onto the crest of the surge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  An anthropocentric perspective transports viewers and  enables drama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' One notable aspect of cinema is how the camera is  able to transport the audience into the scene: people watching sus-  pend disbelief [24] to allow themselves to wholeheartedly imagine,  or “believe”, that they are in the scene, seeing things through the  camera lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' That is, the camera’s perspective becomes the viewer’s  point of view, they are one and the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The height, angle, and  distance of a camera in relation to objects in the scene all play a role  in how a viewer will interpret and respond to the frame that they  ultimately see [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' When a camera is placed high above a setting,  the viewer feels like they are also high above it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' When a camera  is placed at eye level, a viewer feels as if they are standing there  watching the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' For example, both CV1 and CV26 utilize unit  visualizations and concrete scales to visualize quantities in relation  to the size of Manhattan, but each uses perspective to impact the  viewer’s experience in a different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In CV1 the data being dis-  played (plastic bottle usage) is not directly related to the locations  being used as concrete scale referents, and an overview shot is  used, letting the viewer absorb the scale of the data rather than  the details and textures of the city itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In contrast, CV26 begins  with a shot from a camera placed at eye-level, looking at several  of the city’s ubiquitous yellow taxis, transporting viewers to the  city at street level, and forcing them to reckon with the data being  displayed (New York City’s annual green house gas emissions) in a  much more visceral way [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Some cinematic visualizations place the camera perspective  somewhere that is humanly impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' However, if the audience  suspends disbelief, the camera can carry the viewer through these  otherwise inaccessible spaces, for example, CV12 shows an anima-  tion of the Cassini spacecraft as it orbited and eventually crashed  into Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Choice and Chance (CV11), visualizes the events of the  2016 Pulse night club shooting in Tampa Bay, positions a camera  looking “through” the roof of a nightclub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Because the scene is  shot using a digital model instead of a real location, the roof of the  club can simply be removed and problems of occlusion go away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Changing perspectives can also shift the subject of the scene or  add emotional content, for example, when the camera moves to  A  SUBSCRIBE  B  c  Rige!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Cinematic Techniques in Narrative Visualization  , ,  Figure 4: New York City’s greenhouse gas emissions as one-ton spheres of carbon dioxide gas, a cinematic visualization produced  by Carbon Visuals and released online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The cinematic visualization uses a variety of different camera views, along with stark  colors to guide viewers through an explanation of the scale of the city’s greenhouse gas emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The number of instances  of the blue sphere is driven by the rate of emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' As this number grows the city buildings serve as a concrete scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  reveal something that wasn’t already in the frame, the audience  experiences seeing it for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In Choice and Chance the  camera moves to different vantage points throughout the model as  the story progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The camera remains in an overview shot for  the majority of the article, but moves to ground level at the climax,  elevating the intensity of the shot by placing the viewer into the  perspective of a bystander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Author-defined camera trajectories can be played, paused,  and (lightly) modified by viewers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The cinematic visualizations  that we analyzed tended to use author-driven narrative structures [57],  with most user interactions consisting of the user clicking or scrolling  to trigger the visualization to continue to the next stage (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=', CV2,  5-17, 21-22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Operationally, this requires animating the position  and orientation of a digital camera model along a track specified  by the author, and has been used heavily by cinematic visualiza-  tions embedded in articles (16 out of 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The other way in which  (constrained) interactivity was employed was allowing the manipu-  lation of 3D models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In most cases this means the user can position  the camera at a particular location around the model (see CV17 for a  stereotypical example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' These models might be scientific (CV13,17)  or cultural (CV5) objects that would be otherwise inaccessible to  the audience viewing the visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' It is common for orbital  cameras to be used, constraining the camera’s focus to remain on  a particular object of interest while allowing the user to exercise  control over viewing angle and zoom level (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 7D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Cinematic  visualizations that support these interactions must be rendered in  real-time, limiting the fidelity at which the models may be rendered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Visualization techniques are combined toward resolution  of scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' While we traditionally think of 3D graphics as ineffective  for encoding quantities [16], a recurring theme in our examples is  the use of 3D graphics to visualize and communicate quantities of  a massive scale (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=', CV1, 6, 8, 26-28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Quantities at a scale beyond  what we experience in daily life (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' hyperobjects [47]), like amount  of carbon dioxide emitted from NYC annually (CV26), may be es-  pecially difficult for people to picture because we rarely, if ever,  interact with quantities of such a size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Cinematic visualizations can  convey a quantity of scale in a concrete and affecting way by using  cinematography to establish the viewer’s point of view from the  ground, a position which often serves as the implicit zero point  of a y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We observed that several visualization techniques are  naturally expressed in cinematic visualizations, including data vis-  ceralizations [41], unit visualization [51] and concrete scales [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  For example, in CV27 the viewer sees a city park, including trees,  people standing in a grassy field, and a ten meter tall blue sphere  representing the actual size of one metric ton of CO2 (data vis-  ceralization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' As the scene progresses, many more spheres appear,  each representing one metric ton of CO2 (unit visualization), until  so many appear that the camera must zoom out, above the park,  observing the growing pile of spheres in comparison to the city  buildings (concrete scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Objects which are used as backdrops—for example a city skyline  (CV11) or parked car (CV42, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 1)—may serve double duty as  concrete scale referents and contextual elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The use of 3D  graphics affords designers the ability to use concrete scales (CV1,  26) and visual analogies (CV29, 36) to (re-)contextualize the size of  objects, and digital sets are constructed to facilitate comparisons  that are impossible to make directly in the physical world (CV1,  27) and use point-of-view perspective to impart a visceral sense of  magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The visual medium is rich with possibilities for analogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  For example, in [REALISTIC] Elephant rocket fuel - Saturn V (CV29,  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 2), designer Maxim Sachs renders the launch of the Saturn V  rocket, except that the rocket expels elephants behind it as it travels,  rather than exhaust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The elephants represent the mass of fuel that  is being expended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' By juxtaposing these images, Sachs is able to re-  frame an abstract quantity of rocket fuel in terms that people may  have more familiarity with, and do it with a sense of humor that  may make the visualization overall more memorable or engaging  for its audience [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In a more typical case, the narrator of CV40  asks the audience to imagine if Earth were the size of a tennis ball,  and then, using this new scale, shows the relative size of different  planets, moons, and stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' These planets are compared against one  another, rendered into real-world footage including a narrator who  provides guidance and relevant facts about the celestial objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  , ,  Conlen, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Figure 5: How Much is a Gigatonne?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' shows one gigatonne of ice in Central Park, New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' A digital set (A) is designed including  multiple cameras, lighting, and data-driven and contextual elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Footage from the various cameras is composed to create  the final sequence (B-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' This was one of several videos that we developed for an article published on NASA’s climate website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  View the full videos at https://cinematic-visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='io/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  They are shown embedded into several settings, for example an  office, a Los Angeles street, and the New York City skyline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='2  Constraints  The time-based format does not support a high data density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Traditional information graphics often present a data-dense display  with minimal “non-data ink” [64] to remove possible distractions  and optimize the display for tasks such as value look-up and com-  parison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In some cases, designers may choose to add additional  illustrative features to increase the memorability of the visualiza-  tion [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In contrast, cinematic visualizations utilize diegetic data,  embedded in a three dimensional scene with other elements which  contextualize the scene (see CV35 for a striking example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In cine-  matic visualizations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' CV40,42) the elements surrounding the  data fulfill a dual role as both data and non-data ink;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' they add  spatial presence to the visualization [12], supporting a sense of  transportation to the virtual world for viewers, while simultane-  ously serving as guides and axes, points of reference for concrete  scales [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Rather than densely packing data, we see that cinematic  visualizations often only show one or a few data points in the frame,  favoring to include additional contextual elements that help add  emotional resonance to the data-story being told.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Designers trade-off between perceptual effectiveness and  dramatic narrative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Visualizations that employ 3D graphics are  often ineffective perceptually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' These graphics may use sub-optimal  encoding channels like volume and can further bias judgement  through distortion and occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Cinematic visualizations are not  appropriate when the task is centered around value judgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Instead, we see cinematic visualizations effectively used when a  rough estimate of values is sufficient and the precise value is not  of central importance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' CV29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Many of the cinematic visual-  izations that we analyzed use a volume encoding to display data  (CV1,6,26,27,35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Volume is a less effective encoding channel com-  pared to position and may cause the audience to misestimate the  true quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' This trade-off may be acceptable depending on the  data being presented and the precision with which the author hopes  it will be apprehended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  4  CASE STUDY: HOW MUCH IS A  GIGATONNE?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  We collected and studied the aforementioned cinematic visualiza-  tions while exploring designs to support the communication ob-  jectives of NASA’s Earth Science Communications Team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Climate  change is a complex, multi-faceted issue of global importance [49]  and the team is tasked with maintaining climate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='gov, a website  that tracks vital statistics about Earth’s climate, and delivers up-  dates about global warming to a diverse global audience of millions  of readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The team uses traditional information graphics [48], as  well as narrative visualizations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=', [53]), to highlight how scien-  tists know that anthropogenic global warming is truly happening,  what changes have taken place in Earth’s climate so far, and why it  is an important topic for readers to understand even if it does not  seem to be affecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' the team sought data-driven  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='stories that more viscerally engaged their audience and connect  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Digital set design  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content="Cam1 (God's eye view)  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content="Cam2 (bird's eye view)  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Lighting: Global Illumination  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Data-driven element  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Geographic elements  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Cam3 (point-of-view)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Texture from satellite images  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Rendered output  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Central Park  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='ewYorkCit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content="God's eye view (Establishing shot)  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Point-of-view (Establishing)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Point-of-view (Initial action)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Medium-long shot (Peak)Cinematic Techniques in Narrative Visualization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Figure 6: We explored many different designs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' these were left on the cutting room floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The designs were dropped for reasons  including poor perceptual effectiveness (A-C), locations too small for the scale of the data (D-F), and designs too illustrative  and not physically accurate enough (G-H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' It was particularly difficult to identify locations that were broadly recognizable  from a 3D reconstruction but also suitable to server as a concrete scale referent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  the planetary scale data of climate change to a human scale that  readers can readily understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Within the domain of climate change communication is a range  of research investigating how to effectively communicate the latest  science to a broad audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' High level principles of climate change  communication have been synthesized by the Center for Research  on Environmental Decisions [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We think cinematic visualizations  are well suited to satisfy principles “Get Your Audience’s Attention“  and “Translate Scientific Data Into Concrete Experience.” Here  we describe how our work creates connections between ongoing  investigations in narrative visualization, computer graphics, and  film art to achieve this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Guided by editorial priorities set by NASA’s Earth Science Com-  munication team, we produced an article consisting of a several  cinematic visualizations to communicate massive quantities related  to climate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We endeavoured to make them interpretable and  meaningful to a broad public audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' These visualizations were  eventually published to an audience of millions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Here we describe  our design process to create cinematic visualizations, identifying a  general workflow of use to practitioners who wish to create this  type of visualization themselves, and to tool-builders who wish to  provide better support for authoring cinematic visualizations in  the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' As with visualization production in general, these steps  are not necessarily linear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' rather, the process is iterative and error  prone, and may require going back to earlier steps if it becomes  apparent that a design is not working.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We experienced many failed  attempts (see Figure 6) before arriving at our final designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='1  Pre-Production  Narrative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Quantities of ice loss are measured in gigatonnes, a  unit of mass corresponding to one million metric tons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Statistics  about ice loss are often reported using this unit, for example Earth’s  polar ice caps are losing about 426 gigatonnes of ice per year, at  the time of writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The scale of the unit here hides the fact that  426 gigatonnes is a massive amount of ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Our goal was to provide  a visualization that would allow our audience to better interpret  these statistics going forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We collected statistics on ice loss in  Greenland and Antarctica (the two ice sheets) over the course of  significant periods, such as the amount of ice lost between 2002-  2017 when NASA’s Grace satellite was actively observing the polar  ice caps, or since the start of the 20th century (5,000 and 49,000  gigatonnes, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  We settled on cinematic visualization because it is a natural fit  for the use of concrete scales, we wanted to draw people’s attention,  there is a relatively small amount of data that we are showing, and  we wanted to display the data in a context that conveyed corporeal  urgency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Given the affordances identified in Section 3, a cinematic  visualization was an appropriate choice for our task of visualizing  quantities related to climate change in a way that would capture  the attention of our audience and allow them to comprehend the  data in a concrete way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We ultimately chose the form factor for our  visualization to be an interactive article containing a series of short  cinematic visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The visualizations were embedded as pre-  rendered videos, which could be loaded dynamically, allowing for  a certain amount of interactivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Depending on the use case, one  must determine whether real-time rendering is needed or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Using  real-time rendering limits the level of photorealism [52], but enables  another level of interactivity, letting the user control the camera  and interact with elements in the scene (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 7D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We intended  the narrative structure of our visualization to be largely author-  driven [57], and decided that real-time rendering was not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  After determining that a cinematic visualization was appropri-  ate, we began outlining possible scripts and creating storyboards  in which we sketched ideas for locations, cinematography, and se-  quencing of shots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We first sought to identify locations that would  serve as effective backdrops, allowing people to gain a concrete  understanding of the size of data in familiar locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We consid-  ered natural locations like the Grand Canyon, Monument Valley,  Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Everest, and Uluru, urban environments like Houston, New  York City, San Francisco, and St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Louis, and other man-made sites  like football stadiums and the Hoover Dam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Within each of these  environments we created sketches to help determine the camera  placement, mise-en-scène, data, and annotations that the visualiza-  tions would require, and wrote rough scripts to define the narrative  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  While we wanted to place data in a variety of different envi-  ronments so that our diverse audience would be able to connect,  2000  1979  2009  Carbon  Emissions  7021  M, ,  Conlen, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  ultimately many of these locations were not used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' See Figure 6 for  examples of some of the locations that were not able to support  both focus and context at an anthropocentric perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The final  article consisted of videos visualizing one, then 5,000, then 49,000  gigatonnes of ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The videos were embedded throughout the text  of an article which provided context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In the first and last videos the  user could click to choose to play videos displaying the relevant  quantity of ice in different locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Here we look closely at the  design process for the first video, showing one gigatonne of ice in  Central Park, New York City.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='2  Principal Photography  With the storyboards and scripts ready, the source footage that  would make up the final video needed to be created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We chose to  use Blender for this process, which provides both an interactive  GUI-based interface as well as a Python API that allowed us to  load, transform, and bind data to objects in a 3D scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We created  renders for many different scenes, although ultimately ended up  using a small number of them in our published pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Mise-en-scène.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The elements that constitute the mise-en-scène  of a cinematic visualization need to be created and arranged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Be-  cause many of our scenes take place in real-world locations, we  were able to utilize existing open data sets to import geographic  data, including 3D models of buildings and terrain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In addition  to elements derived from real-world locations, we added elements  which would be parameterized by data, for example the large block  of ice placed in Central Park (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' After the models have been  created, they need to be assigned a material, which (along with  lighting) will determine how they appear in final renders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We chose  to use a flat shading for the buildings and other environmental  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' This gave these elements less visual weight while still al-  lowing them to be easily identifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We considered using a similar  flat style for the data elements, but ultimately decided to add a more  photorealistic ice material which would allow the data to stand out  against the buildings and reinforce the idea that we were showing  a concrete amount of ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' While many of the examples that we saw  utilize a studio lighting setup to control shadows and reflection, we  opted to use simple global illumination to emulate the sun shining  in our outdoor scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' This meant our lighting was realistic for the  location and the setup was quite simple, but we were limited in our  ability to use lighting as a tool to guide attention, as we saw it used  (for example) in CV15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  With the scene constructed, the next step was to bind the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  This was the point at which we realized that many of the set lo-  cations were not going to work with the data we were hoping to  visualize (“data changes everything” [66]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' For example, a gigatonne  of ice placed in a football stadium (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 6D) would extend over 200  kilometers into the sky, making it difficult to view both the diegetic  data and the stadium itself simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' For our visualizations  we were simply assigning the dimensions of a primitive 3D object  based on calculations related to the mass of ice melt over specific  periods, along with the density of ice, in order to create blocks of  ice which were physically representative of the quantity lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Cinematography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' After we incorporated our data into the scene  it was time to add animation and cinematography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Blender supports  a keyframe-based animation system which made it simple to add  basic animations to the size and locations of elements in the scene,  as well as the position and perspective of cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Working off of  the storyboards that we had created, we placed cameras (shown  in Figure 5) that would be physically realistic and familiar: we use  three cameras, one a human point-of-view, one a bird’s eye view  (as if it were taken from a helicopter circling the city), and one a  "god’s eye view" taken from the perspective of a satellite overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  The satellite camera allowed us to create an initial establishing shot,  while the other cameras provided views that supported a ground-  level view as well as an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' When sequenced together, these  camera perspectives allow us to present focus plus context [13] to  the viewer, and support our narrative goals [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='3  Post-Production  Once the source material was created, we needed to edit it to form  a coherent narrative, for example by combining multiple videos in  sequence, adding annotations on top of the video to add context,  and adding sound to add presence, guide attention, and provide  details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Any visual effects must be added at this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' For example,  in the case of embedding digital data objects into physical footage  of a narrator, a “match moving” process to align the digital and  physical scenes would need to be performed [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We combined footage from multiple cameras, compos-  ing shots into a narrative structure, starting with establishing shots,  then initial action, peak, and finally release [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The sequence of  images is important to advance the role of narrative, pacing, and  mood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Narrative visualizations often include annotations to provide  additional context and explain to viewers what it is they are seeing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  In the case of cinematic visualizations these annotations can be  composited over the source footage using standard video editing  software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Some examples that we saw embed annotations directly  into the 3D scene itself, which requires them to be embedded in the  source footage directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We chose to composite annotations rather  than include them “in-situ” as it facilitated more rapid iteration dur-  ing the editing process, allowing us to change the timing, location,  and content of annotations, without needing to re-render any of  the source footage — a potentially time-consuming process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In our work we ultimately did not use audio, instead  opting to embed the videos in a larger text article, which would  serve to provide viewers with context for the visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' This is  a limitation and something to be explored more in future work, as  audio can be a useful tool in cinematic visualization to set tone and  drive narrative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='4  Publication  Once the article was completed and approved for publication, it  was posted to NASA’s climate website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We did not collect detailed  metrics on how readers interacted with the videos on the article  itself, but can see how users responded to posts on the NASA  Climate Facebook, Instagram, and Twitter pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' These posts—  which contained a link to the article and (in some cases) directly  embedded the video set in New York City—were collectively viewed  tens of thousands of times,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' received thousands of engagements  (likes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' comments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' shares),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' and the article was subsequently shared  by other organizations such as the United States Department of  Cinematic Techniques in Narrative Visualization  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Agriculture and the World Meteorological Organization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' as well as  by individual scientists and meteorologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Across all of the social platforms users left 94 direct comments,  with topics ranging from positive (for example, some explicitly  expressing that they like this type of visualization “We need more  of these types of comparisons in the media”, “This is an amazing  visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Thanks NASA!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=', or asking for similar visualizations  of different quantities “It would be very interesting to see this illus-  tration but with the predicted sea level after all the ice in Greenland  and Antarctica melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Can you show that?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=') to concern about the  data being visualized (“Oh my God.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Come to our aid.”, “Thanks for  helping us comprehend the enormity of this sad news!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=', a GIF of  a cartoon rodent crying) to climate change denial (“Where’s your  proof”, “Wow, as much as 2 millimetres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Measured by satellite too”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  The comments were distributed roughly uniformly across the three  types (positive attitude toward visualization, concern about climate  change, and climate change denial), but varied heavily across plat-  forms, with users on Facebook expressing concern or denying that  there is a climate problem, users on Instagram leaving both positive  and concerned comments, and users on Twitter expressing a range  of concern, denial, and a positive attitude toward the graphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  5  DISCUSSION  Cinematic visualizations can engage viewers with dramatic and  visceral presentations of data, highlighting particularly important  data points, and presenting an author-guided tour through data  embedded in a relevant context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' On the other hand, they may be  poor choices for communicating large amounts of data and are  not optimal in terms of perceptual effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' If a cinematic  visualization is appropriate, it will require a broad range of skills —  such as cinematography, narrative, 3D modeling, video editing, and  possibly acting — and a time-consuming iterative design process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='1  Challenges of Creating Cinematic  Visualizations  While cinematic visualizations can capture the attention of their  audience and help viewers relate to the data in a concrete way,  they can be challenging and time-consuming to produce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Here we  discuss some of the challenges inherent in creating an effective  cinematic visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  One of the most apparent difficulties of cinematic visualization  is the potentially overwhelming size of the design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Works  in this genre typically use three visual dimensions, plus time and  sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The methods that allow us to analyze and critique cinematic  visualizations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=', [5]) do not necessarily help us to create them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  That is, they are difficult to use generatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' While information  designers are familiar with the attention to detail that is required  when placing objects in a frame in order to achieve an effective  visual hierarchy, in cinematic visualizations there are also objects  outside of the frame that affect the style and tone of the visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  For example, the placement of the camera in relation to the subjects,  the focal length of the camera, and the placement and strength of  light sources are all instrumental in creating a shot which can easily  be decoded by viewers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  There is a diversity of tasks that need to be completed in order  to create a cinematic visualization, each requiring a separate set of  skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' For example, in addition to skills required for traditional visu-  alization (data analysis, transformation, and visualization) and nar-  rative visualization (understanding audience, storytelling, graphic  design), cinematic visualization will often make use of animation,  cinematography, lighting, motion graphics, 3D modeling, sound  design, video editing, and (sometimes) acting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The skills that make  one a good 3D modeler are not necessarily the same skills that make  one a good storyteller, and so graphics of this type often require  a diverse team to create.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Furthermore, for ray-tracing renderers,  there is a large gap between prototypes and final rendered output,  challenging the iterative design process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='2  Considerations for Cinematic Visualization  Creators  While cinematic visualizations share many of the same design goals  of more traditional narrative visualization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=', guide the viewers’  attention), the way in which these goals are operationalized differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Here we highlight ways that these design goals were operational-  ized across the four dimensions of style, both in our own work and  in the examples analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' For a full breakdown of the techniques  used in each example, see Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Mise-en-scène.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Objects’ sizes, colors, shapes, textures, and place-  ment in relation to one another can all be used create an effective  visual hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' For example, to guide a user’s attention in a cin-  ematic visualization, a designer might choose to use lighting to  cast a glow around an object (CV11), or change the object’s color  (CV2, CV13) so that it stands out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In How Much is a Gigatonne,  the ice’s large size, color, and shine draw a viewers attention to  it in contrast with the surrounding buildings, which are smaller,  grayscale, and matte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The mise-en-scène is designed both to com-  municate information—including using narrators (CV42), diegetic  data (CV35), and visual analogies (CV6)—and to add dramatic affect  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' CV11, CV40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Cinematography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Perspective can be used both to drive narra-  tive and to set tone, as well as to provide focus plus context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The  position (CV26), angle (CV28), or focus (CV2) of a camera can be  modified so that the object becomes the focal point of the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  To help narrow the large space of possible cinematic visualizations,  and make effective use of the frame, designers of cinematic visual-  ization may study how shots are composed and sequenced in films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  In How Much is a Gigatonne?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=', we rendered footage from multiple  cameras in order to create close-up, medium, and wide shots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Some  cinematic visualizations enable limited user-control of the camera,  for example letting the user trigger the next stage of animation  (CV9) or rotate their perspective (CV13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Often the camera needs  to track a particular object in the scene (CV12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' If this object is in  motion you may need to set your camera to track it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Planning the  path of the camera so that the object of interest is not occluded by  other objects and so that motion is smooth and visually pleasing  can be difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' This may be done algorithmically [15, 68] or by  hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Putting the footage into a particular order progressively  reveals information to convey the authors’ intended message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Edi-  tors may use footage from one camera at one location (CV29), or  multiple cameras at multiple locations (CV40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The editing tech-  niques used in data videos—particularly the use of establishing,  , ,  Conlen, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  initial, peak, and release shots—has been studied in more depth by  Amini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Similar to movie makers, creators of cinematic visu-  alizations may use the technique of storyboarding to prototype and  communicate their scenes in a lo-fidelity form before endeavouring  on the time intensive task of 3D modeling and rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In How  Much is a Gigatonne we use establishing shots to situate the viewer  before initiating action from the perspective of the ground level (an  anthropocentric perspective), before cutting to the vantage point  of a helicopter, using the city skyline as a concrete scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Audio can set tone (CV25), cue attention (CV28), and  impart additional details through narration on (CV40) or off-screen  (CV45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Music (CV29) and ambient sound (CV26) can affect the tone  of the visualization and add presence to the scene, for example  CV29 uses combines techno music and a visual analogy of of the  weight of rocket fuel (measured in elephants) to create a humorous  juxtaposition which may make the visualization more approachable  and less dry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' CV26 uses diegetic sound (taxi cabs honking) to rein-  force the anthropocentric perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In How Much is a Gigatonne  we did not use sound (neither did most of the other visualizations  that we analyzed which used an “article” format), but effective use  of both the visual and auditory channels has been shown to lead to  improved outcomes in multimedia learning contexts [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='3  Implications for Authoring Tools  As cinematic visualization is a newly emerging genre, there is rel-  atively little tool support to facilitate authoring of this type of  visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Instead, creators turn to general purpose 3D software  that was designed to support a breadth of use cases such as architec-  tural design, modeling, and narrative animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' These tools, while  powerful and expressive, may overwhelm users with complexity  that is incidental to the task of creating a cinematic visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  For example, objects are assigned materials which are powered by  low-level shader code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' One can not choose, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=', between “realistic”  or “cartoon” aesthetics but instead must compose low level shader  components to achieve the desired look.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  These tools do not support the basic building blocks of visualiza-  tion, such as easily ingesting data and binding data values to objects  in a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Instead, users must write custom scripts to handle any  such task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The interfaces in general are multi-modal: most 3D mod-  eling work is done directly through a GUI, but data-driven work  needs to be done in code;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' shaders are described using a directed  graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Authors are forced to context switch between drastically  different environments, arguably making it harder to iterate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  The task of 3D rendering can be computationally intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' De-  pending on the output resolution, complexity of the scene, and  computing power available, a short (30 seconds) animation could  take several hours to render.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' There is a large gap between the  fidelity of the final renders and what a designer sees while con-  structing the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' This setup makes it important to create test  renders frequently, but makes it hard to have a rapid feedback loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='4  Limitations of our Work  Our survey was limited to 50 examples, taken from a limited set of  sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' While not exhaustive, the examples implement a range of  design techniques across a variety of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We do not pro-  vide an empirical evaluation of the work surveyed, instead choosing  to use techniques of film criticism in order to analyze patterns used  and identify the communication intentions of their producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We  similarly did not empirically evaluate our own work, and instead  provide an account of our design process and detail our reasoning  for important decisions that were made along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Our work  does not fully utilize the design space of cinematic visualizations  that we identified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' for example, we did not use sound at all, and all  narration was done through written text with a few small overlays  in the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The experience might be improved by incorporating  narration either on-screen or off [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  6  CONCLUSION  We presented cinematic visualization, a genre of narrative visu-  alization that uses techniques from cinema in order to enhance the  presentation of data-driven stories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' A central contribution of this  work is to identify a new genre of narrative visualization that we  then analyze in depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The importance of genre is clear in other art  forms like literature and cinema;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' however, it is invoked less often  in the context of visualization research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' We believe that this type of  work is crucial for understanding the design of narrative visualiza-  tions, and thinking rigorously about how they can be constructed  and deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' While past work on narrative visualization looked  specifically at the narrative structure, here we look at both narra-  tive and style as formal systems that contribute to the dramatic  experience of watching a cinematic visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' To do this, we  turned to theory from another form of art, film, in order to provide  grounding in the features of style, and used analysis techniques  established in that domain to deconstruct our case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  We analyzed a variety of examples of cinematic visualization and  the techniques that they employ towards certain narrative applica-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Many of these visualizations show a relatively small amount  of data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=', focusing on a single rate or quantity) as opposed to  being data-dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The non-data elements of the scene play an im-  portant role: they are used to set the location in which the shot is  taking place and provide cues to viewers about where they are, what  they are looking at, and why it is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' This approach is quite  different from typical information visualizations, where data may  be reduced to a minimal form, such as a line or a bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Cinematic  visualization instead tends to be more maximal in its approach,  such that the non-data ink is not reduced or omitted, but rather  used to build up entire digital worlds around data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' This style  encourages viewers to feel present in locations augmented with  data objects, or to viscerally experience events that happened in  the past, or are happening far away in the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Rendering data in 3D is a fraught endeavor, as the values being  rendered can be obscured by humans’ relatively poor ability to  estimate and compare volume, and because the 3D projection can  introduce distortion when trying to read values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Why would the cre-  ators choose to follow a cinematic path over one that more clearly  and directly communicates the underlying data with precision?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  We argue that in choosing to treat a visualization as a cinematic  experience, its authors might be looking beyond the immediate  data, in order to viscerally ground that data in meaningful context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  In other words, analytic precision is only one of several objectives  that a visualization might help accomplish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In choosing 3D, we  might diminish precision in service of other objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Cinematic Techniques in Narrative Visualization  , ,  ACKNOWLEDGEMENTS  We would like to thank Susan Callery, Holly Shaftel, Randal Jackson,  Daniel Bailey, Michael Gunson, Josh Willis, Joe Witte, and the  Earth Science Communications Team at NASA’s Jet Propulsion  Laboratory for their support of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' A portion of this research  was carried out at the Jet Propulsion Laboratory, California Institute  of Technology, under a contract with the National Aeronautics and  Space Administration (80NM0018D0004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Microsoft Timeline  Storyteller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' https://timelinestoryteller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Immersion in digital games:  review of gaming experience research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Readings in information visualiza-  tion: using vision to think.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Morgan Kaufmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Using concrete scales:  A practical framework for effective visual depiction of complex measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' IEEE  transactions on visualization and computer graphics 19, 12 (2013), 2426–2435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' A constraint-based approach to camera  path planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In International Symposium on Smart Graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Springer, 172–181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' Graphical perception: Theory, ex-  perimentation, and application to the development of graphical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' An evaluation of cone trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In  People and Computers XIV—Usability or Else!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Springer, 425–436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Idyll: A Markup Language for Authoring  and Publishing Interactive Articles on the Web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In ACM User Interface Software &  Technology (UIST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content='washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='edu/papers/idyll  [19] Matthew Conlen, Alex Kale, and Jeffrey Heer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Capture & Analysis of Active  Reading Behaviors for Interactive Articles on the Web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Computer Graphics Forum  (Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content='washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' The Reyes image  rendering architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' ACM SIGGRAPH Computer Graphics 21, 4 (1987), 95–102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' IATK: An Immersive  Analytics Toolkit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In 2019 IEEE Conference on Virtual Reality and 3D User Interfaces  (VR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' IEEE, 200–209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Metaphoric mappings: The art of visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Aesthetic computing (2006), 89–114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Matchmoving: the invisible art of camera tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' John Wiley  & Sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  [24] Anthony J Ferri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Willing suspension of disbelief: Poetic faith in film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Lexington  Books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The photographer’s eye: composition and design for better  digital photos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' CRC Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' Technique of the drama: An exposition of dramatic composi-  tion and art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Scott, Foresman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' Mise-en-scène: film style and interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' Wallflower  Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Simply bells and whistles?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Cognitive effects of visual aesthetics in digital longforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Digital Journalism 7, 2  (2019), 273–293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' The Fallen of World War II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content='fallen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Animated Transitions in Statistical Data  Graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Visualization & Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' The making of  toy story [computer animation].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In COMPCON’96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Technologies for the Information  Superhighway Digest of Papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' IEEE, 463–468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Sound for film and television.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Taylor & Francis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  [33] Jessica Hullman and Nick Diakopoulos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Visualization rhetoric: Framing  effects in narrative visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' IEEE transactions on visualization and computer  graphics 17, 12 (2011), 2231–2240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' A deeper understanding of sequence in narrative  visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' IEEE Transactions on visualization and computer graphics 19, 12  (2013), 2406–2415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Immersive  Visual Data Stories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In Immersive Analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Springer, 165–184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  A Walk Among the Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' IEEE computer graphics and applications 39, 3 (2019),  19–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  [37] Yvonne Jansen, Pierre Dragicevic, Petra Isenberg, Jason Alexander, Abhijit Karnik,  Johan Kildal, Sriram Subramanian, and Kasper Hornbæk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Opportunities  and challenges for data physicalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In Proceedings of the 33rd Annual ACM  Conference on Human Factors in Computing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 3227–3236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' A realistic camera model for  computer graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In SIGGRAPH, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Storytelling: The next step for visual-  ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' Principles of traditional animation applied to 3D computer  animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In ACM Siggraph Computer Graphics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' ACM, 35–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Data Visceralization: Enabling Deeper Understanding of  Data Using Virtual Reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The modality principle in multimedia learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  The Cambridge handbook of multimedia learning 147 (2005), 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Scientific storytelling using visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' IEEE Computer Graphics  and Applications 32, 1 (2011), 12–19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Immersive  Analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Information is beautiful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Collins London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content='  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Visual narrative flow: Exploring factors shaping data visualization story  reading experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' Wiley Online Library,  377–387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' Hyperobjects: Philosophy and Ecology after the End of the  World.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' U of Minnesota Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Vital signs of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' https://climate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='gov/vital-signs/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Climate change 2014: synthesis report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Contribution of Working  Groups I, II and III to the fifth assessment report of the Intergovernmental Panel on  Climate Change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Ipcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Visual Data Storytelling Contest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' https://vimeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='com/pviscontest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Atom: A grammar for unit visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' IEEE transactions on visualization and  computer graphics 24, 12 (2017), 3032–3043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Physically based rendering:  From theory to implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Morgan Kaufmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  [53] Carol  Rasmussen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  A  Partnership  Forged  by  Fire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  https://climate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='gov/news/2899/a-partnership-forged-by-fire/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' The technique of film editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' Visualization Authoring for Data-driven Storytelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' UC Santa Barbara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  [56] Arvind Satyanarayan and Jeffrey Heer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' Wiley Online Library, 361–370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' IEEE transactions on visualization and computer graphics 16, 6 (2010),  1139–1148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The psychology of climate change com-  munication: a guide for scientists, journalists, educators, political aides, and the  interested public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Center for Research on Environmental Decisions, New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Dxr: A toolkit for building immersive data  visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' IEEE transactions on visualization and computer graphics 25, 1  (2018), 715–725.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Emerging  and recurring data-driven storytelling techniques: Analysis of a curated collection  of recent stories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' The illusion of life:  Disney animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Hyperion New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Understanding the Design Space  and Authoring Paradigms for Animated Data Graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' Wiley Online Library, 207–218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' IEEE  Computer Graphics and Applications 9, 4 (1989), 30–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Data Changes Everything: Chal-  lenges and Opportunities in Data Visualization Design Handoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' IEEE transactions  on visualization and computer graphics 26, 1 (2019), 12–22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+page_content=' Information visualization: perception for design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  [68] I-Cheng Yeh, Chao-Hung Lin, Hung-Jen Chien, and Tong-Yee Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Effi-  cient camera path planning algorithm for human motion overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Computer  Animation and Virtual Worlds 22, 2-3 (2011), 239–250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  [69] Qiyu Zhi, Alvitta Ottley, and Ronald Metoyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Linking and Layout: Ex-  ploring the Integration of Text and Visualization in Storytelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In Computer  Graphics Forum, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Wiley Online Library, 675–685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Cinematic Techniques in Narrative Visualization  , ,  Figure 7: We analyzed the style of 50 cinematic visualizations using the features of mise-en-scène, cinematography, editing, and  sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' An HTML version of this table including URLs for each row can be found at https://cinematic-visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='io/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Cinematography  Mise-en-scene  Editing  Audio  rative Text  Composited Annotations  Jser-controlled camera  Realistic Background  Opacity  mera  er-triggered steps  3617 - L  n-Situ Annotations  Composited Narra  Point-of-view can  nera  Visual Analogy  rator  it Visualizatiol  Scale  punos oeba  Reality  n-situ Narra  notation  Annotation  Annotation  dded  ncrete  erview  pax  ISIC  0  Author  Publishel  Titte  cV1  Drowning in plastic  Simon Scarr, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Reuters  CV2  Is This the Neighborhood New York Deserves?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Michael Kimmelman  NYT  CV3  Krigsskipet som krasjet og sank  B  Stangvik, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  VG  CV4  How China Turned a City Into a Prison  Chris Buckley, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content="  NYT  CV5  The Forbidden City's unique architecture  Marco Hernandez  South China Morning Post  CV6  Here are 120 million Monopoly pieces, roughly one." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Confessore, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content="  NYT  CV7  Building Katie's New Face  Jason Treat  NatGeo  CV8  Mass Exodus: The scale of the Rohingya crisis  Christian Inton  Reuters  This 3-D Simulation Shows Why Social Distancing." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  CV9  Parshina-Kottas, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content="  NYT  CV10 Tracking China's Muslim Gulag  Simon Scarr  Reuters  CV11  Choice and Chance  Staff  Tampa Bay Times  cV12 Cassini's Grand Tour  Nadia Drake, et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  NatGeo  CV13 Resurecting a Dragon  Brian T Jacobs  NatGeo  CV14Want to fireproof your house?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=" Here's where to start  Kyle Kim, et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  LATimes  CV15Apoll 11 -As They Shot It  Jonathan Corum, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=" NYT  CV16 The Atlas of Moons  NatGeo Staff  NatGeo  CV17 Explore a Toad's Digital Clone  Brian T." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Jacobs  NatGeo  CV18 THE THOMAS FIRE: 40 DAYS OF DEVASTATION  Joe Fox  LATimes  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  CV19 Is the Nasdag in Another Bubble?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Roger Kenny, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Wall Street Journal  CV20 A 3-D View of a Chart That Predicts The Econom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Gregor Aisch, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  NYT  CV21  Inside the Taser  Simon Scarr  Reuters  CV22 Seeing Earth from Outer Space  Matthew Conlen  The Pudding  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  CV23 Of Catastrophes and Rescues: Making the Invisib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='.  Peter Mindek, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  PacificVis  CV24A Visualization of Two-stage Autoignition of n-dod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='.  Yucong Ye, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  PacificVis  CV25 How Mariano Rivera Dominates Hitters  Graham Roberts, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' NYTimes  Real World Visuals  Real World Visuals  CV27 CCS:A2 degree solution  Real World Visuals  Real world Visuals  CV28CARS  Real World Visuals  Real world Visuals  CV29[REALISTIC] Elephant rocket fuel - Saturn V  Maxim Sachs  YOUTUBE  CV30 University of Exeter greenhouse gas emissions in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='.  Real World Visuals  Real world Visuals  CV31if The World Were 100 People  Gabriel Reilich, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Good Magazine  CV32 Up - and down - from Ground Zero  Graham Roberts, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' NYT  CV33 The Birth of a Virtual Cell  Peter Mindek, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  PacificVis  CV34  The Nuclear Threat - The Shadow Peace  Neil Halloran  Youtube  CV35What f Carbon Left Your Tailpipe as Solid Chunks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Sukee Bennett  PBS Nova  CV36Stay Home,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Flatten the Curve  keta  Youtube  CV37 Chart Party: We decided to erase the three-pointer  Jon Bois  Youtube  cV38 200 Countries,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 200 Years,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' 4 Minutes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Hans Rosling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Youtube  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content="CV39 The best stats you've ever seen  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Hans Rosling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='TED  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='CV40 VFX Artist Reveals the True Scale of the Universe  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Wren Weichman  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Corridor Crew  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='CV41  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Helge Ingstad  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Hallvard Sandberg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='NRKbeta  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='CV42The dangers of storm surge  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='The Weather Channel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='The Weather Channel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='CV43 Survive the Tornado  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='The Weather Channel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='The Weather Channel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='CV44  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Television Elections Coverage  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='KING5 TV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='KING5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='CV45 Powers of Ten  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='Eames & Eames  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='IBM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='CV46 Strange things happen when you rotate in 4 dimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Hamish Todd  Youtube  CV47 Discovering Gale Crater  Armand Emamdjomeh LATimes  A  CV48 Taiwan earthquake: Survivors found in rubble of Ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=".  Malachy Brown,  SketchFab / Australian  cV49 Four of the Best Olympians, as You've Never Seen." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  John Branch  NYT  CV50 How We Created a Virtual Crime Scene to Investig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content='  Malachy Brown,  NYT  CV13  CV40  CV26  CV35  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Author-Guided  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' In-situ Narrator  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Anthropocentric Perspective  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
+page_content=' Resolving Scale  Interactive Camera' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfWAR_/content/2301.03109v1.pdf'}
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+Do entangled states correspond to entangled measurements under local transformations?
+Florian Pimpel,1, ∗ Martin J. Renner,2, 3, ∗ and Armin Tavakoli4
+1Atominstitut, Technische Universität Wien, Stadionallee 2, 1020 Vienna, Austria
+2University of Vienna, Faculty of Physics, Vienna Center for Quantum Science and Technology (VCQ), Boltzmanngasse 5, 1090 Vienna, Austria
+3Institute for Quantum Optics and Quantum Information - IQOQI Vienna,
+Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
+4Physics Department, Lund University, Box 118, 22100 Lund, Sweden
+We investigate whether pure entangled states can be associated to a measurement basis in which all vectors are
+local unitary transformations of the original state. We prove that for bipartite states with a local dimension that is
+either 2, 4 or 8, every state corresponds to a basis. Via numerics we strongly evidence the same conclusion also
+for two qutrits and three qubits. However, for some states of four qubits we are unable to find a basis, leading
+us to conjecture that not all quantum states admit a corresponding measurement. Furthermore, we investigate
+whether there can exist a set of local unitaries that transform any state into a basis. While we show that such a
+state-independent construction cannot exist for general quantum states, we prove that it does exist for real-valued
+n-qubit states if and only if n = 2, 3, and that such constructions are impossible for any multipartite system
+of an odd local dimension. Our results suggest a rich relationship between entangled states and iso-entangled
+measurements with a strong dependence on both particle numbers and dimension.
+Entanglement is a fundamental, broadly useful and an in-
+tensely studied feature of quantum mechanics.
+However,
+in spite being of arguably similar foundational significance,
+much less is known about the entanglement of joint quan-
+tum measurements than the entanglement of quantum states.
+Entangled measurements are crucial for seminal quantum in-
+formation protocols such as teleportation [1], dense coding
+[2] and entanglement swapping [3], which are instrumen-
+tal for various quantum technologies.
+Typically, they are
+based on the paradigmatic Bell basis, which is composed of
+the four maximally entangled states (|00⟩ ± |11⟩)/
+√
+2 and
+(|01⟩ ± |10⟩)/
+√
+2. In the same way that the Bell basis may
+be thought of as the measurement corresponding to the max-
+imally entangled state, it is natural to ask whether entangled
+states in general can be associated with a corresponding en-
+tangled measurement. Studying the relationship between en-
+tangled states and entangled measurements is not only inter-
+esting for understanding quantum mechanics. It is also an
+invitation to explore, in the context of quantum information
+applications, the largely uncharted terrain of entangled mea-
+surements beyond the Bell basis and its immediate general-
+isations. Most notably, entangled measurements beyond the
+Bell basis are also increasingly interesting for topics such as
+network nonlocality [4] and entanglement-assisted quantum
+communication [5, 6].
+Consider that we are given a pure quantum state |ψ⟩ com-
+prised of n subsystems, each of dimension d. Is it possible
+to find a measurement, namely an orthonormal basis of the
+global dn-dimensional Hilbert space, in which all basis states
+have the same degree of entanglement as |ψ⟩? Specifically, we
+want to decide the existence of dn strings, {Vj}dn
+j=1, of local
+unitary transformations,
+Vj =
+n
+�
+k=1
+U (j)
+k
+(1)
+∗ These authors contributed equally.
+where U (j)
+k
+is a d-dimensional unitary operator, such that the
+set of states |ψj⟩ ≡ Vj |ψ⟩ form a basis, i.e. | ⟨ψj|ψj′⟩ | =
+δjj′. If affirmative, we say that |ψ⟩ admits a basis and we call
+the set of basis vectors {|ψj⟩}dn
+j=1 a |ψ⟩-basis.
+Known examples of entangled measurements can be ac-
+commodated in this picture. For example, the Bell basis can
+be obtained from operating on |ψ⟩ = (|00⟩ + |11⟩)/
+√
+2 with
+the four strings of local unitaries {Vj}4
+j=1 = {11 ⊗ 11, 11 ⊗
+X, Z ⊗ 11, Z ⊗ X}, where X and Z are bit-flip and phase-
+flip Pauli operators. A well-known generalisation of the Bell
+basis to n systems of dimension d can be thought of as a
+|GHZn,d⟩-measurement where the relevant state is the higher-
+dimensional GHZ state |GHZn,d⟩ =
+1
+√
+d
+�d−1
+k=0 |k⟩⊗n. The
+corresponding strings of local unitaries are Vj = Zj1
+d ⊗Xj2
+d ⊗
+. . . ⊗ Xjn
+d |GHZn,d⟩ where j = j1 . . . jn ∈ {0, . . . , d − 1}n
+and where Zd = �d−1
+l=0 e
+2πi
+d l |l⟩⟨l| and Xd = �d−1
+l=0 |l + 1⟩⟨l|
+are generalised Pauli operators. More generally, any state that
+is locally maximally entanglable (for example graph states)
+is known to admit a basis via suitable unitaries of the form
+Vj = U j1
+1 ⊗ . . . ⊗ U jn
+n
+[7]. These states are characterised
+by the property that if each qubit is supplemented with a
+qubit ancilla and controlled unitary gates are performed on
+the state-ancilla pairs, then a maximally entangled bipartite
+state can be constructed between the collection of state-qubits
+and the collection of ancilla-qubits.
+However, this is far
+from a complete characterisation of the states that admit a
+basis, which is seen already in the restrictive form of the
+strings of unitaries. For example, the three-qubit W-state,
+|W3⟩ = (|001⟩+|010⟩+|100⟩)/
+√
+3, is not locally maximally
+entanglable but is neverthelss known to admit a basis [8]. In
+what follows, we set out to systematically explore whether
+entangled states admit a corresponding basis and then, as we
+will introduce later, whether such bases can be constructed
+even without prior knowledge of the state.
+Let us begin with considering the simplest situation, namely
+when |ψ⟩ is a state of two qubits. We constructively show that
+every such state admits a basis. To this end, we first apply
+the state-dependent local unitaries W A
+ψ ⊗ W B
+ψ that map |ψ⟩,
+arXiv:2301.13285v1  [quant-ph]  30 Jan 2023
+
+2
+via a Schmidt decomposition, into the computational basis,
+|ψS⟩ = λ |00⟩+
+√
+1 − λ2 |11⟩ for some coefficient 0 ≤ λ ≤ 1.
+Then, we consider the action of the following four strings of
+local unitaries
+�
+�
+�
+�
+�
+11 ⊗ 11
+11 ⊗ XZ
+XZ ⊗ Z
+XZ ⊗ X
+�
+�
+�
+�
+�
+.
+(2)
+One can verify that this transforms |ψS⟩ into a |ψ⟩-basis. No-
+tice that once the state has been rotated into the Schmidt form
+|ψS⟩, the subsequent unitaries (2) do not depend on λ. This
+construction can be extended to bipartite (n = 2) states of
+local dimension d = 4 and d = 8. Again via Schmidt de-
+composition, we can find state-dependent local unitaries that
+transform |ψ⟩ into |ψS⟩ = �d−1
+l=0 λl |ll⟩ for some Schmidt co-
+efficients �
+l λ2
+l = 1. In Appendix A, we show that there is a
+set of local unitaries that indeed leads to a |ψ⟩-basis indepen-
+dently of the specific Schmidt coefficients.
+It is natural to consider also the simplest case that is not
+of the above convenient form, namely that of two qutrits,
+(n, d) = (2, 3). This appears to be considerably different be-
+cause we fail to find strings of local unitaries that bring the
+Schmidt decomposition |ψS⟩ into a basis without explicit de-
+pendence on the Schmidt coefficients. Nevertheless, a basis
+might still be possible to construct by taking the Schmidt co-
+efficients into account when choosing the local unitaries. Ac-
+tually, this seems to always be possible. To arrive at this, we
+have used a numerical method. Let {|φj⟩}m
+j=1 be a set of states
+in a given Hilbert space. These states are pairwise orthogonal
+if and only if they realise the global minimum (zero) of the
+following objective function
+f({φj}) ≡
+�
+j̸=j′
+| ⟨φj|φj′⟩ |2.
+(3)
+For a given state |ψ⟩, we numerically minimise f({ψj}) over
+all possible strings {Vj}dn
+j=1 of local unitaries. To this end,
+we parameterise the local unitaries U (j)
+k
+using the scheme of
+Ref. [9]. For the two-qutrit case, we have randomly chosen
+1000 pairs of Schmidt coefficients (λ1, λ2) which (up to local
+unitaries) fully specifies the state. In each case we numerically
+minimise f({ψj}). Without exception, we find strings of lo-
+cal unitaries that yield a result below our selected precision
+threshold of f ≤ 10−6.
+Furthermore, we have also numerically investigated the
+case of three qubits, (n, d) = (3, 2). This scenario requires
+a different approach than the previous cases since multipar-
+tite states have no Schmidt decomposition. Instead, for any
+given three-qubit state |ψ⟩, there exists local unitary transfor-
+mations that map it onto the canonical form a |000⟩+b |011⟩+
+c |101⟩ + d |110⟩ + e |111⟩ where (b, c, d, e) are real num-
+bers and a is a complex number [10, 11]. Hence, up to lo-
+cal unitaries, the state space (after normalisation) is charac-
+terised by five real numbers. Later, we will provide an analyt-
+ical construction of a |ψ⟩-basis for the four-parameter family
+corresponding to restricting a to be real. However, we have
+not found an analytical basis construction for general three-
+qubit states, but we nevertheless conjecture that it exists. To
+evidence this, we have employed the previously introduced
+numerical search method. Again, we have randomly chosen
+1000 normalised sets of coefficients (a, b, c, d, e) and searched
+for the minimal value of f over all the strings of local qubit
+unitaries. In all cases, we find that f vanishes up to our se-
+lected precision of f ≤ 10−6.
+Given the above case studies, one might suspect that ev-
+ery pure quantum state admits a basis.
+Interestingly, this
+seems not to be true.
+While some states of four qubits,
+(n, d) = (4, 2), are found to admit a basis, for example
+a W state and doubly-excited Dicke state [23], it appears
+that most four-qubit states do not admit a basis. We have
+sampled many different four-qubit states and repeatingly at-
+tempted to numerically find a basis via the minimisation of
+(3), also using several different search algorithms. It was reg-
+ularly found that the estimated minimum is multiple orders
+of magnitude above our given precision threshold for a basis.
+For example, we searched for the minimum of f for the state
+2
+√
+6 |W⟩ +
+√
+2
+√
+6 |GHZ4,2⟩, with 100 randomised initial points,
+and never reached below f = 10−1, five orders of magnitude
+above our precision threshold. We have attempted to prove
+that no basis exists by employing semidefinite outer relax-
+ations of f over the set of dimensionally-restricted quantum
+correlations [12] combined with a modified sampling of the
+state and measurement space [13] and symmetrisation tech-
+niques [14] to efficiently treat the large number of single-qubit
+unitaries featured in this problem. However, the conjecture
+has resisted our efforts. A guiding intuition for the impossi-
+bility of a basis is to note that the number of free parameters is
+3n(2n − 1) whereas the number of orthogonality constraints
+(counting both the real and imaginary part) is 22n − 2n, and
+the latter is larger than the former only when n ≥ 4.
+Furthermore, if an n-qubit state |ψ⟩ does not admit a ba-
+sis, then the (n + 1)-qubit state |ψ′⟩ = |ψ⟩ ⊗ |0⟩ also does
+not admit a basis. By contradiction, suppose there are 2n+1
+unitaries V ′
+j = Vj ⊗ U (j)
+n+1 such that |⟨ψ′|(V ′
+j )†V ′
+k|ψ′⟩| =
+δjk ∀j, k ∈ {1, ..., 2n+1}. Divide the 2n+1 states U (j)
+n+1 |0⟩
+into two sets such that two orthogonal vectors are not in
+the same set (e.g. the northern and southern hemisphere
+of the Bloch ball).
+Consider the set that contains at least
+as many elements as the other one, hence, at least 2n el-
+ements.
+By construction, these states cannot be distin-
+guished on the last qubit, |⟨0|U (j)†
+n+1U (k)
+n+1|0⟩| ̸= 0.
+Since
+|⟨ψ′|(V ′
+j )†V ′
+k|ψ′⟩| = |⟨ψ|V †
+j Vk|ψ⟩| · |⟨0|U (j)†
+n+1U (k)
+n+1|0⟩|, we
+must have |⟨ψ|V †
+j Vk|ψ⟩| = δjk for all of those pairs, which
+contradicts that |ψ⟩ does not admit a basis. By induction, this
+argument shows that if our above conjecture holds, namely
+that some four-qubit states do not admit a basis, then the same
+holds for any number of qubits.
+Since not all pure quantum states admit a basis, and this
+seems to be typical rather than exceptional for four qubits, it
+is interesting to ask whether some distinguished families of
+n-qubit states can nevertheless admit a basis. This is well-
+known to be the case for n-qubit GHZ-states and graph-states
+
+3
+since they are locally maximally entanglable. More interest-
+ingly, a positive answer is also possible for states that are not
+of this kind: we construct a basis for the n-qubit W-state,
+|Wn⟩ =
+1
+√n
+�
+σ σ(|0⟩⊗n−1 |1⟩) where σ runs over all permu-
+tations of the position of “1”. Note that |W1⟩ = |1⟩ and that a
+|W1⟩-basis is obtained from the unitaries {11, X}. Now we ap-
+ply induction. Consider that the strings {V (n)
+j
+}2n
+j=1 generate a
+|Wn⟩-basis. One can then construct a basis for n+1 qubits as
+follows. For half of the basis elements, namely j = 1, . . . , 2n,
+define V (n+1)
+j
+= V (n)
+j
+⊗ 11 and for the other half, namely j =
+2n +1, . . . , 2n+1, define V (n+1)
+j
+= �n
+k=1 U (j)
+k Z ⊗X. As we
+detail in Appendix B, one can verify that {V (n+1)
+j
+|Wn+1⟩}j
+is a W-basis. We note that for the purpose of entanglement
+distillation, a different construction of a W-basis was given in
+Ref. [8].
+So far, we have considered whether a specific state can be
+associated to a specific measurement. In other words, the uni-
+tary constructions have been state-dependent. We now go fur-
+ther and introduce a complementary concept, namely whether
+there exist strings of local unitaries {Vj} that can transform
+any state in a space of states S into a basis, i.e. strings of local
+unitaries that satisfy
+∀ψ ∈ S,
+|⟨ψ|V †
+j Vj′|ψ⟩| = δjj′.
+(4)
+Naturally, this state-independent notion of basis construc-
+tion is much stronger than the previously considered state-
+dependent notion.
+In the most ambitious case, when we
+choose the space S to be the entire Hilbert space of n sub-
+systems of dimension d, i.e. S ≃ (Cd)⊗n, then a state-
+independent construction cannot exist. In fact, not even two
+orthogonal vectors can be state-independently constructed for
+the full quantum state space. To show this, we can w. l. g. set
+V1 = 11 and assume that there exists local unitaries {Uk}
+such that |ψ1⟩ = |ψ⟩ and |ψ2⟩ = �n
+k=1 Uk |ψ⟩ are orthog-
+onal for all |ψ⟩. Focus now on the particular state |ψ⟩ =
+�n
+k=1 |µk⟩ where |µk⟩ is some eigenvector of the unitary Uk.
+Since the eigenvalues of a unitary are complex phases, writ-
+ten eiϕk for Uk and |µk⟩, we obtain |ψ1⟩ = �n
+k=1 |µk⟩ and
+|ψ2⟩ = ei �n
+k=1 ϕk �n
+k=1 |µk⟩. These two states are evidently
+not orthogonal and hence we have a contradiction.
+Interestingly, the situation changes radically if we limit our
+state-independent investigation to all quantum states in a real-
+valued Hilbert space. That is, S ≃ (Rd)⊗n. Such real quan-
+tum systems have also been contrasted in the literature with
+their complex counterparts [15–17]. Let us momentarily ig-
+nore the n-partition structure of our Hilbert space and sim-
+ply consider two real states |ψ1⟩ = |ψ⟩ and |ψ2⟩ = U |ψ⟩
+obtained from a given real target state |ψ⟩ and a fixed (ψ-
+independent) unitary U.
+It holds that ψ1 and ψ2 are or-
+thogonal if and only if U is skew-symmetric.
+To prove
+this, assume first the skew-symmetry property U = −U T .
+Since for real states ⟨ψ1|ψ2⟩ = ⟨ψ2|ψ1⟩∗ is equivalent to
+⟨ψ|U|ψ⟩ = ⟨ψ|U †|ψ⟩∗ = ⟨ψ|U T |ψ⟩, skew-symmetry im-
+plies that ⟨ψ1|ψ2⟩ = 0. Conversely, assume that ⟨ψ|U|ψ⟩ = 0
+for all real-valued ψ. Choosing in particular |ψ⟩ = |k⟩ for
+k = 0, . . . , d − 1, it follows that all diagonal elements of U
+must vanish. Then, choose |ψ⟩ =
+1
+√
+2(|i⟩ + |j⟩) for any pair
+i ̸= j. This yield Uii+Ujj+Uij+Uji = 0, but since we know
+that the diagonals vanish we are left with just Uij = −Uji
+which defines a skew-symmetric operator.
+Returning to our n-partitioned real Hilbert space, and still
+w. l. g. taking V1 = 11, the above result demands that we find
+local unitaries such that
+U1 ⊗ . . . ⊗ Un = −U T
+1 ⊗ . . . ⊗ U T
+n .
+(5)
+This is only possible if U T
+k = ±Uk. Hence, all local unitaries
+must be either symmetric or skew-symmetric, and the number
+of the latter must be odd. When extended from two orthogonal
+states to a whole basis, we require that this property holds for
+every pair of distinct labels (j, j′) in the basis. In other words,
+we require that every string (Vj)†Vj′ with j ̸= j′ is skew-
+symmetric.
+The question becomes whether the above condition can
+be satisfied for a given scenario. Consider it first for qubit
+systems (d = 2). In Appendix C we show that the set of
+complex qubit unitaries that are either symmetric or skew-
+symmetric and whose products are again either symmetric
+or skew-symmetric, must obey a simple structure; they are
+essentially equivalent to the four Pauli-type operators P ≡
+{11, X, Z, XZ}. Thus, if a state-independent construction ex-
+ists, we can restrict to selecting one of these four operators for
+each of our local unitaries U (j)
+k . Interestingly, for the case of
+two qubits, (n, d) = (2, 2), a state-independent construction
+is possible. It is in fact given by Eq. (2). One can straightfor-
+wardly verify that the above criterion is satisfied, i.e. all local
+unitaries are selected from P and all pairs of products of uni-
+tary strings in (2) are skew-symmetric. Alternatively, one can
+easily verify that (2) maps every state �
+i,j=0,1 αij |ij⟩ into a
+basis, for any real coefficients αij. Furthermore, by the same
+token, a state-independent basis is also possible for every real
+state of three qubits, (n, d) = (3, 2). One explicit construc-
+tion that satisfies our necessary and sufficient criterion is the
+following set of eight strings of local unitaries
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+11 ⊗ 11 ⊗ 11
+Z ⊗ Z ⊗ XZ
+Z ⊗ XZ ⊗ 11
+XZ ⊗ 11 ⊗ 11
+Z ⊗ X ⊗ XZ
+X ⊗ 11 ⊗ XZ
+X ⊗ XZ ⊗ Z
+X ⊗ XZ ⊗ X
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+Again,
+one
+may
+easily
+verify
+that
+every
+real
+state
+�
+i,j,k=0,1 αijk |ijk⟩ is mapped into a basis.
+Two- and three-qubits are interesting cases because they
+are exceptional.
+As we now show, there exists no state-
+independent construction for real states of four or more qubits.
+We first prove this for n = 4 and then show that this im-
+plies impossibility also for n > 4. The four-qubit case con-
+tains 16 strings of unitaries and we know that each local uni-
+tary can w. l. g. be selected from P. Since we seek a state-
+independent construction, we can momentarily consider only
+the state |0000⟩. In order for it to be mapped into a basis, we
+
+4
+(2,2,R) (2,2,C) (3,2,R) (3,2,C) (4,2,R) (2,3,C) (2,4 or 8,C) (n, 2m + 1,R)
+State-dependent
+construction
+
+
+
+()
+()
+()
+
+− − −
+State-independent
+construction
+
+
+
+
+
+
+
+
+TABLE I: Overview of results. The first row indicates the scenario: (n, d, S) gives particle number, dimension and the type of state space
+respectively. The symbol indicates the existence of a basis under local unitaries. The symbol indicates that there in general can be no basis
+under local unitaries, i.e. at least one state admits no basis. Paranthesis indicates that the result is obtained from numerical search. The
+symbol − − − indicates that no investigation was made.
+see that Z acts trivially on every register and therefore each
+one of the 16 combinations of bit-flip or identity operators,
+{Xc1 ⊗ Xc2 ⊗ Xc3 ⊗ Xc4} for c1, c2, c3, c4 ∈ {0, 1}, must
+be featured in exactly one of the 16 unitary strings {Vj}16
+j=1.
+Let us now look only at six of these strings, namely those
+corresponding to having zero bit-flips (1 case), one bit-flip (4
+cases) and four bit-flips (1 case). W. l. g. fixing V1 = 11 (zero
+bit-flips), the strings take the form
+V1
+11
+⊗
+11
+⊗
+11
+⊗
+11
+V2
+XZr11 ⊗
+Zr12
+⊗
+Zr13
+⊗
+Zr14
+V3
+Zr21
+⊗ XZr22 ⊗
+Zr23
+⊗
+Zr24
+V4
+Zr31
+⊗
+Zr32
+⊗ XZr33 ⊗
+Zr34
+V5
+Zr41
+⊗
+Zr42
+⊗
+Zr43
+⊗ XZr44
+V6
+XZr51 ⊗ XZr52 ⊗ XZr53 ⊗ XZr54
+,
+(6)
+where rij
+∈ {0, 1} represent our freedom to insert a Z
+operator and thus realise the two relevant elements of P.
+Since every row must be skew-symmetric and the only skew-
+symmetric element in P is XZ, we must have r11 = r22 =
+r33 = r44 = 1 and r51 + r52 + r53 + r54 = 1 where ad-
+dition is modulo two. Moreover, every product of two rows
+must be skew-symmetric, i.e. the product must have an odd
+number of XZ operations. For the four middle rows, this im-
+plies rij + rji = 1 for distinct indices i, j ∈ {1, 2, 3, 4}. For
+the products V †
+6 Vj for j = 2, 3, 4, 5, the conditions for skew-
+symmetry respectively become
+r12 + r13 + r14 + r52 + r53 + r54 = 1
+r21 + r23 + r24 + r51 + r53 + r54 = 1
+r31 + r32 + r34 + r51 + r52 + r54 = 1
+r41 + r42 + r43 + r51 + r52 + r53 = 1.
+(7)
+Summing these four equations and using the previously es-
+tablished skew-symmetry conditions, one can cancel out all
+degrees of freedom rij and arrive at the contradiction 1 = 0.
+Hence, we conclude that the state-independent basis construc-
+tion for four qubits is impossible.
+For the case of five qubits, we can again assume w. l. g. that
+the 32 combinations of bit-flip or identity operators, {Xc1 ⊗
+Xc2⊗Xc3⊗Xc4⊗Xc5} for c1, c2, c3, c4, c5 ∈ {0, 1} must be
+featured in exactly one of the 32 unitary strings since the state
+|00000⟩ has to be mapped into an orthonormal basis. Suppose
+there is a state-independent construction that maps every real-
+valued five-qubit state into a basis, in especially any state of
+the form |ψ⟩ ⊗ |0⟩, where |ψ⟩ is an arbitrary real-valued four
+qubit state. Now consider the 16 strings where c5 = 0. Since
+the fifth qubit is always mapped to itself, it has to hold that the
+first four qubits are pairwise distinguishable. However, this
+implies a state-independent construction for four qubits which
+is in contradiction to the above. By induction, this implies that
+no state-independent construction can exist whenever n ≥ 4.
+The possibility of state-independent constructions for real-
+valued bi- and tri-partite systems draws heavily on the sim-
+ple structure of skew-symmetric qubit unitaries. If we con-
+sider real-valued systems of dimension d > 2, the situa-
+tion changes considerably.
+Using our necessary and suffi-
+cient condition, it follows immediately that state-independent
+constructions are impossible in all odd dimensions, i.e. when
+(n, d) = (n, 2m+1). This stems from the fact that there exists
+no skew-symmetric unitary matrix in odd dimensions. To see
+that, simply note that if A is skew-symmetric then det(A) =
+det
+�
+AT �
+= det(−A) = (−1)2m+1 det(A) = − det(A) and
+hence det(A) = 0, but that contradicts unitarity because the
+determinant of a unitary has unit modulus.
+In summary, we have investigated the correspondence be-
+tween entangled states and entangled measurements under lo-
+cal unitary transformations, both when the local transforma-
+tion can and cannot explicitly depend on the target state. Per-
+haps surprisingly, we have found that this problem is not so
+straightforward and has a strong dependence on both the num-
+ber of subsystems involved and their dimension. Our analyt-
+ical and numerical results and conjectures are summarised in
+Table I.
+The conspicuous open problem left by our work is to prove
+our conjecture that there exists states that do not admit a ba-
+sis under local unitaries. An interesting related question is if
+one can bound the relative volume of four-qubit states that do
+not admit a basis. Our numerical investigations suggest that
+nearly all four-qubit states should belong to this class. Fur-
+thermore, it would be useful to find analytical solutions for
+the three-qubit and two-qutrit state-dependent cases. More-
+over, for the state-independent considerations, we focused on
+real Hilbert spaces. A natural question is whether there ex-
+ists state-independent basis constructions for other interest-
+ing spaces. For example, if one restricts to bipartite states
+of a known entanglement entropy, can one construct a state-
+independent basis? The answer is clearly positive for the lim-
+iting cases of product states and maximally entangled states.
+Another interesting space to consider is the symmetric sub-
+space of n-qubit Hilbert space.
+
+5
+Our results may also have prospects in quantum informa-
+tion as one may now construct entangled measurements asso-
+ciated to entangled states. Recently there has been proposals
+of two-qubit entangled projections; the so-called Elegant Joint
+Measurements [18, 19] which have also been realised in vari-
+ous experiments [20? , 21]. The Elegant Joint Measurements
+can be seen as a particular type of |ψ⟩-basis where |ψ⟩ is a
+partially entangled two-qubit state. However, the basis addi-
+tionally has the feature that the collections of reduced states
+form a tetrahedron. This requirement goes beyond our prob-
+lem formulation, as we do not impose any structure on the
+reduced states of our bases. However, it suggests an avenue
+to identifying interesting and highly symmetric measurements
+by finding the particular |ψ⟩-basis that maximises the Hilbert
+space volume spanned its collection of reduced states.
+Finally, one of the notable shortcommings of traditional,
+GHZ based, multiqubit entanglement swapping protocols is
+that the loss of one particle renders the measurement separa-
+ble. However, some other states that are inequivalent to GHZ
+under LOCC can preserve their entanglement under reduc-
+tions. The existence of an iso-entangled basis composed of
+such states may constitute an avenue to more noise-resiliant
+entanglement swapping protocols which have natural quan-
+tum information applications.
+Note added.— During the late stage of our work, we be-
+came aware of the previous work [23] where i. a. bases are
+found for some Dicke states.
+ACKNOWLEDGMENTS
+We thank Hayata Yamasaki,
+Marcus Huber,
+Jakub
+Czartowski and Karol ˙Zyczkowski for discussions. A. T. ac-
+knowledges support from the Wenner-Gren Foundation and
+from the Wallenberg Centre for Quantum Technology. M. J.
+R. acknowledges financial support from the Austrian Science
+Fund (FWF) through BeyondC (F7103-N38), the Project No.
+I-2906, as well as support by the John Templeton Founda-
+tion through Grant 61466, The Quantum Information Struc-
+ture of Spacetime (qiss.fr), the Foundational Questions Insti-
+tute (FQXi) and the research platform TURIS. The opinions
+expressed in this publication are those of the authors and do
+not necessarily reflect the views of the John Templeton Foun-
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+Appendix A: Basis construction for every bipartite state of local dimension d = 4 and d = 8
+Let the local dimension be a power of two, d = 2m, and index the d2 basis elements as (˜j, j) where ˜j = 0, 1, . . . , d − 1
+and j = 1, 2, . . . , d. Let W A
+ψ ⊗ W B
+ψ be the state-dependent local unitaries that transform the general state |ψ⟩ into the Schmidt
+basis, i.e. |ψS⟩ ≡ W A
+ψ ⊗ W B
+ψ |ψ⟩ = �d−1
+l=0 λl |l, l⟩, with the Schmidt coefficients λl ∈ R satisfying �
+l λ2
+l = 1. We now
+further decompose the individual d-dimensional registers as a string of m qubits, writing |l⟩ = |l1 . . . lm⟩. Thus, the Schmidt
+decomposed state reads
+|ψS⟩ =
+�
+l1,...,lm=0,1
+λl |l1 . . . lm, l1 . . . lm⟩ .
+(A1)
+Once the state has been put in the form (A1), we apply a set of local unitaries that is independent of the Schmidt coefficients.
+For d = 4 and ˜j = 0, the two sets of unitaries read as follows:
+˜j j
+U (˜j,j)
+1
+U (˜j,j)
+2
+U (˜j,j)
+1
+⊗ U (˜j,j)
+2
+|ψS⟩
+0 1
+11 ⊗ 11
+11 ⊗ 11
+λ00 |00, 00⟩ + λ01 |01, 01⟩ + λ10 |10, 10⟩ + λ11 |11, 11⟩
+0 2
+11 ⊗ X
+11 ⊗ XZ
+λ00 |01, 01⟩ − λ01 |00, 00⟩ + λ10 |11, 11⟩ − λ11 |10, 10⟩
+0 3
+X ⊗ 11
+XZ ⊗ Z
+λ00 |10, 10⟩ − λ01 |11, 11⟩ − λ10 |00, 00⟩ + λ11 |01, 01⟩
+0 4
+X ⊗ X XZ ⊗ X
+λ00 |11, 11⟩ + λ01 |10, 10⟩ − λ10 |01, 01⟩ − λ11 |00, 00⟩
+(A2)
+In addition, we define U (˜j,j)
+1
+:= X
+˜j
+4 U (˜j=0,j)
+1
+and U (˜j,j)
+2
+:= U (˜j=0,j)
+2
+, where Xd is the d-dimensional shift-operator Xd =
+�d−1
+l=0 |l + 1⟩⟨l|. Note that, the unitaries U (˜j,j)
+2
+coincide with the state-independent set for two qubits given in Eq. (2) and
+do not depend on ˜j. At the same time, U (˜j=0,j)
+1
+are the same as U (˜j,j)
+2
+where the Z gates are left out. We now show that
+{U (˜j,j)
+1
+⊗ U (˜j,j)
+2
+|ψS⟩}˜j,j is a basis of the bipartite Hilbert space. One can check directly that the four states with ˜j = 0 stated in
+Eq. (A2) above are pairwise orthogonal. We want to mention that we are exploiting the fact that U (˜j=0,j)
+2
+are the elements of a
+state-independent construction. To see the connection, note that the calculation for the state-independent two-qubit construction
+reads as follows:
+(11 ⊗ 11)(λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩) = λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩ ,
+(A3)
+(11 ⊗ XZ)(λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩) = λ00 |01⟩ − λ01 |00⟩ + λ10 |11⟩ − λ11 |10⟩ ,
+(A4)
+(XZ ⊗ Z)(λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩) = λ00 |10⟩ − λ01 |11⟩ − λ10 |00⟩ + λ11 |01⟩ ,
+(A5)
+(XZ ⊗ X)(λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩) = λ00 |11⟩ + λ01 |10⟩ − λ10 |01⟩ − λ11 |00⟩ .
+(A6)
+Since these states are pairwise orthogonal for arbitrary real coefficients λl1l2, the same holds true for the states in Eq. (A2).
+In addition, all of the states where ˜j = 0 are elements of the subspace spanned by |00, 00⟩, |01, 01⟩, |10, 10⟩ and |11, 11⟩.
+Hence, they form a basis of this four-dimensional subspace.
+By shifting now the first system we obtain a basis for the
+remaining orthogonal subspaces.
+More precisely, since we defined U (˜j,j)
+1
+= X
+˜j
+4 U (˜j=0,j)
+1
+the states where ˜j = 1 are
+esentially the same states as the ones in Eq. (A2) but with the first system shifted by one l → l ⊕ 1 (mod 4). For example,
+λ00 |11, 10⟩ − λ01 |00, 11⟩ − λ10 |01, 00⟩ + λ11 |10, 01⟩ is the state that corresponds to ˜j = 1 and j = 3. In this way, the
+four states where ˜j = 1 form a basis of the subspace spanned by |01, 00⟩, |10, 01⟩, |11, 10⟩ and |00, 11⟩ (or all states where
+|l + 1, l⟩). Analogously, the four states where ˜j = 2 (˜j = 3) form a basis of the subspaces spanned by the vectors with |l + 2, l⟩
+(|l + 3, l⟩). Altogether, the sixteen states {U (˜j,j)
+1
+⊗ U (˜j,j)
+2
+|ψS⟩}˜j,j form a basis of the entire sixteen dimensional Hilbert space.
+A similar construction can be found for d = 8 by using the state-independent construction of three qubits. Similar as above,
+
+7
+the set for ˜j = 0 reads as follows:
+˜j j
+U (˜j,j)
+1
+U (˜j,j)
+2
+U (˜j,j)
+1
+⊗ U (˜j,j)
+2
+|ψS⟩
+0 1
+11 ⊗ 11 ⊗ 11
+11 ⊗ 11 ⊗ 11
++λ000 |000, 000⟩ + λ001 |001, 001⟩ + λ010 |010, 010⟩ + λ011 |011, 011⟩
++λ100 |100, 100⟩ + λ101 |101, 101⟩ + λ110 |110, 110⟩ + λ111 |111, 111⟩
+0 2
+11 ⊗ 11 ⊗ X
+Z ⊗ Z ⊗ XZ
++λ000 |001, 001⟩ − λ001 |000, 000⟩ − λ010 |011, 011⟩ + λ011 |010, 010⟩
+−λ100 |101, 101⟩ + λ101 |100, 100⟩ + λ110 |111, 111⟩ − λ111 |110, 110⟩
+0 3
+11 ⊗ X ⊗ 11
+Z ⊗ XZ ⊗ 11
++λ000 |010, 010⟩ + λ001 |011, 011⟩ − λ010 |000, 000⟩ − λ011 |001, 001⟩
+−λ100 |110, 110⟩ − λ101 |111, 111⟩ + λ110 |100, 100⟩ + λ111 |101, 101⟩
+0 4
+X ⊗ 11 ⊗ 11
+XZ ⊗ 11 ⊗ 11
+(...)
+0 5
+11 ⊗ X ⊗ X
+Z ⊗ X ⊗ XZ
+(...)
+0 6
+X ⊗ 11 ⊗ X
+X ⊗ 11 ⊗ XZ
+(...)
+0 7
+X ⊗ X ⊗ 11
+X ⊗ XZ ⊗ Z
+(...)
+0 8
+X ⊗ X ⊗ X X ⊗ XZ ⊗ X
+(...)
+(A7)
+Again, we define U (˜j,j)
+1
+= X
+˜j
+8 U (˜j=0,j)
+1
+and U (˜j,j)
+2
+= U (˜j=0,j)
+2
+. The proof that this forms a basis of the 64-dimension Hilbert
+space is completely analogous to the case of d = 4 before. The eight states for ˜j = 0 form a basis of the eight-dimensional
+subspace spanned by |l1l2l3, l1l2l3⟩ (for li = 0, 1). Applying the shift operator X8 to the first system, one obtains bases of the
+other eight-dimensional orthogonal subspaces spanned by the vectors with
+��l + ˜j, l
+�
+. This approach cannot (immediately) be
+generalized to higher dimensions d = 2n, due to the lack of state-independent constructions for n ≥ 4 qubits. However, there is
+in principle no reason to restrict the unitaries on the second system to tensor products of single qubit Pauli gates as we do here.
+In principle, we could also consider general permutations with suitably chosen signs such that all terms cancel in this pairwise
+sense as above. Even when considering this larger class of possibilities, we made an exhaustive search and could not find any
+additional construction. Due to this, it seems unlikely that a construction exists in which the unitaries do not depend on the
+Schmidt coefficients.
+Appendix B: An n-qubit basis of W-states
+We define the n-qubit W-state as
+|W1⟩ ≡ |1⟩
+|W2⟩ ≡
+1
+√
+2 (|01⟩ + |10⟩)
+|W3⟩ ≡
+1
+√
+3 (|001⟩ + |010⟩ + |100⟩)
+|W4⟩ ≡ 1
+2 (|0001⟩ + |0010⟩ + |0100⟩ + |1000⟩)
+...
+(B1)
+Note that for one and two qubits, the definition is only introduced for sake of convenience. In general, we write
+|Wn⟩ ≡
+1
+√n
+�
+σ
+σ(|0⟩⊗n−1 |1⟩),
+(B2)
+where σ runs over all permutations of the position of “1”. It is also useful to write the state recursively as
+|Wn+1⟩ =
+�
+n
+n + 1 |Wn⟩ ⊗ |0⟩ +
+1
+√n + 1 |0⟩n ⊗ |1⟩
+(B3)
+Clearly, if we apply the local unitaries U (1)
+1
+= 11 and U (2)
+1
+= X to |W1⟩ we generate the trivial one-qubit W-basis {|0⟩ , |1⟩}.
+Assume now that the local unitaries {U (j)
+k } for k = 1, . . . n and j = 1, . . . , 2n yield a |Wn⟩-basis. We will now show that under
+this assumption we can construct a basis for |Wn+1⟩ and hence it follows from induction that a W-basis exists for any number
+of qubits.
+
+8
+We illustrate the induction step as follows,
+U (1)
+1
+⊗
+U (1)
+2
+⊗ . . . ⊗
+U (1)
+n
+⊗
+11
+U (2)
+1
+⊗
+U (2)
+2
+⊗ . . . ⊗
+U (2)
+n
+⊗
+11
+...
+...
+...
+U (2n)
+1
+⊗
+U (2n)
+2
+⊗ . . . ⊗
+U (2n)
+n
+⊗
+11
+U (1)
+1 Z
+⊗
+U (1)
+2 Z
+⊗ . . . ⊗
+U (1)
+n Z
+⊗ X
+U (2)
+1 Z
+⊗
+U (2)
+2 Z
+⊗ . . . ⊗
+U (2)
+n Z
+⊗ X
+...
+...
+...
+U (2n)
+1
+Z ⊗ U (2n)
+2
+Z ⊗ . . . ⊗ U (2n)
+n
+Z ⊗ X
+.
+(B4)
+We see that for the first 2n basis elements, we extend the unitaries for n qubits by tensoring with 11 for qubit number n + 1.
+For the latter 2n basis elements, we extend the unitaries for n qubits by multiplying all of them from the right by Z and finally
+tensoring with X for qubit number n + 1. As usual, we now write the string of unitaries associated to each row as V (n+1)
+j
+for
+n = 1, . . . , 2n+1. We similarly use V (n)
+j
+for the unitary strings for the case of n qubits.
+To see that this yields a basis, we first show that the first 2n basis elements (upper block of table, j = 1, . . . , 2n) are orthogonal.
+For this purpose, we use the recursion formula (B3) to write for j ̸= j′
+⟨Wn+1|(V (n+1)
+j′
+)†V (n+1)
+j
+|Wn+1⟩ =
+n
+n + 1⟨Wn0|(V (n)
+j′
+)†V (n)
+j
+⊗ 11|Wn0⟩ +
+1
+n + 1⟨0 . . . 01|(V (n)
+j′
+)†V (n)
+j
+⊗ 11|0 . . . 01⟩
++
+√n
+n + 1⟨Wn0|(V (n)
+j′
+)†V (n)
+j
+⊗ 11|0 . . . 01⟩ +
+√n
+n + 1⟨0 . . . 01|(V (n)
+j′
+)†V (n)
+j
+⊗ 11|Wn0⟩ = 0
+The first term is zero for all j′ ̸= j due to the induction hypothesis. The third and fourth terms are zero due to orthogonality in
+the last qubit register. The second term is zero for every j′ ̸= j there exists at least one qubit register k for which U (j′)
+k
+and U (j)
+k
+are composed of different numbers of bit-flips (X). The latter follows from the initial condition of using {11, X} to construct the
+|W1⟩-basis.
+The same procedure will analogously show that the latter 2n basis elements (lower block of the table, j = 2n + 1, . . . , 2n+1)
+are orthogonal. We are left with showing that every overlap between the upper and lower block, i.e. with any j′ = 1, . . . , 2n and
+any j = 2n + 1, . . . , 2n+1, also vanishes. For this we have
+⟨Wn+1|(V (n+1)
+j′
+)†V (n+1)
+j
+|Wn+1⟩ =
+n
+n + 1⟨Wn0|
+�
+(V (n)
+j′
+)†V (n)
+j
+⊗ X
+�
+n
+�
+k=1
+Z ⊗ 11|Wn0⟩
++
+1
+n + 1⟨0 . . . 01|
+�
+(V (n)
+j′
+)†V (n)
+j
+⊗ X
+�
+n
+�
+k=1
+Z ⊗ 11|0 . . . 01⟩
++
+√n
+n + 1⟨Wn0|
+�
+(V (n)
+j′
+)†V (n)
+j
+⊗ X
+�
+n
+�
+k=1
+Z ⊗ 11|0 . . . 01⟩
++
+√n
+n + 1⟨0 . . . 01|
+�
+(V (n)
+j′
+)†V (n)
+j
+⊗ X
+�
+n
+�
+k=1
+Z ⊗ 11|Wn0⟩
+Note that �n
+k=1 Z ⊗ 11 |Wn0⟩ = − |Wn0⟩ and �n
+k=1 Z ⊗ 11 |0 . . . 01⟩ = |0 . . . 01⟩. The first and second terms are both zero due
+to orthogonality in the final qubit register. We thus have
+⟨Wn+1|(V (n+1)
+j′
+)†V (n+1)
+j
+|Wn+1⟩ =
+√n
+n + 1⟨Wn|(V (n)
+j′
+)†V (n)
+j
+|0 . . . 0⟩ −
+√n
+n + 1⟨0 . . . 0|(V (n)
+j′
+)†V (n)
+j
+|Wn⟩
+=
+√n
+n + 1⟨Wn|(V (n)
+j′
+)†V (n)
+j
+− (V (n)
+j
+)†V (n)
+j′
+|0 . . . 0⟩ = 0.
+(B5)
+The last equality follows from the fact that it is sufficient, for given (j, j′), that there exist some register index k such that
+(U (j′))†
+kU (j)
+k
+− (U (j))†
+kU (j′)
+k
+= 0 in order for the overlap to vanish. This is always the case because due to our construction (see
+initial condition and the table), for every two unitaries there is at least one register k where the single-qubit unitaries differ by
+X, meaning that either (U (j)
+k , U (j′)
+k
+) = (11, X)/(Z, XZ), or the same with j ↔ j′ is true. The condition above is satisfied by
+all of these combinations. Hence we conclude that the proposed construction satisfies
+⟨Wn+1|(V (n+1)
+j
+)†V (n+1)
+j′
+|Wn+1⟩ = δjj′
+(B6)
+
+9
+and therefore yields a W-state basis for any number of qubits.
+Appendix C: The Pauli structure for state-independent qubit unitary constructions
+We consider the set of local unitaries P that are applied to the i-th qubit in the state-independent construction and show that
+without loss of generality, the set can be chosen to be the Pauli-type gates P ≡ {11, X, Z, XZ}. First, note that the set is finite
+since there are exactly 2n basis states. Next, we observe that the identity 11 has to be within the set P since we demand that
+V1 = 11. Furthermore, we can argue that the gate
+XZ =
+�
+0 −1
+1
+0
+�
+has to be within the set as well, since it is the only gate that maps every real qubit state to its orthogonal state. More precisely,
+if it is not used on the i-th qubit at least once, one can choose a real qubit state |φi⟩ such that none of the gates in P map
+|φi⟩ to its orthogonal vector. Hence if we apply the state-independent construction to the real-valued product state |φ⟩ =
+|0⟩1 ⊗ . . . |0⟩i−1 ⊗ |φi⟩ ⊗ |0⟩i+1 ⊗ . . . ⊗ |0⟩n none of the resulting 2n states are distinguishable on the i-th qubit, which is
+impossible if these states should form a basis of product states. Therefore, the gate XZ has to be within the set P. Apart from
+the gates 11 and XZ we can constrain which other qubit unitaries can be in the set P. We know that if we demand V1 = 11, every
+string of local unitaries (Vj) and their products (Vj)†Vj′ with j ̸= j′ have to be skew-symmetric. As a result, the local unitaries
+on each subsystem (hence, the unitaries in the set P) and also all their products have to be either symmetric or skew-symmetric.
+By neglecting a global phase, the general form of a unitary operator can be written as:
+U =
+�
+cos (θ)eiα
+sin (θ)eiβ
+− sin (θ)e−iβ cos (θ)e−iα
+�
+.
+(C1)
+The only skew-symmetric 2×2 unitary is, up to an irrelevant global phase, the Pauli-type operator XZ, which we already found
+to be necessarily in the set P. All the symmetric matrices of this form can be written as:
+U =
+�
+cos (θ)eiα
+i sin (θ)
+i sin (θ) cos (θ)e−iα
+�
+.
+(C2)
+If the gate U is in P, it is at some point multiplied with the gate XZ since the operator XZ is used at least once on the i-th qubit.
+Since we know that the result of this product has to be again either symmetric or skew-symmetric, we obtain that α = π/2, 3π/2
+due to:
+(XZ)†U =
+�
+0 1
+−1 0
+� �
+cos (θ)eiα
+i sin (θ)
+i sin (θ) cos (θ)e−iα
+�
+=
+�
+i sin (θ) cos (θ)e−iα
+− cos (θ)eiα
+−i sin (θ)
+�
+.
+(C3)
+The two possibilities for α = π/2, 3π/2 correspond to the two solutions
+U1 =
+�
+cos (θ)
+sin (θ)
+sin (θ) − cos (θ)
+�
+,
+U2 =
+�
+sin (θ) − cos (θ)
+− cos (θ) − sin (θ)
+�
+.
+(C4)
+We left the irrelevant global factor i for simplicity. Considering the additional degree of freedom of θ, we can restrict to the first
+class of solutions U1 since the second class U2 can be obtained by shifting θ by π/2. Hence, if we add a gate U to the set P, it
+has to be of the form given by U1 above. Now if we add two such gates to the set P, the product of U1 with another valid matrix
+U ′
+1 is
+U †
+1U ′
+1 =
+�
+cos (θ)
+sin (θ)
+sin (θ) − cos (θ)
+� �
+cos (θ′)
+sin (θ′)
+sin (θ′) − cos (θ′)
+�
+=
+=
+�
+cos (θ) cos (θ′) + sin (θ) sin (θ′)
+cos (θ) sin (θ′) − sin (θ) cos (θ′)
+sin (θ) cos (θ′) − cos (θ) sin (θ′)
+cos (θ) cos (θ′) + sin (θ) sin (θ′)
+�
+=
+�
+cos (θ − θ′)
+− sin (θ − θ′)
+sin (θ − θ′)
+cos (θ − θ′)
+�
+If both, U1 and U ′
+1, are in P, this product has to be again either symmetric, which is true if θ = θ′ or skew-symmetric, which is
+true if θ = θ′ + π/2. (Note that, also θ = θ′ + π and θ = θ′ + 3π/2 are possible solutions but we do not have to consider them
+
+10
+since they just differ by an irrelevant global factor of (−1) in one of the two unitaries.) Hence, U ′
+1 is either U1 or the unitary U2
+stated above. Hence, for each single-qubit subsystem, we can only use a set of operators P ≡ {11, U1, U2, XZ} for our basis
+construction.
+In a final step, we can show that we can restrict also θ. To see this, suppose a state-independent construction exists where we
+use the gates from the set P ≡ {11, U1, U2, XZ}. Now consider the construction where each gate U1 is replaced with W †U1W,
+each gate U2 with W †U2W, each gate XZ with W †XZW and each gate 11 with W †11W, where:
+W =
+�
+cos (α) − sin (α)
+sin (α)
+cos (α)
+�
+(C5)
+for some freely chosen parameter α. This also has to be a state-independent construction for any state with real coefficients,
+since W is a map from real states to real states, and all inner products between the basis states remain the same under this
+local transformation. Hence, if a state-independent construction exists with the gate set P ≡ {11, U1, U2, XZ}, another state-
+independent construction with the gate set P′ ≡ {W †11W, W †U1W, W †U2W, W †XZW} has to exist as well. Choosing
+α = θ/2, the set P′ ≡ {W †11W, W †U1W, W †U2W, W †XZW} becomes exactly P′ ≡ {11, X, Z, XZ}, which concludes the
+proof.
+
diff --git a/E9FQT4oBgHgl3EQfRDbD/content/tmp_files/load_file.txt b/E9FQT4oBgHgl3EQfRDbD/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d04b6ace4740a9d6d7556c20aca1f3816dae5555
--- /dev/null
+++ b/E9FQT4oBgHgl3EQfRDbD/content/tmp_files/load_file.txt
@@ -0,0 +1,755 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf,len=754
+page_content='Do entangled states correspond to entangled measurements under local transformations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Florian Pimpel,1, ∗ Martin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Renner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' ∗ and Armin Tavakoli4  1Atominstitut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Technische Universität Wien,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Stadionallee 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 1020 Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Austria  2University of Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Faculty of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Vienna Center for Quantum Science and Technology (VCQ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Boltzmanngasse 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 1090 Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Austria  3Institute for Quantum Optics and Quantum Information - IQOQI Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Austrian Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Boltzmanngasse 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 1090 Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Austria  4Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Lund University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Box 118,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 22100 Lund,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Sweden  We investigate whether pure entangled states can be associated to a measurement basis in which all vectors are  local unitary transformations of the original state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' We prove that for bipartite states with a local dimension that is  either 2, 4 or 8, every state corresponds to a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Via numerics we strongly evidence the same conclusion also  for two qutrits and three qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' However, for some states of four qubits we are unable to find a basis, leading  us to conjecture that not all quantum states admit a corresponding measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Furthermore, we investigate  whether there can exist a set of local unitaries that transform any state into a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' While we show that such a  state-independent construction cannot exist for general quantum states, we prove that it does exist for real-valued  n-qubit states if and only if n = 2, 3, and that such constructions are impossible for any multipartite system  of an odd local dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Our results suggest a rich relationship between entangled states and iso-entangled  measurements with a strong dependence on both particle numbers and dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Entanglement is a fundamental, broadly useful and an in-  tensely studied feature of quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  However,  in spite being of arguably similar foundational significance,  much less is known about the entanglement of joint quan-  tum measurements than the entanglement of quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Entangled measurements are crucial for seminal quantum in-  formation protocols such as teleportation [1], dense coding  [2] and entanglement swapping [3], which are instrumen-  tal for various quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Typically, they are  based on the paradigmatic Bell basis, which is composed of  the four maximally entangled states (|00⟩ ± |11⟩)/  √  2 and  (|01⟩ ± |10⟩)/  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' In the same way that the Bell basis may  be thought of as the measurement corresponding to the max-  imally entangled state, it is natural to ask whether entangled  states in general can be associated with a corresponding en-  tangled measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Studying the relationship between en-  tangled states and entangled measurements is not only inter-  esting for understanding quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' It is also an  invitation to explore, in the context of quantum information  applications, the largely uncharted terrain of entangled mea-  surements beyond the Bell basis and its immediate general-  isations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Most notably, entangled measurements beyond the  Bell basis are also increasingly interesting for topics such as  network nonlocality [4] and entanglement-assisted quantum  communication [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Consider that we are given a pure quantum state |ψ⟩ com-  prised of n subsystems, each of dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Is it possible  to find a measurement, namely an orthonormal basis of the  global dn-dimensional Hilbert space, in which all basis states  have the same degree of entanglement as |ψ⟩?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Specifically, we  want to decide the existence of dn strings, {Vj}dn  j=1, of local  unitary transformations,  Vj =  n  �  k=1  U (j)  k  (1)  ∗ These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  where U (j)  k  is a d-dimensional unitary operator, such that the  set of states |ψj⟩ ≡ Vj |ψ⟩ form a basis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' | ⟨ψj|ψj′⟩ | =  δjj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' If affirmative, we say that |ψ⟩ admits a basis and we call  the set of basis vectors {|ψj⟩}dn  j=1 a |ψ⟩-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Known examples of entangled measurements can be ac-  commodated in this picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' For example, the Bell basis can  be obtained from operating on |ψ⟩ = (|00⟩ + |11⟩)/  √  2 with  the four strings of local unitaries {Vj}4  j=1 = {11 ⊗ 11, 11 ⊗  X, Z ⊗ 11, Z ⊗ X}, where X and Z are bit-flip and phase-  flip Pauli operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' A well-known generalisation of the Bell  basis to n systems of dimension d can be thought of as a  |GHZn,d⟩-measurement where the relevant state is the higher-  dimensional GHZ state |GHZn,d⟩ =  1  √  d  �d−1  k=0 |k⟩⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' The  corresponding strings of local unitaries are Vj = Zj1  d ⊗Xj2  d ⊗  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' ⊗ Xjn  d |GHZn,d⟩ where j = j1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' jn ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' , d − 1}n  and where Zd = �d−1  l=0 e  2πi  d l |l⟩⟨l| and Xd = �d−1  l=0 |l + 1⟩⟨l|  are generalised Pauli operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' More generally, any state that  is locally maximally entanglable (for example graph states)  is known to admit a basis via suitable unitaries of the form  Vj = U j1  1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' ⊗ U jn  n  [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' These states are characterised  by the property that if each qubit is supplemented with a  qubit ancilla and controlled unitary gates are performed on  the state-ancilla pairs, then a maximally entangled bipartite  state can be constructed between the collection of state-qubits  and the collection of ancilla-qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  However, this is far  from a complete characterisation of the states that admit a  basis, which is seen already in the restrictive form of the  strings of unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' For example, the three-qubit W-state,  |W3⟩ = (|001⟩+|010⟩+|100⟩)/  √  3, is not locally maximally  entanglable but is neverthelss known to admit a basis [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' In  what follows, we set out to systematically explore whether  entangled states admit a corresponding basis and then, as we  will introduce later, whether such bases can be constructed  even without prior knowledge of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Let us begin with considering the simplest situation, namely  when |ψ⟩ is a state of two qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' We constructively show that  every such state admits a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' To this end, we first apply  the state-dependent local unitaries W A  ψ ⊗ W B  ψ that map |ψ⟩,  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='13285v1  [quant-ph]  30 Jan 2023  2  via a Schmidt decomposition, into the computational basis,  |ψS⟩ = λ |00⟩+  √  1 − λ2 |11⟩ for some coefficient 0 ≤ λ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Then, we consider the action of the following four strings of  local unitaries  �  �  �  �  �  11 ⊗ 11  11 ⊗ XZ  XZ ⊗ Z  XZ ⊗ X  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  (2)  One can verify that this transforms |ψS⟩ into a |ψ⟩-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' No-  tice that once the state has been rotated into the Schmidt form  |ψS⟩, the subsequent unitaries (2) do not depend on λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' This  construction can be extended to bipartite (n = 2) states of  local dimension d = 4 and d = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Again via Schmidt de-  composition, we can find state-dependent local unitaries that  transform |ψ⟩ into |ψS⟩ = �d−1  l=0 λl |ll⟩ for some Schmidt co-  efficients �  l λ2  l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' In Appendix A, we show that there is a  set of local unitaries that indeed leads to a |ψ⟩-basis indepen-  dently of the specific Schmidt coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  It is natural to consider also the simplest case that is not  of the above convenient form, namely that of two qutrits,  (n, d) = (2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' This appears to be considerably different be-  cause we fail to find strings of local unitaries that bring the  Schmidt decomposition |ψS⟩ into a basis without explicit de-  pendence on the Schmidt coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Nevertheless, a basis  might still be possible to construct by taking the Schmidt co-  efficients into account when choosing the local unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Ac-  tually, this seems to always be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' To arrive at this, we  have used a numerical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Let {|φj⟩}m  j=1 be a set of states  in a given Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' These states are pairwise orthogonal  if and only if they realise the global minimum (zero) of the  following objective function  f({φj}) ≡  �  j̸=j′  | ⟨φj|φj′⟩ |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  (3)  For a given state |ψ⟩, we numerically minimise f({ψj}) over  all possible strings {Vj}dn  j=1 of local unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' To this end,  we parameterise the local unitaries U (j)  k  using the scheme of  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' For the two-qutrit case, we have randomly chosen  1000 pairs of Schmidt coefficients (λ1, λ2) which (up to local  unitaries) fully specifies the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' In each case we numerically  minimise f({ψj}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Without exception, we find strings of lo-  cal unitaries that yield a result below our selected precision  threshold of f ≤ 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Furthermore, we have also numerically investigated the  case of three qubits, (n, d) = (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' This scenario requires  a different approach than the previous cases since multipar-  tite states have no Schmidt decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Instead, for any  given three-qubit state |ψ⟩, there exists local unitary transfor-  mations that map it onto the canonical form a |000⟩+b |011⟩+  c |101⟩ + d |110⟩ + e |111⟩ where (b, c, d, e) are real num-  bers and a is a complex number [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Hence, up to lo-  cal unitaries, the state space (after normalisation) is charac-  terised by five real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Later, we will provide an analyt-  ical construction of a |ψ⟩-basis for the four-parameter family  corresponding to restricting a to be real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' However, we have  not found an analytical basis construction for general three-  qubit states, but we nevertheless conjecture that it exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' To  evidence this, we have employed the previously introduced  numerical search method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Again, we have randomly chosen  1000 normalised sets of coefficients (a, b, c, d, e) and searched  for the minimal value of f over all the strings of local qubit  unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' In all cases, we find that f vanishes up to our se-  lected precision of f ≤ 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Given the above case studies, one might suspect that ev-  ery pure quantum state admits a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Interestingly, this  seems not to be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  While some states of four qubits,  (n, d) = (4, 2), are found to admit a basis, for example  a W state and doubly-excited Dicke state [23], it appears  that most four-qubit states do not admit a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' We have  sampled many different four-qubit states and repeatingly at-  tempted to numerically find a basis via the minimisation of  (3), also using several different search algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' It was reg-  ularly found that the estimated minimum is multiple orders  of magnitude above our given precision threshold for a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  For example, we searched for the minimum of f for the state  2  √  6 |W⟩ +  √  2  √  6 |GHZ4,2⟩, with 100 randomised initial points,  and never reached below f = 10−1, five orders of magnitude  above our precision threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' We have attempted to prove  that no basis exists by employing semidefinite outer relax-  ations of f over the set of dimensionally-restricted quantum  correlations [12] combined with a modified sampling of the  state and measurement space [13] and symmetrisation tech-  niques [14] to efficiently treat the large number of single-qubit  unitaries featured in this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' However, the conjecture  has resisted our efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' A guiding intuition for the impossi-  bility of a basis is to note that the number of free parameters is  3n(2n − 1) whereas the number of orthogonality constraints  (counting both the real and imaginary part) is 22n − 2n, and  the latter is larger than the former only when n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Furthermore, if an n-qubit state |ψ⟩ does not admit a ba-  sis, then the (n + 1)-qubit state |ψ′⟩ = |ψ⟩ ⊗ |0⟩ also does  not admit a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' By contradiction, suppose there are 2n+1  unitaries V ′  j = Vj ⊗ U (j)  n+1 such that |⟨ψ′|(V ′  j )†V ′  k|ψ′⟩| =  δjk ∀j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=', 2n+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Divide the 2n+1 states U (j)  n+1 |0⟩  into two sets such that two orthogonal vectors are not in  the same set (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' the northern and southern hemisphere  of the Bloch ball).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Consider the set that contains at least  as many elements as the other one, hence, at least 2n el-  ements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  By construction, these states cannot be distin-  guished on the last qubit, |⟨0|U (j)†  n+1U (k)  n+1|0⟩| ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Since  |⟨ψ′|(V ′  j )†V ′  k|ψ′⟩| = |⟨ψ|V †  j Vk|ψ⟩| · |⟨0|U (j)†  n+1U (k)  n+1|0⟩|, we  must have |⟨ψ|V †  j Vk|ψ⟩| = δjk for all of those pairs, which  contradicts that |ψ⟩ does not admit a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' By induction, this  argument shows that if our above conjecture holds, namely  that some four-qubit states do not admit a basis, then the same  holds for any number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Since not all pure quantum states admit a basis, and this  seems to be typical rather than exceptional for four qubits, it  is interesting to ask whether some distinguished families of  n-qubit states can nevertheless admit a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' This is well-  known to be the case for n-qubit GHZ-states and graph-states  3  since they are locally maximally entanglable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' More interest-  ingly, a positive answer is also possible for states that are not  of this kind: we construct a basis for the n-qubit W-state,  |Wn⟩ =  1  √n  �  σ σ(|0⟩⊗n−1 |1⟩) where σ runs over all permu-  tations of the position of “1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Note that |W1⟩ = |1⟩ and that a  |W1⟩-basis is obtained from the unitaries {11, X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Now we ap-  ply induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Consider that the strings {V (n)  j  }2n  j=1 generate a  |Wn⟩-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' One can then construct a basis for n+1 qubits as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' For half of the basis elements, namely j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' , 2n,  define V (n+1)  j  = V (n)  j  ⊗ 11 and for the other half, namely j =  2n +1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' , 2n+1, define V (n+1)  j  = �n  k=1 U (j)  k Z ⊗X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' As we  detail in Appendix B, one can verify that {V (n+1)  j  |Wn+1⟩}j  is a W-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' We note that for the purpose of entanglement  distillation, a different construction of a W-basis was given in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  So far, we have considered whether a specific state can be  associated to a specific measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' In other words, the uni-  tary constructions have been state-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' We now go fur-  ther and introduce a complementary concept, namely whether  there exist strings of local unitaries {Vj} that can transform  any state in a space of states S into a basis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' strings of local  unitaries that satisfy  ∀ψ ∈ S,  |⟨ψ|V †  j Vj′|ψ⟩| = δjj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  (4)  Naturally, this state-independent notion of basis construc-  tion is much stronger than the previously considered state-  dependent notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  In the most ambitious case, when we  choose the space S to be the entire Hilbert space of n sub-  systems of dimension d, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' S ≃ (Cd)⊗n, then a state-  independent construction cannot exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' In fact, not even two  orthogonal vectors can be state-independently constructed for  the full quantum state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' To show this, we can w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' set  V1 = 11 and assume that there exists local unitaries {Uk}  such that |ψ1⟩ = |ψ⟩ and |ψ2⟩ = �n  k=1 Uk |ψ⟩ are orthog-  onal for all |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Focus now on the particular state |ψ⟩ =  �n  k=1 |µk⟩ where |µk⟩ is some eigenvector of the unitary Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Since the eigenvalues of a unitary are complex phases, writ-  ten eiϕk for Uk and |µk⟩, we obtain |ψ1⟩ = �n  k=1 |µk⟩ and  |ψ2⟩ = ei �n  k=1 ϕk �n  k=1 |µk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' These two states are evidently  not orthogonal and hence we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Interestingly, the situation changes radically if we limit our  state-independent investigation to all quantum states in a real-  valued Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' That is, S ≃ (Rd)⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Such real quan-  tum systems have also been contrasted in the literature with  their complex counterparts [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Let us momentarily ig-  nore the n-partition structure of our Hilbert space and sim-  ply consider two real states |ψ1⟩ = |ψ⟩ and |ψ2⟩ = U |ψ⟩  obtained from a given real target state |ψ⟩ and a fixed (ψ-  independent) unitary U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  It holds that ψ1 and ψ2 are or-  thogonal if and only if U is skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  To prove  this, assume first the skew-symmetry property U = −U T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Since for real states ⟨ψ1|ψ2⟩ = ⟨ψ2|ψ1⟩∗ is equivalent to  ⟨ψ|U|ψ⟩ = ⟨ψ|U †|ψ⟩∗ = ⟨ψ|U T |ψ⟩, skew-symmetry im-  plies that ⟨ψ1|ψ2⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Conversely, assume that ⟨ψ|U|ψ⟩ = 0  for all real-valued ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Choosing in particular |ψ⟩ = |k⟩ for  k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' , d − 1, it follows that all diagonal elements of U  must vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Then, choose |ψ⟩ =  1  √  2(|i⟩ + |j⟩) for any pair  i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' This yield Uii+Ujj+Uij+Uji = 0, but since we know  that the diagonals vanish we are left with just Uij = −Uji  which defines a skew-symmetric operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Returning to our n-partitioned real Hilbert space, and still  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' taking V1 = 11, the above result demands that we find  local unitaries such that  U1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' ⊗ Un = −U T  1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' ⊗ U T  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  (5)  This is only possible if U T  k = ±Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Hence, all local unitaries  must be either symmetric or skew-symmetric, and the number  of the latter must be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' When extended from two orthogonal  states to a whole basis, we require that this property holds for  every pair of distinct labels (j, j′) in the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' In other words,  we require that every string (Vj)†Vj′ with j ̸= j′ is skew-  symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  The question becomes whether the above condition can  be satisfied for a given scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Consider it first for qubit  systems (d = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' In Appendix C we show that the set of  complex qubit unitaries that are either symmetric or skew-  symmetric and whose products are again either symmetric  or skew-symmetric, must obey a simple structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' they are  essentially equivalent to the four Pauli-type operators P ≡  {11, X, Z, XZ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Thus, if a state-independent construction ex-  ists, we can restrict to selecting one of these four operators for  each of our local unitaries U (j)  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Interestingly, for the case of  two qubits, (n, d) = (2, 2), a state-independent construction  is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' It is in fact given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' One can straightfor-  wardly verify that the above criterion is satisfied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' all local  unitaries are selected from P and all pairs of products of uni-  tary strings in (2) are skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Alternatively, one can  easily verify that (2) maps every state �  i,j=0,1 αij |ij⟩ into a  basis, for any real coefficients αij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Furthermore, by the same  token, a state-independent basis is also possible for every real  state of three qubits, (n, d) = (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' One explicit construc-  tion that satisfies our necessary and sufficient criterion is the  following set of eight strings of local unitaries  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  11 ⊗ 11 ⊗ 11  Z ⊗ Z ⊗ XZ  Z ⊗ XZ ⊗ 11  XZ ⊗ 11 ⊗ 11  Z ⊗ X ⊗ XZ  X ⊗ 11 ⊗ XZ  X ⊗ XZ ⊗ Z  X ⊗ XZ ⊗ X  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Again,  one  may  easily  verify  that  every  real  state  �  i,j,k=0,1 αijk |ijk⟩ is mapped into a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Two- and three-qubits are interesting cases because they  are exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  As we now show, there exists no state-  independent construction for real states of four or more qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  We first prove this for n = 4 and then show that this im-  plies impossibility also for n > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' The four-qubit case con-  tains 16 strings of unitaries and we know that each local uni-  tary can w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' be selected from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Since we seek a state-  independent construction, we can momentarily consider only  the state |0000⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' In order for it to be mapped into a basis, we  4  (2,2,R) (2,2,C) (3,2,R) (3,2,C) (4,2,R) (2,3,C) (2,4 or 8,C) (n, 2m + 1,R)  State-dependent  construction  \x13  \x13  \x13  (\x13)  (\x17)  (\x13)  \x13  − − −  State-independent  construction  \x13  \x17  \x13  \x17  \x17  \x17  \x17  \x17  TABLE I: Overview of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' The first row indicates the scenario: (n, d, S) gives particle number, dimension and the type of state space  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' The symbol \x13indicates the existence of a basis under local unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' The symbol \x17indicates that there in general can be no basis  under local unitaries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' at least one state admits no basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Paranthesis indicates that the result is obtained from numerical search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' The  symbol − − − indicates that no investigation was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  see that Z acts trivially on every register and therefore each  one of the 16 combinations of bit-flip or identity operators,  {Xc1 ⊗ Xc2 ⊗ Xc3 ⊗ Xc4} for c1, c2, c3, c4 ∈ {0, 1}, must  be featured in exactly one of the 16 unitary strings {Vj}16  j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Let us now look only at six of these strings, namely those  corresponding to having zero bit-flips (1 case), one bit-flip (4  cases) and four bit-flips (1 case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' fixing V1 = 11 (zero  bit-flips), the strings take the form  V1  11  ⊗  11  ⊗  11  ⊗  11  V2  XZr11 ⊗  Zr12  ⊗  Zr13  ⊗  Zr14  V3  Zr21  ⊗ XZr22 ⊗  Zr23  ⊗  Zr24  V4  Zr31  ⊗  Zr32  ⊗ XZr33 ⊗  Zr34  V5  Zr41  ⊗  Zr42  ⊗  Zr43  ⊗ XZr44  V6  XZr51 ⊗ XZr52 ⊗ XZr53 ⊗ XZr54  ,  (6)  where rij  ∈ {0, 1} represent our freedom to insert a Z  operator and thus realise the two relevant elements of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Since every row must be skew-symmetric and the only skew-  symmetric element in P is XZ, we must have r11 = r22 =  r33 = r44 = 1 and r51 + r52 + r53 + r54 = 1 where ad-  dition is modulo two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Moreover, every product of two rows  must be skew-symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' the product must have an odd  number of XZ operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' For the four middle rows, this im-  plies rij + rji = 1 for distinct indices i, j ∈ {1, 2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' For  the products V †  6 Vj for j = 2, 3, 4, 5, the conditions for skew-  symmetry respectively become  r12 + r13 + r14 + r52 + r53 + r54 = 1  r21 + r23 + r24 + r51 + r53 + r54 = 1  r31 + r32 + r34 + r51 + r52 + r54 = 1  r41 + r42 + r43 + r51 + r52 + r53 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  (7)  Summing these four equations and using the previously es-  tablished skew-symmetry conditions, one can cancel out all  degrees of freedom rij and arrive at the contradiction 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Hence, we conclude that the state-independent basis construc-  tion for four qubits is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  For the case of five qubits, we can again assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' that  the 32 combinations of bit-flip or identity operators, {Xc1 ⊗  Xc2⊗Xc3⊗Xc4⊗Xc5} for c1, c2, c3, c4, c5 ∈ {0, 1} must be  featured in exactly one of the 32 unitary strings since the state  |00000⟩ has to be mapped into an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Suppose  there is a state-independent construction that maps every real-  valued five-qubit state into a basis, in especially any state of  the form |ψ⟩ ⊗ |0⟩, where |ψ⟩ is an arbitrary real-valued four  qubit state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Now consider the 16 strings where c5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Since  the fifth qubit is always mapped to itself, it has to hold that the  first four qubits are pairwise distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' However, this  implies a state-independent construction for four qubits which  is in contradiction to the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' By induction, this implies that  no state-independent construction can exist whenever n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  The possibility of state-independent constructions for real-  valued bi- and tri-partite systems draws heavily on the sim-  ple structure of skew-symmetric qubit unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' If we con-  sider real-valued systems of dimension d > 2, the situa-  tion changes considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Using our necessary and suffi-  cient condition, it follows immediately that state-independent  constructions are impossible in all odd dimensions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' when  (n, d) = (n, 2m+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' This stems from the fact that there exists  no skew-symmetric unitary matrix in odd dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' To see  that, simply note that if A is skew-symmetric then det(A) =  det  �  AT �  = det(−A) = (−1)2m+1 det(A) = − det(A) and  hence det(A) = 0, but that contradicts unitarity because the  determinant of a unitary has unit modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  In summary, we have investigated the correspondence be-  tween entangled states and entangled measurements under lo-  cal unitary transformations, both when the local transforma-  tion can and cannot explicitly depend on the target state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Per-  haps surprisingly, we have found that this problem is not so  straightforward and has a strong dependence on both the num-  ber of subsystems involved and their dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Our analyt-  ical and numerical results and conjectures are summarised in  Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  The conspicuous open problem left by our work is to prove  our conjecture that there exists states that do not admit a ba-  sis under local unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' An interesting related question is if  one can bound the relative volume of four-qubit states that do  not admit a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Our numerical investigations suggest that  nearly all four-qubit states should belong to this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Fur-  thermore, it would be useful to find analytical solutions for  the three-qubit and two-qutrit state-dependent cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' More-  over, for the state-independent considerations, we focused on  real Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' A natural question is whether there ex-  ists state-independent basis constructions for other interest-  ing spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' For example, if one restricts to bipartite states  of a known entanglement entropy, can one construct a state-  independent basis?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' The answer is clearly positive for the lim-  iting cases of product states and maximally entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Another interesting space to consider is the symmetric sub-  space of n-qubit Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  5  Our results may also have prospects in quantum informa-  tion as one may now construct entangled measurements asso-  ciated to entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Recently there has been proposals  of two-qubit entangled projections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' the so-called Elegant Joint  Measurements [18, 19] which have also been realised in vari-  ous experiments [20?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' , 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' The Elegant Joint Measurements  can be seen as a particular type of |ψ⟩-basis where |ψ⟩ is a  partially entangled two-qubit state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' However, the basis addi-  tionally has the feature that the collections of reduced states  form a tetrahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' This requirement goes beyond our prob-  lem formulation, as we do not impose any structure on the  reduced states of our bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' However, it suggests an avenue  to identifying interesting and highly symmetric measurements  by finding the particular |ψ⟩-basis that maximises the Hilbert  space volume spanned its collection of reduced states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Finally, one of the notable shortcommings of traditional,  GHZ based, multiqubit entanglement swapping protocols is  that the loss of one particle renders the measurement separa-  ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' However, some other states that are inequivalent to GHZ  under LOCC can preserve their entanglement under reduc-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' The existence of an iso-entangled basis composed of  such states may constitute an avenue to more noise-resiliant  entanglement swapping protocols which have natural quan-  tum information applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Note added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='— During the late stage of our work, we be-  came aware of the previous work [23] where i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' bases are  found for some Dicke states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We thank Hayata Yamasaki,  Marcus Huber,  Jakub  Czartowski and Karol ˙Zyczkowski for discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' ac-  knowledges support from the Wenner-Gren Foundation and  from the Wallenberg Centre for Quantum Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' acknowledges financial support from the Austrian Science  Fund (FWF) through BeyondC (F7103-N38), the Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  I-2906, as well as support by the John Templeton Founda-  tion through Grant 61466, The Quantum Information Struc-  ture of Spacetime (qiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='fr), the Foundational Questions Insti-  tute (FQXi) and the research platform TURIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' The opinions  expressed in this publication are those of the authors and do  not necessarily reflect the views of the John Templeton Foun-  dation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Bennett, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Brassard, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Crépeau, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Jozsa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Peres, and  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Wootters, Teleporting an unknown quantum state via dual  classical and einstein-podolsky-rosen channels, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  70, 1895 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Bennett and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
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+page_content=' Shang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
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+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Li, and  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
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+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
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+page_content=' Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='-  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Huang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Li, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Guo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Gisin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Branciard, and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Tavakoli, Entanglement swapping and quantum correla-  tions via symmetric joint measurements, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 129,  030502 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  6  [22] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Bäumer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Gisin, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Tavakoli, Demonstrating the power  of quantum computers, certification of highly entangled mea-  surements and scalable quantum nonlocality, npj Quantum In-  formation 7, 117 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  [23] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Tanaka,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Markham,  and  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Murao,  Local  encoding  of  classical  information  onto  quantum  states,  Journal  of  Modern  Optics  54,  2259  (2007),  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='1080/09500340701403301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Appendix A: Basis construction for every bipartite state of local dimension d = 4 and d = 8  Let the local dimension be a power of two, d = 2m, and index the d2 basis elements as (˜j, j) where ˜j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' , d − 1  and j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Let W A  ψ ⊗ W B  ψ be the state-dependent local unitaries that transform the general state |ψ⟩ into the Schmidt  basis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' |ψS⟩ ≡ W A  ψ ⊗ W B  ψ |ψ⟩ = �d−1  l=0 λl |l, l⟩, with the Schmidt coefficients λl ∈ R satisfying �  l λ2  l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' We now  further decompose the individual d-dimensional registers as a string of m qubits, writing |l⟩ = |l1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' lm⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Thus, the Schmidt  decomposed state reads  |ψS⟩ =  �  l1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=',lm=0,1  λl |l1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' lm, l1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' lm⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  (A1)  Once the state has been put in the form (A1), we apply a set of local unitaries that is independent of the Schmidt coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  For d = 4 and ˜j = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' the two sets of unitaries read as follows:  ˜j j  U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='j)  1  U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='j)  2  U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='j)  1  ⊗ U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='j)  2  |ψS⟩  0 1  11 ⊗ 11  11 ⊗ 11  λ00 |00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 00⟩ + λ01 |01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 01⟩ + λ10 |10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 10⟩ + λ11 |11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 11⟩  0 2  11 ⊗ X  11 ⊗ XZ  λ00 |01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 01⟩ − λ01 |00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 00⟩ + λ10 |11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 11⟩ − λ11 |10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 10⟩  0 3  X ⊗ 11  XZ ⊗ Z  λ00 |10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 10⟩ − λ01 |11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 11⟩ − λ10 |00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 00⟩ + λ11 |01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 01⟩  0 4  X ⊗ X XZ ⊗ X  λ00 |11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 11⟩ + λ01 |10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 10⟩ − λ10 |01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 01⟩ − λ11 |00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 00⟩  (A2)  In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' we define U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='j)  1  := X  ˜j  4 U (˜j=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='j)  1  and U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='j)  2  := U (˜j=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='j)  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' where Xd is the d-dimensional shift-operator Xd =  �d−1  l=0 |l + 1⟩⟨l|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Note that, the unitaries U (˜j,j)  2  coincide with the state-independent set for two qubits given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' (2) and  do not depend on ˜j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' At the same time, U (˜j=0,j)  1  are the same as U (˜j,j)  2  where the Z gates are left out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' We now show that  {U (˜j,j)  1  ⊗ U (˜j,j)  2  |ψS⟩}˜j,j is a basis of the bipartite Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' One can check directly that the four states with ˜j = 0 stated in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' (A2) above are pairwise orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' We want to mention that we are exploiting the fact that U (˜j=0,j)  2  are the elements of a  state-independent construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' To see the connection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' note that the calculation for the state-independent two-qubit construction  reads as follows:  (11 ⊗ 11)(λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩) = λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  (A3)  (11 ⊗ XZ)(λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩) = λ00 |01⟩ − λ01 |00⟩ + λ10 |11⟩ − λ11 |10⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  (A4)  (XZ ⊗ Z)(λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩) = λ00 |10⟩ − λ01 |11⟩ − λ10 |00⟩ + λ11 |01⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  (A5)  (XZ ⊗ X)(λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩) = λ00 |11⟩ + λ01 |10⟩ − λ10 |01⟩ − λ11 |00⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  (A6)  Since these states are pairwise orthogonal for arbitrary real coefficients λl1l2, the same holds true for the states in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  In addition, all of the states where ˜j = 0 are elements of the subspace spanned by |00, 00⟩, |01, 01⟩, |10, 10⟩ and |11, 11⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Hence, they form a basis of this four-dimensional subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  By shifting now the first system we obtain a basis for the  remaining orthogonal subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  More precisely, since we defined U (˜j,j)  1  = X  ˜j  4 U (˜j=0,j)  1  the states where ˜j = 1 are  esentially the same states as the ones in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' (A2) but with the first system shifted by one l → l ⊕ 1 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' For example,  λ00 |11, 10⟩ − λ01 |00, 11⟩ − λ10 |01, 00⟩ + λ11 |10, 01⟩ is the state that corresponds to ˜j = 1 and j = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' In this way, the  four states where ˜j = 1 form a basis of the subspace spanned by |01, 00⟩, |10, 01⟩, |11, 10⟩ and |00, 11⟩ (or all states where  |l + 1, l⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Analogously, the four states where ˜j = 2 (˜j = 3) form a basis of the subspaces spanned by the vectors with |l + 2, l⟩  (|l + 3, l⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Altogether, the sixteen states {U (˜j,j)  1  ⊗ U (˜j,j)  2  |ψS⟩}˜j,j form a basis of the entire sixteen dimensional Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  A similar construction can be found for d = 8 by using the state-independent construction of three qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Similar as above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  7  the set for ˜j = 0 reads as follows:  ˜j j  U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='j)  1  U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='j)  2  U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='j)  1  ⊗ U (˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='j)  2  |ψS⟩  0 1  11 ⊗ 11 ⊗ 11  11 ⊗ 11 ⊗ 11  +λ000 |000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 000⟩ + λ001 |001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 001⟩ + λ010 |010,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 010⟩ + λ011 |011,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 011⟩  +λ100 |100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 100⟩ + λ101 |101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 101⟩ + λ110 |110,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 110⟩ + λ111 |111,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 111⟩  0 2  11 ⊗ 11 ⊗ X  Z ⊗ Z ⊗ XZ  +λ000 |001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 001⟩ − λ001 |000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 000⟩ − λ010 |011,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 011⟩ + λ011 |010,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 010⟩  −λ100 |101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 101⟩ + λ101 |100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 100⟩ + λ110 |111,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 111⟩ − λ111 |110,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 110⟩  0 3  11 ⊗ X ⊗ 11  Z ⊗ XZ ⊗ 11  +λ000 |010,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 010⟩ + λ001 |011,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 011⟩ − λ010 |000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 000⟩ − λ011 |001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 001⟩  −λ100 |110,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 110⟩ − λ101 |111,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 111⟩ + λ110 |100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 100⟩ + λ111 |101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 101⟩  0 4  X ⊗ 11 ⊗ 11  XZ ⊗ 11 ⊗ 11  (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=')  0 5  11 ⊗ X ⊗ X  Z ⊗ X ⊗ XZ  (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=')  0 6  X ⊗ 11 ⊗ X  X ⊗ 11 ⊗ XZ  (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=')  0 7  X ⊗ X ⊗ 11  X ⊗ XZ ⊗ Z  (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=')  0 8  X ⊗ X ⊗ X X ⊗ XZ ⊗ X  (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=')  (A7)  Again, we define U (˜j,j)  1  = X  ˜j  8 U (˜j=0,j)  1  and U (˜j,j)  2  = U (˜j=0,j)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' The proof that this forms a basis of the 64-dimension Hilbert  space is completely analogous to the case of d = 4 before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' The eight states for ˜j = 0 form a basis of the eight-dimensional  subspace spanned by |l1l2l3, l1l2l3⟩ (for li = 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Applying the shift operator X8 to the first system, one obtains bases of the  other eight-dimensional orthogonal subspaces spanned by the vectors with  ��l + ˜j, l  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' This approach cannot (immediately) be  generalized to higher dimensions d = 2n, due to the lack of state-independent constructions for n ≥ 4 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' However, there is  in principle no reason to restrict the unitaries on the second system to tensor products of single qubit Pauli gates as we do here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  In principle, we could also consider general permutations with suitably chosen signs such that all terms cancel in this pairwise  sense as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Even when considering this larger class of possibilities, we made an exhaustive search and could not find any  additional construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Due to this, it seems unlikely that a construction exists in which the unitaries do not depend on the  Schmidt coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Appendix B: An n-qubit basis of W-states  We define the n-qubit W-state as  |W1⟩ ≡ |1⟩  |W2⟩ ≡  1  √  2 (|01⟩ + |10⟩)  |W3⟩ ≡  1  √  3 (|001⟩ + |010⟩ + |100⟩)  |W4⟩ ≡ 1  2 (|0001⟩ + |0010⟩ + |0100⟩ + |1000⟩)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  (B1)  Note that for one and two qubits, the definition is only introduced for sake of convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' In general, we write  |Wn⟩ ≡  1  √n  �  σ  σ(|0⟩⊗n−1 |1⟩),  (B2)  where σ runs over all permutations of the position of “1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' It is also useful to write the state recursively as  |Wn+1⟩ =  �  n  n + 1 |Wn⟩ ⊗ |0⟩ +  1  √n + 1 |0⟩n ⊗ |1⟩  (B3)  Clearly, if we apply the local unitaries U (1)  1  = 11 and U (2)  1  = X to |W1⟩ we generate the trivial one-qubit W-basis {|0⟩ , |1⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Assume now that the local unitaries {U (j)  k } for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' n and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' , 2n yield a |Wn⟩-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' We will now show that under  this assumption we can construct a basis for |Wn+1⟩ and hence it follows from induction that a W-basis exists for any number  of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  8  We illustrate the induction step as follows,  U (1)  1  ⊗  U (1)  2  ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' ⊗  U (1)  n  ⊗  11  U (2)  1  ⊗  U (2)  2  ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' ⊗  U (2)  n  ⊗  11  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  U (2n)  1  ⊗  U (2n)  2  ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' ⊗  U (2n)  n  ⊗  11  U (1)  1 Z  ⊗  U (1)  2 Z  ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' ⊗  U (1)  n Z  ⊗ X  U (2)  1 Z  ⊗  U (2)  2 Z  ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' ⊗  U (2)  n Z  ⊗ X  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  U (2n)  1  Z ⊗ U (2n)  2  Z ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' ⊗ U (2n)  n  Z ⊗ X  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  (B4)  We see that for the first 2n basis elements, we extend the unitaries for n qubits by tensoring with 11 for qubit number n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  For the latter 2n basis elements, we extend the unitaries for n qubits by multiplying all of them from the right by Z and finally  tensoring with X for qubit number n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' As usual, we now write the string of unitaries associated to each row as V (n+1)  j  for  n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' , 2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' We similarly use V (n)  j  for the unitary strings for the case of n qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  To see that this yields a basis, we first show that the first 2n basis elements (upper block of table, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' , 2n) are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  For this purpose, we use the recursion formula (B3) to write for j ̸= j′  ⟨Wn+1|(V (n+1)  j′  )†V (n+1)  j  |Wn+1⟩ =  n  n + 1⟨Wn0|(V (n)  j′  )†V (n)  j  ⊗ 11|Wn0⟩ +  1  n + 1⟨0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 01|(V (n)  j′  )†V (n)  j  ⊗ 11|0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 01⟩  +  √n  n + 1⟨Wn0|(V (n)  j′  )†V (n)  j  ⊗ 11|0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 01⟩ +  √n  n + 1⟨0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 01|(V (n)  j′  )†V (n)  j  ⊗ 11|Wn0⟩ = 0  The first term is zero for all j′ ̸= j due to the induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' The third and fourth terms are zero due to orthogonality in  the last qubit register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' The second term is zero for every j′ ̸= j there exists at least one qubit register k for which U (j′)  k  and U (j)  k  are composed of different numbers of bit-flips (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' The latter follows from the initial condition of using {11, X} to construct the  |W1⟩-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  The same procedure will analogously show that the latter 2n basis elements (lower block of the table, j = 2n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' , 2n+1)  are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' We are left with showing that every overlap between the upper and lower block, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' with any j′ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' , 2n and  any j = 2n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' , 2n+1, also vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' For this we have  ⟨Wn+1|(V (n+1)  j′  )†V (n+1)  j  |Wn+1⟩ =  n  n + 1⟨Wn0|  �  (V (n)  j′  )†V (n)  j  ⊗ X  �  n  �  k=1  Z ⊗ 11|Wn0⟩  +  1  n + 1⟨0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 01|  �  (V (n)  j′  )†V (n)  j  ⊗ X  �  n  �  k=1  Z ⊗ 11|0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 01⟩  +  √n  n + 1⟨Wn0|  �  (V (n)  j′  )†V (n)  j  ⊗ X  �  n  �  k=1  Z ⊗ 11|0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 01⟩  +  √n  n + 1⟨0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 01|  �  (V (n)  j′  )†V (n)  j  ⊗ X  �  n  �  k=1  Z ⊗ 11|Wn0⟩  Note that �n  k=1 Z ⊗ 11 |Wn0⟩ = − |Wn0⟩ and �n  k=1 Z ⊗ 11 |0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 01⟩ = |0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 01⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' The first and second terms are both zero due  to orthogonality in the final qubit register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' We thus have  ⟨Wn+1|(V (n+1)  j′  )†V (n+1)  j  |Wn+1⟩ =  √n  n + 1⟨Wn|(V (n)  j′  )†V (n)  j  |0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 0⟩ −  √n  n + 1⟨0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 0|(V (n)  j′  )†V (n)  j  |Wn⟩  =  √n  n + 1⟨Wn|(V (n)  j′  )†V (n)  j  − (V (n)  j  )†V (n)  j′  |0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' 0⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  (B5)  The last equality follows from the fact that it is sufficient, for given (j, j′), that there exist some register index k such that  (U (j′))†  kU (j)  k  − (U (j))†  kU (j′)  k  = 0 in order for the overlap to vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' This is always the case because due to our construction (see  initial condition and the table), for every two unitaries there is at least one register k where the single-qubit unitaries differ by  X, meaning that either (U (j)  k , U (j′)  k  ) = (11, X)/(Z, XZ), or the same with j ↔ j′ is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' The condition above is satisfied by  all of these combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Hence we conclude that the proposed construction satisfies  ⟨Wn+1|(V (n+1)  j  )†V (n+1)  j′  |Wn+1⟩ = δjj′  (B6)  9  and therefore yields a W-state basis for any number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Appendix C: The Pauli structure for state-independent qubit unitary constructions  We consider the set of local unitaries P that are applied to the i-th qubit in the state-independent construction and show that  without loss of generality, the set can be chosen to be the Pauli-type gates P ≡ {11, X, Z, XZ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' First, note that the set is finite  since there are exactly 2n basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Next, we observe that the identity 11 has to be within the set P since we demand that  V1 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Furthermore, we can argue that the gate  XZ =  �  0 −1  1  0  �  has to be within the set as well, since it is the only gate that maps every real qubit state to its orthogonal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' More precisely,  if it is not used on the i-th qubit at least once, one can choose a real qubit state |φi⟩ such that none of the gates in P map  |φi⟩ to its orthogonal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Hence if we apply the state-independent construction to the real-valued product state |φ⟩ =  |0⟩1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' |0⟩i−1 ⊗ |φi⟩ ⊗ |0⟩i+1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' ⊗ |0⟩n none of the resulting 2n states are distinguishable on the i-th qubit, which is  impossible if these states should form a basis of product states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Therefore, the gate XZ has to be within the set P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Apart from  the gates 11 and XZ we can constrain which other qubit unitaries can be in the set P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' We know that if we demand V1 = 11, every  string of local unitaries (Vj) and their products (Vj)†Vj′ with j ̸= j′ have to be skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' As a result, the local unitaries  on each subsystem (hence, the unitaries in the set P) and also all their products have to be either symmetric or skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  By neglecting a global phase, the general form of a unitary operator can be written as:  U =  �  cos (θ)eiα  sin (θ)eiβ  − sin (θ)e−iβ cos (θ)e−iα  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  (C1)  The only skew-symmetric 2×2 unitary is, up to an irrelevant global phase, the Pauli-type operator XZ, which we already found  to be necessarily in the set P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' All the symmetric matrices of this form can be written as:  U =  �  cos (θ)eiα  i sin (θ)  i sin (θ) cos (θ)e−iα  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  (C2)  If the gate U is in P, it is at some point multiplied with the gate XZ since the operator XZ is used at least once on the i-th qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  Since we know that the result of this product has to be again either symmetric or skew-symmetric, we obtain that α = π/2, 3π/2  due to:  (XZ)†U =  �  0 1  −1 0  � �  cos (θ)eiα  i sin (θ)  i sin (θ) cos (θ)e−iα  �  =  �  i sin (θ) cos (θ)e−iα  − cos (θ)eiα  −i sin (θ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  (C3)  The two possibilities for α = π/2, 3π/2 correspond to the two solutions  U1 =  �  cos (θ)  sin (θ)  sin (θ) − cos (θ)  �  ,  U2 =  �  sin (θ) − cos (θ)  − cos (θ) − sin (θ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  (C4)  We left the irrelevant global factor i for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Considering the additional degree of freedom of θ, we can restrict to the first  class of solutions U1 since the second class U2 can be obtained by shifting θ by π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Hence, if we add a gate U to the set P, it  has to be of the form given by U1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Now if we add two such gates to the set P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' the product of U1 with another valid matrix  U ′  1 is  U †  1U ′  1 =  �  cos (θ)  sin (θ)  sin (θ) − cos (θ)  � �  cos (θ′)  sin (θ′)  sin (θ′) − cos (θ′)  �  =  =  �  cos (θ) cos (θ′) + sin (θ) sin (θ′)  cos (θ) sin (θ′) − sin (θ) cos (θ′)  sin (θ) cos (θ′) − cos (θ) sin (θ′)  cos (θ) cos (θ′) + sin (θ) sin (θ′)  �  =  �  cos (θ − θ′)  − sin (θ − θ′)  sin (θ − θ′)  cos (θ − θ′)  �  If both,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' U1 and U ′  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' are in P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' this product has to be again either symmetric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' which is true if θ = θ′ or skew-symmetric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' which is  true if θ = θ′ + π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' (Note that, also θ = θ′ + π and θ = θ′ + 3π/2 are possible solutions but we do not have to consider them  10  since they just differ by an irrelevant global factor of (−1) in one of the two unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=') Hence, U ′  1 is either U1 or the unitary U2  stated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Hence, for each single-qubit subsystem, we can only use a set of operators P ≡ {11, U1, U2, XZ} for our basis  construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content='  In a final step, we can show that we can restrict also θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' To see this, suppose a state-independent construction exists where we  use the gates from the set P ≡ {11, U1, U2, XZ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Now consider the construction where each gate U1 is replaced with W †U1W,  each gate U2 with W †U2W, each gate XZ with W †XZW and each gate 11 with W †11W, where:  W =  �  cos (α) − sin (α)  sin (α)  cos (α)  �  (C5)  for some freely chosen parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' This also has to be a state-independent construction for any state with real coefficients,  since W is a map from real states to real states, and all inner products between the basis states remain the same under this  local transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Hence, if a state-independent construction exists with the gate set P ≡ {11, U1, U2, XZ}, another state-  independent construction with the gate set P′ ≡ {W †11W, W †U1W, W †U2W, W †XZW} has to exist as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
+page_content=' Choosing  α = θ/2, the set P′ ≡ {W †11W, W †U1W, W †U2W, W †XZW} becomes exactly P′ ≡ {11, X, Z, XZ}, which concludes the  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf'}
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+arXiv:2301.01268v1  [math.CV]  3 Jan 2023
+Proper holomorphic maps in Euclidean spaces
+avoiding unbounded convex sets
+Barbara Drinovec Drnovˇsek and Franc Forstneriˇc
+Abstract We show that if E is a closed convex set in Cn (n > 1) contained in a closed halfspace H such
+that E ∩ bH is nonempty and bounded, then the concave domain Ω = Cn \ E contains images of proper
+holomorphic maps f : X → Cn from any Stein manifold X of dimension < n, with approximation of
+a given map on closed compact subsets of X. If in addition 2 dim X + 1 ≤ n then f can be chosen an
+embedding, and if 2 dim X = n then it can be chosen an immersion. Under a stronger condition on E
+we also obtain the interpolation property for such maps on closed complex subvarieties.
+Keywords Stein manifold, holomorphic embedding, Oka manifold, minimal surface, convexity
+MSC (2010): 32H02, 32Q56; 52A20, 53A10
+Date: 3 January 2023
+In memoriam Nessim Sibony
+1. Introduction
+Let X be a Stein manifold. Denote by O(X, Cn) the Frechet space of holomorphic maps
+X → Cn endowed with the compact-open topology and write O(X, C) = O(X). A theorem of
+Remmert [36] (1956), Narasimhan [35] (1960), and Bishop [7] (1961) states that almost proper
+maps are residual in O(X, Cn) if dim X = n, proper maps are dense if dim X < n, proper
+immersions are dense if 2 dim X ≤ n, and proper embeddings are dense if 2 dim X < n. A
+proof is also given in the monograph [29] by Gunning and Rossi.
+It is natural to ask how much space proper maps need. We pose the following question.
+Problem 1.1. For which domains Ω ⊂ Cn are proper holomorphic maps (immersions,
+embeddings) X → Cn as above, with images contained in Ω, dense in O(X, Ω)?
+It is evident that Ω cannot be contained in a halfspace of Cn since every holomorphic map
+from C to a halfspace lies in a complex hyperplane. In this paper we give an affirmative answer
+for concave domains whose complement E = Cn \ Ω satisfies the following condition.
+Definition 1.2. A closed convex set E in a real or complex Euclidean space V has bounded
+convex exhaustion hulls (BCEH) if for every compact convex set K in V
+(1.1)
+the set h(E, K) = Conv(E ∪ K) \ E is bounded.
+Here, Conv denotes the convex hull. The following is our first main result.
+Theorem 1.3. Let E be an unbounded closed convex set in Cn (n > 1) with bounded convex
+exhaustion hulls. Given a Stein manifold X with dim X < n, a compact O(X)-convex set K in
+X, and a holomorphic map f0 : K → Cn with f0(bK) ⊂ Ω = Cn \ E, we can approximate f0
+uniformly on K by proper holomorphic maps f : X → Cn satisfying f(X \ ˚
+K) ⊂ Ω. The map
+f can be chosen an embedding if 2 dim X < n and an immersion if 2 dim X ≤ n.
+
+2
+B. Drinovec Drnovˇsek and F. Forstneriˇc
+In this paper, a map f : K → Cn from a compact set K is said to be holomorphic if it is the
+restriction to K of a holomorphic map on an open neighbourhood of K.
+In particular, if f0(K) ⊂ Ω then the theorem gives uniform approximation of f0 by proper
+holomorphic maps f : X → Cn with f(X) ⊂ Ω. If bE is of class C 1 and strictly convex
+near infinity, we obtain an analogue of Theorem 1.3 with additional interpolation on a closed
+complex subvariety X′ of X such that f0 : X′ → Cn is proper holomorphic; see Theorem 4.2.
+Without the condition on the range, interpolation of proper holomorphic embeddings X ֒→ Cn
+on a closed complex subvariety was obtained by Acquistapace et al. [1] in 1975.
+The analogue of the BCEH condition for unbounded closed sets in Stein manifolds, with the
+convex hull replaced by the holomorphically convex hull, is used in holomorphic approximation
+theory of Arakelyan and Carleman type; see the survey in [18].
+It is evident that a closed convex set E ⊂ Rn has BCEH if and only if there is an increasing
+sequence K1 ⊂ K2 ⊂ · · · of compact convex sets exhausting Rn such that the set h(E, Kj) (see
+(1.1)) is bounded for every j = 1, 2, . . .. In particular, BCEH is a condition at infinity which is
+invariant under perturbations supported on a compact subset. For compact convex sets E ⊂ Cn,
+Theorem 1.3 was proved in [24]; in this case BCEH trivially holds.
+We show in Section 3 that a closed convex set E in Rn has BCEH if and only if E is
+continuous in the sense of Gale and Klee [26]; see Proposition 3.3. If E has BCEH then
+Conv(E ∪ K) is closed for any compact convex set K ⊂ Rn (see [26, Theorem 1.5]). If such E
+is unbounded, which is the main case of interest, there are affine coordinates (x, y) ∈ Rn−1 × R
+such that E = Eφ = {(x, y) ∈ Rn : y ≥ φ(x)} is the epigraph of a convex function
+φ : Rn−1 → R+ = [0, +∞) growing at least linearly near infinity (see Proposition 3.4). In
+particular, an unbounded closed convex set E ⊂ Cn with BCEH is of the form
+(1.2)
+E = Eφ = {z = (z′, zn) ∈ Cn : ℑzn ≥ φ(z′, ℜzn)}
+in some affine complex coordinates z = (z′, zn) on Cn, with φ as above. (Here, ℜ and ℑ denote
+the real and the imaginary part.) For a convex function φ of class C 1 we give a differential
+characterization of the BCEH condition on its epigraph Eφ; see Proposition 3.8. The BCEH
+property holds if the radial derivative of φ tends to infinity; see Corollary 3.9. On the other hand,
+there are convex functions of linear growth whose epigraphs have BCEH; see Example 3.10. By
+Proposition 3.11, a convex function φ with at least linear growth at infinity can be approximated
+uniformly on compacts by functions ψ ≤ φ of the same kind whose epigraphs Eψ have BCEH.
+This allows us to extend Theorem 1.3 as follows; see Section 4 for the proof.
+Corollary 1.4. The conclusion of Theorem 1.3 holds for any convex epigraph Eφ of the form
+(1.2) such that φ ≥ 0 and the set {φ = 0} is nonempty and compact.
+A closed convex set E ⊂ Cn with BCEH does not contain any affine real line (see Proposition
+3.4), and for n > 1 its complement Ω = Cn \ E is an Oka domain according to Wold and the
+second named author; see [25, Theorem 1.8]. This fact plays an important role in our proof
+of Theorem 1.3, given in Section 4. (The precise result from Oka theory which we shall use
+is stated as Theorem 4.1.) Among closed convex epigraphs (1.2), the class of sets with Oka
+complement is strictly bigger than the class of sets with BCEH. In particular, the former class
+contains many sets containing boundary lines, which is impossible for a set with BCEH.
+Problem 1.5. Is there a (not necessarily convex) set Eφ ⊂ Cn of the form (1.2) with φ ≥ 0 of
+sublinear growth for which Theorem 1.3 holds? Is there a set of this kind in C2 such that C2\Eφ
+contains the image of a proper holomorphic disc D = {z ∈ C : |z| < 1} → C2?
+
+Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
+3
+Theorem 1.3 is the first general result in the literature providing proper holomorphic maps
+X → Cn from any Stein manifold of dimension < n whose images avoid large convex sets in
+Cn close to a halfspace, and with approximation of a given map on a compact holomorphically
+convex set in X. Without the approximation condition and assuming that dim X ≤ n − 2, there
+are proper holomorphic maps of X into a complex hyperplane in Cn \ E.
+On the other hand, there are many known results concerning proper holomorphic maps in
+Euclidean spaces and in more general Stein manifolds whose images avoid certain small closed
+subsets, such as compact or complete pluripolar ones, and results in which the source manifold
+is the disc D = {z ∈ C : |z| < 1}. Proper holomorphic discs in C2 avoiding closed complete
+pluripolar sets of the form E = E′ × C, with E′ ⊂ C, were constructed by Alexander [5] in
+1977. The first named author showed in [13] (2004) that for every closed complete pluripolar
+set E in a Stein manifold Y with dim Y > 1 and point p ∈ Y \ E there is a proper holomorphic
+disc f : D → Y with p ∈ f(D) ⊂ Y \ E. If Y = C2 there also exist embedded holomorphic
+discs with this property according to Borell et al. [8] (2008), and for dim Y ≥ 3 this holds
+by the general position argument. Proper holomorphic discs in C2 with images contained in
+certain concave cones were constructed by Globevnik and the second named author [23] in
+2001. They also constructed proper holomorphic discs in C2 with images in (C \ {0})2, and
+hence proper harmonic discs D → R2, disproving a conjecture by Schoen and Yau [37, p. 18].
+(Another construction of such maps was given by Boˇzin [9].) More generally, it was shown
+by Alarc´on and L´opez [4, Corollary 1.1] in 2012 that every open Riemann surface X admits a
+proper harmonic map to R2 which is the projection of a conformal minimal immersion X → R3.
+The aforementioned result from [23] was used by the first named author in [12] (2002) to classify
+closed convex sets in C2 whose complement is filled by images of holomorphic discs which are
+proper in C2. More recently, Forstneriˇc and Ritter [24] (2014) proved Theorem 1.3 in the case
+when E ⊂ Cn is a compact polynomially convex set and 2 dim X ≤ n (for immersions) or
+2 dim X < n (for embeddings), and for proper holomorphic maps X → Cn when dim X < n
+and E is a compact convex set. A further development in this direction is the analogue of
+Theorem 1.3 when Cn is replaced by a Stein manifold Y with the density property and E ⊂ Y
+is a compact O(Y )-convex set; see [22, Remark 4.5] and the references therein. However, in all
+mentioned results except those in [23, 12], the avoided sets are thin or compact.
+Without insisting on approximation, the theorem of Remmert, Bishop, and Narasimhan is not
+optimal with respect to the dimension of the target space. Indeed, it was shown by Eliashberg and
+Gromov [17] in 1992, with an improvement for odd dimensional Stein manifolds by Sch¨urmann
+[38] in 1997, that a Stein manifold X of dimension m ≥ 2 embeds properly holomorphically
+in Cn with n =
+�3m
+2
+�
++ 1, and for m ≥ 1 it immerses properly holomorphically in Cn with
+n =
+� 3m+1
+2
+�
+. (See also [20, Sect. 9.3].) However, the construction method in these papers,
+which relies on the Oka principle for sections of certain stratified holomorphic fibre bundles,
+does not give the density statement, and we do not know whether Theorem 1.3 holds for maps to
+these lower dimensional spaces. It is also an open problem whether every open Riemann surface
+embeds properly holomorphically in C2; see [20, Secs. 9.10-9.11] and the survey [21].
+Theorem 1.3 is proved in Section 4. The proof relies on two main ingredients. One is the
+result of Wold and the second named author [25, Theorem 1.8] which shows in particular that the
+complement Ω = Cn \E of a closed convex set E having BCEH is an Oka domain. The second
+main technique comes from the work of Dor [10, 11] (1993-95), following earlier papers by
+Stensønes [39] (1989) and Hakim [30] (1990). Dor constructed proper holomorphic immersions
+and embeddings of any smoothly bounded, relatively compact, strongly pseudoconvex domain
+D in a Stein manifold X into any pseudoconvex domain Ω in Cn under the dimension conditions
+
+4
+B. Drinovec Drnovˇsek and F. Forstneriˇc
+in Theorem 1.3.
+Previously, Hakim [30] constructed proper holomorphic maps to balls in
+codimension one. The main idea is to inductively lift the image of bD under a holomorphic
+map f : ¯D → Ω to a given higher superlevel set of a strongly plurisubharmonic exhaustion
+function ρ : Ω → R+ in a controlled way, taking care not to decrease the value of ρ ◦ f very
+much anywhere on D during the process. When D is a finite bordered Riemann surface, this can
+be achieved by using approximate solutions of a Riemann-Hilbert boundary value problem (see
+[14]). In higher dimensions the proof is more subtle and uses carefully controlled holomorphic
+peak functions on ¯D to push a given map f : ¯D → Ω locally at a point z ∈ f(bD) in the direction
+of the zero set Sz of the holomorphic (quadratic) Levi polynomial of the exhaustion function
+ρ : Ω → R. At a noncritical point z ∈ Ω of ρ, Sz is a smooth local complex hypersurface and
+the restricted function ρ|Sz increases quadratically as we move away from z. If ρ is a strictly
+convex function, this can be achieved by pushing the image of f(bD) in the direction of suitably
+chosen affine complex hyperplanes. Dor’s construction was extended by the authors to maps
+from strongly pseudoconvex domains in Stein manifolds to an arbitrary Stein manifold Ω, and
+also to q-convex complex manifolds for suitable values of q ∈ N; see the papers [14, 15] from
+2007 and 2010, respectively. In those papers we introduced the technique of gluing holomorphic
+sprays of manifold-valued maps on a strongly pseudoconvex Cartan pair with control up to the
+boundary (a nonlinear version of the Cousin-I problem) and a systematic approach for avoiding
+critical points of a q-convex Morse exhaustion function on Ω.
+Earlier constructions of this type, using simpler holomorphic peak functions and higher
+codimension, were given in 1985 by Løw [34] and Forstneriˇc [19] who showed that every
+relatively compact strongly pseudoconvex domain D in a Stein manifold embeds properly
+holomorphically in a high dimensional Euclidean ball. A related result with interpolation on
+a suitable subset of the boundary of D is due to Globevnik [27] (1987). This peak function
+technique was inspired by the construction of inner functions on the ball of Cn by Løw [33] in
+1982, based on the work of Hakim and Sibony [31].
+We apply this technique to push the boundary f0(bD) ⊂ Ω = Cn \ E of a holomorphic map
+f0 : ¯D → Cn in Theorem 1.3 out of a certain compact convex cap C attached to E along a part
+of bC contained in bE and such that the set E1 = E ∪ C is convex and has bounded convex
+exhaustion hulls. At the same time, we ensure that the new map g : ¯D → Cn still sends D \ K
+to Ω. For a precise result, see Proposition 2.1. In the next step, we use that Ω1 = Cn \ E1 is
+an Oka domain (see Corollary 3.6). Since g(bD) ⊂ Ω1, we can apply the Oka principle (see
+Theorem 4.1) to approximate g by a holomorphic map f1 : X → Cn with f1(X \ D) ⊂ Ω1.
+Continuing inductively, we obtain a sequence of holomorphic maps X → Cn converging to a
+proper map satisfying Theorem 1.3. The details are given in Section 4.
+The analogues of Theorem 1.3 and Corollary 1.4 also hold for minimal surfaces in Rn.
+Theorem 1.6. Let n ≥ 3, and let φ : Rn−1 → R+ be a convex function such that the set {φ = 0}
+is nonempty and compact. Given an open Riemann surface X, a compact O(X)-convex set K
+in X, and a conformal minimal immersion f0 : U → Rn from a neighbourhood of K with
+f0(bK) ⊂ Ω = {y < φ(x)}, we can approximate f0 uniformly on K by proper conformal
+minimal immersions f : X → Rn (embeddings if n ≥ 5) satisfying f(X \ ˚
+K) ⊂ Ω.
+If in addition φ is of class C 1, strictly convex at infinity, and the epigraph Eφ = {y ≥ φ(x)}
+has BCEH then one can add to this statement the interpolation of the map on discrete sets, in
+analogy to Theorem 4.2.
+Theorem 1.6 is obtained by following the proof of Theorem 1.3, replacing Proposition 2.1
+by the analogous result obtained by the Riemann–Hilbert deformation method for conformal
+
+Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
+5
+minimal surfaces (see [2] or [3, Chapter 6]). Furthermore, it has recently been shown by the
+authors [16, Corollary 1.5] that the complement of a closed convex set E ⊂ Rn (n ≥ 3) is
+flexible for minimal surfaces (an analogue of the Oka property in complex geometry) if and only
+if E is not a halfspace or a slab; clearly this includes all sets with BCEH.
+Another method for constructing proper minimal surfaces, which yields the same result in
+some examples not covered by Theorem 1.6, was developed by Alarc´on and L´opez [4] in 2012.
+They showed that Theorem 1.6 holds for any wedge domain Γ × R ⊂ R3, where Γ ⊂ R2 is an
+open cone with angle > π; see [4, Theorem 5.6]. The complement of this set is convex but it
+fails to satisfy the hypotheses of Theorem 1.6 due to the presence of lines in the boundary. An
+important difference between these two fields, which affects the possible construction methods,
+is that every open Riemann surface admits a proper harmonic map to the plane R2 (see [4,
+Theorem I]), while only few such surfaces admit proper holomorphic maps to C.
+The analogue of Problem 1.5 for minimal surfaces asks whether there is a domain in R3 of
+the form {x3 < φ(x1, x2)}, where φ : R2 → R+ is a function with sublinear growth, which
+contains minimal surfaces of hyperbolic type that are proper in R3, or just a proper hyperbolic
+end of a minimal surface. In particular, it would be interesting to know whether the domain
+below the upper half of a vertical catenoid has this property. On the other hand, the strong
+halfspace theorem of Hoffman and Meeks [32] says that the only proper minimal surfaces in R3
+contained in a halfspace are planes.
+2. Pushing a strongly pseudoconvex boundary out of a strictly convex cap
+Let O be a convex domain in Cn for some n > 1. Recall that a continuous function ρ : O → R
+is said to be strictly convex if for any pair of points a, b ∈ O we have that
+ρ(ta + (1 − t)b) < tρ(a) + (1 − t)ρ(b) for all 0 < t < 1.
+Assume now that ρt : O → R (t ∈ [0, 1]) is a continuous family of C 1 functions satisfying
+the following conditions:
+(a) For every t ∈ [0, 1] the function ρt is strictly convex. Note that dρt ̸= 0 on Mt := {ρt = 0}.
+(b) If 0 ≤ s < t ≤ 1 then ρt ≤ 0 on Ms.
+(c) There is an open relatively compact subset ω0 of M0 such that for every pair of numbers
+0 ≤ s < t ≤ 1 we have that Mt ∩ M0 = Mt ∩ Ms = M0 \ ω0.
+This means that the hypersurfaces Mt coincide on the subset M0 \ ω0, and as t ∈ [0, 1]
+increases the domains ωt = Mt \ M0 ⊂ Mt are pairwise disjoint and move into the convex
+direction. Each compact set of the form
+(2.1)
+Ct =
+�
+s∈[0,t]
+ωs for t ∈ [0, 1]
+is called a strictly convex cap with the base ω0. Note that bCt = ω0 ∪ ωt, Ct is strictly convex
+along ωt, strictly concave along ω0, and it has corners along ω0 ∩ ωt. As t ∈ [0, 1] increases to
+1, the caps Ct monotonically increase to C1 and they share the same base ω0. Likewise, for any
+0 ≤ s < t ≤ 1 the set Cs,t = �
+u∈[s,t] ωu is a strictly convex cap with the base ωs. The sets
+(2.2)
+Et = {z ∈ O : ρt(z) ≤ 0} for t ∈ [0, 1]
+are strictly convex along bEt = {ρt = 0}, they form a continuously increasing family in t, and
+Et = E0 ∪ Ct for every t ∈ [0, 1].
+
+6
+B. Drinovec Drnovˇsek and F. Forstneriˇc
+Under these assumptions, we have the following result.
+Proposition 2.1. Let D be a smoothly bounded, relatively compact, strongly pseudoconvex
+domain in a Stein manifold X with dim X < n. Let the sets Et ⊂ O ⊂ Cn (t ∈ [0, 1]) be
+given by (2.2), and let f0 : ¯D → O be a map of class A ( ¯D) such that f0(bD) ∩ E0 = ∅. Given
+a compact set K ⊂ D such that f0(D \ K) ∩ E0 = ∅ and a number ǫ > 0, there is a map
+f : ¯D → O of class A ( ¯D) satisfying the following conditions:
+(i) f(bD) ∩ E1 = ∅,
+(ii) f(D \ K) ∩ E0 = ∅, and
+(iii) maxx∈K |f(x) − f0(x)| < ǫ.
+Recall that a map f : ¯D → O is said to be of class A ( ¯D) if it is continuous on ¯D and
+holomorphic on D. In our application of Proposition 2.1 in the proof of Theorem 1.3, the set O
+will be a ball (or the entire Euclidean space) and the hypersurfaces Mt = {ρt = 0} = bEt will
+be convex graphs over the coordinate hyperplane Cn−1 × R ⊂ Cn.
+In the proof of Proposition 2.1 we shall need the following lemma.
+Lemma 2.2. Assume that O is a convex open subset of Cn for n > 1, L is a compact subset of
+O, and ρ : O → R is a C 1 smooth strictly convex function. Then there is a number δ > 0 with
+the following property. If D is a smoothly bounded strongly pseudoconvex domain in a Stein
+manifold X of dimension dim X = m < n, K is a compact subset of D, and f : ¯D → O is a
+map of class A ( ¯D) such that
+(2.3)
+ρ(f(z)) > −δ for all z ∈ bD
+and
+ρ(f(z)) > 0 if z ∈ bD and f(z) /∈ L,
+then given η > 0 there is a map g : ¯D → O of class A ( ¯D) satisfying the following conditions:
+(i) ρ(g(z)) > 0 for z ∈ bD,
+(ii) ρ(g(z)) > δ for those z ∈ bD for which g(z) ∈ L,
+(iii) ρ(g(z)) > ρ(f(z)) − η for z ∈ D \ K, and
+(iv) |f(z) − g(z)| < η for z ∈ K.
+For m = 1, i.e., when D is a finite bordered Riemann surface, this is a simplified version of
+[14, Lemmas 6.2 and 6.3], which is proved by using approximate solutions of a Riemann–Hilbert
+boundary value problem. This method was employed in several earlier papers mentioned in [14].
+When ρ is strictly convex, C 1 smoothness suffices since in the proof we may take a continuous
+family of tangential linear discs to the sublevel set of ρ.
+For m ≥ 2, Lemma 2.2 is a simplified and slightly modified version of [15, Lemma 5.3].
+Besides the fact that we are considering single maps ¯D → O instead of sprays of maps, the only
+difference is that the assumption in [15, Lemma 5.3] that the set {ρ = 0} is compact is replaced
+by the assumption (2.3) saying that ρ(f(z)) for z ∈ bD may be negative only if f(z) lies in the
+compact set L ⊂ O. This hypothesis ensures that the lifting for a relatively big amount (the role
+of the constant δ) only needs to be made on a compact subset of O, while elsewhere it suffices to
+pay attention not to decrease ρ ◦ f by more than a given amount and to approximate sufficiently
+closely on K (the role of the constant η). The proof requires only a minor adaptation of [15,
+proof of Lemma 5.3], using its local version [15, Lemma 5.2] in a finite induction with respect
+to a covering of bD by small open sets on which there are good systems of local holomorphic
+peak functions. In fact, Lemma 2.2 corresponds to a simplified version of [15, Sublemma 5.4],
+which explains how to lift the image of bD with respect to ρ for a sufficiently large amount at
+those points in bD which the map f sends to a certain coordinate chart Ui in the target manifold.
+
+Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
+7
+In our situation, the role of Ui is played by an open relatively compact neighbourhood of the set
+L ∩ {ρ = 0} in O, and there is no need to use the rest of the proof of [15, Lemma 5.3].
+Proof of Proposition 2.1. For t ∈ [0, 1] let δt > 0 be a number for which the conclusion of
+Lemma 2.2 holds for the function ρt and the compact set L = C1 (see (2.1)). The open sets
+Ut = {z ∈ O : −δt < ρt(z) < δt} for t ∈ [0, 1]
+form an open covering of C1, so there exists a finite subcovering {Utj} for 0 ≤ t1 < t2 < . . . <
+tk ≤ 1. Applying Lemma 2.2 we inductively find maps f1, . . . , fk ∈ A ( ¯D) such that for every
+j = 1, . . . , k we have that
+(a) fj(bD) ∩ Etj = ∅ (where Et is given by (2.2)),
+(b) fj(D \ K) ∩ E0 = ∅, and
+(c) |fj − fj−1| < ǫ/k on K.
+Note that conditions (a) and (b) hold for f0 and (c) is void. Assume inductively that for some
+j ∈ {1, . . . , k} we have maps f0, . . . , fj−1 satisfying these conditions. Applying Lemma 2.2
+with f = fj−1 and taking fj = g, condition (a) follows from part (i) in Lemma 2.2, (b) follows
+from (ii) provided that the number η > 0 in Lemma 2.2 is chosen small enough, and (c) follows
+from (iii) in Lemma 2.2 provided that η ≤ ǫ/k . This gives the map fj satisfying conditions
+(a)–(c) and the induction may continue. The map f = fk then satisfies the proposition.
+□
+Remark 2.3. Proposition 2.1 also holds, with the same proof, if ρt (t ∈ [0, 1]) are strongly
+plurisubharmonic functions of class C 2 satisfying dρt ̸= 0 on Mt = {ρt = 0}. Indeed, the
+results from [15], which are used in the proof, pertain to this case. In the present paper we shall
+only use the convex case under C 1 smoothness, which comes naturally in the construction.
+3. Closed convex sets with BCEH
+In the context of convex analysis, closed unbounded convex sets that share several important
+properties with compact convex sets were studied by Gale and Klee [26] in 1959.
+They
+introduced the class of continuous sets, and we show that this class coincides with the class
+of sets having BCEH, introduced in Definition 1.2; see Proposition 3.3. We then develop further
+properties of these sets which are relevant to the proof of our main theorems.
+By a ray in Rn, we shall mean a closed affine halfline. Let E be a closed convex subset of
+Rn. A boundary ray of E is a ray contained in the boundary of E. An asymptote of E is a ray
+L ⊂ Rn \ E such that dist(L, E) = inf{|x − y| : x ∈ L, y ∈ E} = 0. The function
+σ : {u ∈ Rn : |u| = 1} → R ∪ {+∞},
+σ(u) = sup{x · u : x ∈ E}
+is called the the support function of E. (Here, x · u denotes the Euclidean inner product.) A
+closed convex set E is said to be continuous in the sense of Gale and Klee [26] if the support
+function of E is continuous. Note that every compact convex set is continuous.
+The following result is a part of [26, Theorem 1.3] due to Gale and Klee; we only list those
+conditions that will be used. The last item (iv) uses also [26, Theorem 1.5].
+Theorem 3.1. For a closed convex subset E in Rn the following conditions are equivalent:
+(i) E is continuous.
+(ii) E has no boundary ray nor asymptote.
+(iii) For each point p ∈ Rn the convex hull Conv(E ∪ {p}) is closed.
+(iv) For every compact convex set K ⊂ Rn the set Conv(E ∪ K) is closed.
+
+8
+B. Drinovec Drnovˇsek and F. Forstneriˇc
+Condition (iii) implies that the closed convex hull Conv(E ∪ {p}) is the union of the line
+segments connecting p to the points in E. It also shows that an unbounded continuous closed
+convex subset E of Rn is not contained in any affine hyperplane.
+Let us record the following observation which will be used in the sequel.
+Lemma 3.2. Let E ⊂ Rn be a closed convex set, p ∈ Rn\E, and L ⊂ Rn be an affine subspace
+containing p. Then, Conv(E ∪ {p}) ∩ L = Conv((E ∩ L) ∪ {p}).
+Proof. Set E′ = E ∩ L. It is obvious that Conv(E′ ∪ {p}) ⊂ Conv(E ∪ {p}) ∩ L. Conversely,
+since E is convex, every point q ∈ Conv(E ∪ {p}) belongs to a line segment from p to a point
+q′ ∈ E. If in addition q ∈ L and q ̸= p then q′ ∈ E′, and hence q ∈ Conv(E′ ∪ {p}).
+□
+Proposition 3.3. A closed convex set E ⊂ Rn has BCEH if and only if it is continuous in the
+sense of Gale and Klee [26].
+Proof. Since all closed bounded convex sets have BCEH and are continuous, it suffices to
+consider the case when the set E is unbounded.
+If E is not continuous then by Theorem 3.1 it has a boundary ray or an asymptote. Denote it
+by L, and let ℓ be the affine line containing L. Pick any affine 2-plane H ⊂ Rn containing ℓ.
+There is a point p ∈ H \(ℓ∪E). By considering rays from p to points q ∈ E approaching L and
+going to infinity (if L is a boundary ray, we can choose points q ∈ L), we see that the closure
+of the set h(E, p) = Conv(E ∪ {p}) \ E contains the parallel translate L′ ⊂ H+ of L passing
+through p, so h(E, p) is unbounded and hence E does not have BCEH.
+Assume now that E is a continuous and let us prove that it has BCEH. We need to show that
+for any closed ball B ⊂ Rn the set h(E, B) = Conv(E ∪ B) \ E is bounded. Assume to the
+contrary that there is a sequence xm ∈ h(E, B) with |xm| → ∞ as m → ∞. Since the sets E
+and B are convex, we have that
+xm = tmbm + (1 − tm)em for tm ∈ [0, 1], bm ∈ B, em ∈ E, and m ∈ N.
+Note that (1 − tm)|em| → ∞ as m → ∞. By compactness of the respective sets we may
+assume, passing to a subsequence, that em ̸= 0 for all m and the sequences tm, bm, and
+1
+|em|em
+are convergent. Denote their respective limits by t, b, and f. We have that
+xm = tmbm + (1 − tm)em = bm + (1 − tm)|em|
+� em
+|em| − bm
+|em|
+�
+= bm + (1 − tm)|em|fm
+where fm =
+� em
+|em|− bm
+|em|
+�
+. Note that limm→∞ fm = f. Pick a number α ≥ 0 and set p = b+αf.
+If m is large enough then (1−tm)|em| > α, so the point ym = bm+αfm lies on the line segment
+connecting bm and xm. Since xm ∈ Conv(E ∪ {bm}), it follows that ym ∈ Conv(E ∪ {bm}).
+Note that the sequence ym converges to p. Since E is continuous, Conv(E ∪ {b}) is closed by
+Theorem 3.1, so p = limm→∞ ym ∈ Conv(E ∪ {b}). Since this holds for every α ≥ 0, the ray
+L = {b + αf : α ∈ [0, ∞)} lies in Conv(E ∪ {b}). By Lemma 3.2 there is α0 ∈ [0, ∞) such
+that the ray L′ = {b + αf : α ≥ α0} lies in E. Since E is continuous, L is not a boundary ray
+of E by Theorem 3.1, thus L contains a point q = b+α1f ∈ E \bE for some α1 ≥ α0. Choose
+a neighbourhood Uq ⊂ E of q. For any large enough m we then have pm := bm + α1fm ∈ Uq.
+Let Lm = {bm + αfm : α ≥ 0}. Note that Lm ∩ Conv(E ∪ {bm}) = Conv((Lm ∩ E) ∪ {bm})
+by Lemma 3.2. However, for m large enough the point xm ∈ Lm lies on the opposite side of pm
+than bm, so xm belongs to Lm ∩ Conv(E ∪ {bm}) but not to Conv((Lm ∩ E) ∪ {bm}). This
+contradiction proves that E has BCEH.
+□
+
+Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
+9
+Given a function φ : Rn−1 → R, the epigraph of φ is the set
+(3.1)
+E = Eφ = {(x, y) ∈ Rn−1 × R : y ≥ φ(x)}.
+Note that a function is convex if and only if its epigraph is convex.
+Proposition 3.4. If E ⊊ Rn is a closed unbounded convex set with BCEH then
+(i) E does not contain any affine real line, and
+(ii) for every affine line ℓ intersecting E in a ray and any hyperplane H transverse to ℓ, E is
+the epigraph of a convex function on H. In particular, there are affine coordinates (x, y)
+on Rn in which E is of the form (3.1) for a convex function φ : Rn−1 → R+ satisfying
+(3.2)
+lim inf
+|x|→+∞
+φ(x)
+|x|
+> 0.
+The condition (3.2) says that φ grows at least linearly at infinity. We show in Example 3.10
+that linear growth is possible.
+Proof. (i) Assume that ℓ ⊂ E is an affine line and let us prove that E does not have BCEH.
+Since E is a proper subset of Rn, there is a parallel translate ℓ′ of ℓ which is not contained in E,
+and hence ℓ′ \E contains a ray L. Let p be the endpoint of L, and let p′ ∈ L be an arbitrary other
+point. Since E ∩L = ∅, there is a ball B around p′ such that Conv(B ∪{p})∩E = ∅. Clearly,
+there is a point q ∈ B such that the ray Lq with the endpoint p and containing q intersects the
+line ℓ, so the line segment from p to q belongs to Conv(E ∪ {p}) \ E = h(E, p). By moving
+p′ ∈ L to infinity we see that h(E, p) is unbounded, so E does not have BCEH.
+(ii) Since E is unbounded, it contains a ray L. Denote by ℓ the affine line containing L. Let
+ℓ′ be any parallel translate of ℓ. Since E contains no affine lines by part (i), there is a point
+p ∈ ℓ′ \E. The closed convex hull of the union of L and p contains the parallel translate L′ ⊂ ℓ′
+of L passing through p. Since E has BCEH, we conclude that L′ ⊂ Conv(E ∪ {p}) and L′ \ E
+is bounded. Since E ∩ L′ is convex, L′ ∩ E is a closed ray with the endpoint on bE. This shows
+that E is a union of closed rays contained in parallel translates of the line ℓ, so it is an epigraph
+of a convex function defined on any hyperplane H ⊂ Rn transverse to ℓ. Choosing H such that
+H ∩ E = ∅ there are affine coordinates (x, y) on Rn with H = {y = 0} and ℓ = {x = 0}. In
+these coordinates, E is of the form (3.1) for a positive convex function φ.
+Finally, if condition (3.2) fails then there is a sequence (xk, yk) ∈ E with |xk| → +∞ and
+yk/|xk| → 0 as k → ∞. The union of the line segments Lk connecting p = (0, −1) ∈ Rn−1×R
+to (xk, yk), intersected with the lower halfspace y ≤ 0, is then an unbounded subset of
+h(E, p) = Conv(E ∪ {p}) \ E, contradicting the assumption that E has BCEH.
+□
+Remark 3.5. The growth condition (3.2) for an epigraph can always be achieved in suitable
+linear coordinates (even without the BCEH property) if there is a supporting hyperplane H ⊂ Rn
+for E such that the set E∩H is nonempty and compact. Indeed, we may then choose coordinates
+(x, y) on Rn such that H = {y = 0}, E ⊂ {y ≥ 0}, and 0 ∈ E. If the condition (3.2) fails,
+there is a sequence (xk, yk) ∈ E with |xk| → +∞ and yk/|xk| → 0 as k → ∞. After passing
+to a subsequence, a ray in E ∩ H lies in the closure of the union of the line segments Lk ⊂ E
+connecting the origin to (xk, yk), contradicting the assumption that the latter set is compact.
+Corollary 3.6. If E is a closed convex set in Cn (n > 1) having BCEH then Cn \ E is Oka.
+Proof. By Proposition 3.4 the set E does not contain any affine real line, and hence Cn \ E is
+Oka by [25, Theorem 1.8].
+□
+
+10
+B. Drinovec Drnovˇsek and F. Forstneriˇc
+The following lemma shows that the BCEH condition is stable under uniform approximation.
+Lemma 3.7. Assume that φ : Rn−1 → R is a convex function whose epigraph Eφ (3.1) has
+BCEH. Then for any ǫ > 0 and convex function ψ : Rn−1 → R satisfying |φ − ψ| < ǫ the
+epigraph Eψ also has BCEH.
+Proof. If Eψ fails to have BCEH then by Theorem 3.1 and Proposition 3.3 it has a boundary
+ray or an asymptote, L. Since dist(L, Eψ) = 0 and Eψ is convex, dist(x, Eψ) converges
+to zero as x ∈ L goes to infinity.
+Thus, by making L shorter if necessary, we have that
+L ⊂ Eφ−2ǫ \ Eφ+2ǫ. Hence, L lies out of Eφ+2ǫ but the vertical translation of L for 4ǫ pushes it
+in Eφ+2ǫ. Since Eφ+2ǫ, being a translate of Eφ, has BCEH, this contradicts Proposition 3.4 (ii).
+The contradiction shows that Eψ has BCEH as claimed.
+□
+We now give a differential characterization of the BCEH property of an epigraph (3.1).
+Proposition 3.8. If φ : Rn−1 → R is a convex function of class C 1 satisfying condition (3.2),
+then the epigraph E = {(x, y) ∈ Rn : y ≥ φ(x)} has BCEH if and only if
+(3.3)
+lim
+|x|→∞ |x|
+�
+1 −
+φ(x)
+x · ∇φ(x)
+�
+= +∞.
+Proof. We first consider the case n = 2. Then, x is a single variable and (3.3) is equivalent to
+(3.4)
+lim
+x→+∞
+�
+x − φ(x)
+φ′(x)
+�
+= +∞
+and
+lim
+x→−∞
+�
+x − φ(x)
+φ′(x)
+�
+= −∞.
+For every x ∈ R such that φ′(x) ̸= 0 the number
+(3.5)
+ξ(x) = x − φ(x)
+φ′(x)
+is the first coordinate of the intersection of the tangent line to the graph of φ at the point (x, φ(x))
+with the first coordinate axis y = 0. By (3.2) and convexity of φ we have that |φ′(x)| is bounded
+away from zero for all sufficiently big |x|. This shows that conditions (3.4) are invariant under
+translations, so we may assume that φ ≥ 0 and φ(0) = 0. It is easily seen that the function ξ is
+increasing. If φ is of class C 2, we have that ξ′(x) = φ(x)φ′′(x)/φ′(x)2 ≥ 0.
+Assume now that conditions (3.4) hold.
+Pick a pair of sequences aj < bj in R with
+limj→∞ aj = −∞ and limj→∞ bj = +∞. The intervals Ij = [ξ(aj), ξ(bj)] then increase
+to R as j → ∞. We identify Ij with Ij × {0} ⊂ R2. Since φ is convex, its epigraph lies above
+the tangent line at any point. It follows that the set h(E, Ij) (see (1.1)) is the bounded region in
+R×R+ whose boundary consists of Ij, the two line segments Lj and L′
+j connecting the endpoints
+(ξ(aj), 0) and (ξ(bj), 0) of Ij to the respective points Aj = (aj, φ(aj)) and Bj = (bj, φ(bj))
+on bE, and the graph of φ over [aj, bj]. The supporting lines of Lj and L′
+j intersect at a point
+Cj in the lower halfspace y < 0, and we obtain a closed triangle ∆j with the endpoints Aj, Bj,
+and Cj. Note that ∆j ∩ (R × {0}) = Ij. Since φ grows at least linearly (see (3.2)), the triangles
+∆j ⊂ R2 exhaust R2 as j → ∞, and the set h(E, ∆j) (1.1) is bounded for every j. Hence,
+E has BCEH. This argument furthermore shows that for any point p = (0, −c) /∈ E there is
+a unique pair of tangent lines to bE passing through p such that, denoting by q1, q2 ∈ bE the
+respective points where these lines intersect bE, the convex hull Conv(E ∪ {p}) is the union of
+E and the triangle with vertices p, q1, q2.
+Conversely, if (3.3) fails then it is easily seen that E has a boundary ray or an asymptote, so
+it does not have BCEH. We leave the details to the reader.
+
+Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
+11
+The case with n ≥ 3 now follows easily. Pick a unit vector v ∈ Rn−1, |v| = 1, and let Lv
+denote the 2-plane in Rn passing through the origin and spanned by v and en = (0, . . . , 0, 1).
+Then, Ev := E ∩ Lv = {(t, y) ∈ R2 : y ≥ φ(tv)} and the first condition in (3.4) reads
+(3.6)
+lim
+t→+∞
+�
+t −
+φ(tv)
+�n−1
+j=1 vj
+∂φ
+∂xj (tv)
+�
+= +∞.
+Writing x = tv with t ≥ 0 and v = x/|x|, this is clearly equivalent to (3.3). As before, let
+p = (0, . . . , 0, −c) /∈ E. If (3.3) holds then Conv(Ev ∪ {p}) ⊂ Lv is obtained by adding to
+Ev the triangle in Lv obtained by the two tangent lines to bEv passing through p as described in
+the case n = 2. The sizes of these triangles are uniformly bounded with respect to the direction
+vector |v| = 1, and condition (3.2) implies that these triangles increase to Lv as c → +∞,
+uniformly with respect to v. Since �
+|v|=1 Lv = Rn, Lemma 3.2 shows that
+Conv(E ∪ {p}) =
+�
+|v|=1
+Conv(Ev ∪ {p}),
+and hence E has BCEH. The converse is seen as in the special case n = 2.
+□
+Corollary 3.9. If φ : Rn−1 → R+ is a convex function of class C 1 such that
+lim
+|x|→+∞
+x · ∇φ(x)
+|x|
+= +∞,
+then the epigraph E = {(x, y) ∈ Rn : y ≥ φ(x)} has BCEH.
+Proof. By restricting to planes as in the above proof, it suffices to consider the case n = 2. We
+may assume that φ ≥ 0 and φ(0) = 0. Since φ is convex, g(x) = φ′(x) is an increasing function
+and the above condition reads limx→±∞ g(x) = ±∞. For any x0 > 0 and x ≥ x0 we have that
+ξ(x) := x −
+1
+g(x)
+� x
+0
+g(t)dt =
+� x
+0
+�
+1 − g(t)
+g(x)
+�
+dt ≥
+� x0
+0
+�
+1 − g(t)
+g(x)
+�
+dt.
+Letting x → +∞ we have that g(t)
+g(x) → 0 uniformly on t ∈ [0, x0], and hence the last integral
+converges to x0. Letting x0 → ∞ we see that limx→+∞ ξ(x) = +∞. The analogous argument
+applies when x → −∞. Hence, conditions (3.3) hold and therefore E has BCEH.
+□
+Example 3.10. There exist convex epigraphs (3.1) having BCEH where the function φ grows
+linearly, although it cannot be too close to linear near infinity in the absence of boundary rays
+and asymptotes. We give such an example in R2. Let g : R → (−1, 1) be an odd, continuous,
+increasing function with limx→+∞ g(x) = 1 and
+� ∞
+0 (1−g(x))dx = +∞. (An explicit example
+is g(x) = 2
+πArctan x.) Its integral φ(x) =
+� x
+0 g(t)dt for x ∈ R then clearly satisfies φ(x) ≥ 0,
+φ′(x) = g(x) ∈ (−1, +1) (hence φ grows linearly), and φ is convex. We now show that (3.3)
+holds. Let x > 0 be large enough so that g(x) > 0. We have that
+ξ(x) = x −
+1
+g(x)
+� x
+0
+g(t)dt =
+� x
+0
+�
+1 − g(t)
+g(x)
+�
+dt.
+Fix x0 > 0 and let x ≥ x0. Then, ξ(x) ≥
+� x0
+0 (1 − g(t)/g(x))dt. Since limx→+∞ g(x) = 1
+and ξ is increasing for large enough |x|, it follows that limx→+∞ ξ(x) ≥
+� x0
+0 (1 − g(t))dt.
+Sending x0 → +∞ gives limx→+∞ ξ(x) ≥
+� ∞
+0 (1 − g(t))dt = +∞. Similarly we see that
+limx→−∞ ξ(x) = −∞. Thus, (3.3) holds, and hence the epigraph of φ has BCEH.
+By using the idea in the above example we now prove the following approximation result,
+which extends Theorem 1.3 to a much bigger class of convex epigraphs (see Corollary 1.4).
+
+12
+B. Drinovec Drnovˇsek and F. Forstneriˇc
+Proposition 3.11. Assume that φ : Rn−1 → R+ is a convex function such that the set {φ = 0}
+is nonempty and compact. Given numbers ǫ > 0 (small) and R > 0 (big) there is a smooth
+convex function ψ : Rn−1 → R such that ψ < φ on Rn−1, φ(x) − ψ(x) < ǫ for all |x| ≤ R,
+and the epigraph Eψ = {y ≥ ψ} has BCEH.
+Proof. By Remark 3.5 the function φ grows at least linearly near infinity (see (3.2)). Set
+(3.7)
+A = lim inf
+|x|→∞
+φ(x)
+|x| > 0.
+Since the set φ = 0 does not contain any affine line, Azagra’s result [6, Theorem 1 and
+Proposition 1] implies that for every ǫ > 0 there is a smooth strictly convex function ψ on
+Rn−1 satisfying φ − ǫ < ψ < φ. Replacing φ by ψ − minx ψ(x) ≥ 0 we may therefore assume
+that φ is smooth. By increasing the number R > 0 if necessary, we may assume that
+(3.8)
+φ(x)
+|x|
+≥ A
+2
+for all |x| ≥ R.
+Pick a number r ∈ (0, 1) close to 1 such that
+(3.9)
+(1 − r) sup
+|x|≤R
+φ(x) < ǫ.
+Choose a smooth increasing function h : R → R+ such that
+h(t) = 0 for t ≤ R,
+lim
+t→+∞ h(t) = 1,
+and
+� ∞
+0
+(1 − h(t))dt = +∞.
+(We can take a smoothing of the Arctan function used in Example 3.10.) Set
+H(x) =
+� |x|
+0
+h(s)ds
+for x ∈ Rn−1.
+Clearly, H ≥ 0 is a radially symmetric smooth convex function that vanishes on |x| ≤ R and
+satisfies H(x) ≤ |x| for all x ∈ Rn−1. With A and r as in (3.7) and (3.9) we set
+δ = A(1 − r)
+2
+.
+We claim that the function
+ψ(x) = rφ(x) + δH(x)
+for x ∈ Rn−1
+satisfies the conditions in the theorem. Clearly, ψ ≥ rφ is a smooth convex function. For
+|x| ≤ R we have H(x) = 0, so ψ(x) = rφ(x) ≤ φ(x) and φ(x) − ψ(x) = (1 − r)φ(x) < ǫ by
+(3.9). If |x| > R then φ(x)/|x| ≥ A/2 by (3.8) and H(x) < |x|, which implies
+ψ(x)
+|x|
+≤ rφ(x)
+|x| + δ ≤ φ(x)
+|x| .
+Indeed, we have that φ(x)
+|x| − r φ(x)
+|x| = (1 − r)φ(x)
+|x| ≥ A(1−r)
+2
+= δ. Hence, ψ ≤ φ on Rn−1.
+It remains to show that the epigraph Eψ satisfies BCEH. We shall verify (3.3), which is
+equivalent to (3.6) with uniform convergence with respect to the vector v = x/|x|. Write
+gv(t) = r∂φ(tv)
+∂t
+,
+k(t) = δh(t),
+˜gv(t) = ∂ψ(tv)
+∂t
+= gv(t) + k(t).
+
+Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
+13
+The quantity in (3.6) associated to the function ψ is given by
+ξv(t)
+=
+t − ψ(tv)
+˜gv(t) =
+� t
+0
+�
+1 − gv(s) + k(s)
+gv(t) + k(t)
+�
+ds
+=
+� t
+0
+gv(t) − gv(s)
+gv(t) + k(t) ds +
+� t
+0
+k(t) − k(s)
+gv(t) + k(t)ds
+≥
+� t
+0
+gv(t) − gv(s)
+gv(t) + δ
+ds +
+� t
+0
+k(t) − k(s)
+gv(t) + δ ds,
+where the last inequality holds since the functions gv and k are nonnegative and increasing and
+k < δ. Pick c > 0. We will show that for large enough t > 0 and any unit vector v ∈ Rn−1 the
+above expression is bigger than or equal to c. Choose positive numbers t0, a, t1 as follows:
+t0 = 3c,
+a = max{3 max
+|v|=1 gv(t0), 3δ},
+� t1
+0
+(k(t1) − k(s))ds > ac.
+Such t1 exists since limt→+∞
+� t
+0(k(t) − k(s))ds = δ
+� ∞
+0 (1 − h(s))ds = +∞. Since the
+integrands in the bound for ξv(t) are nonnegative, we have for t ≥ max{t0, t1} and |v| = 1 that
+(3.10)
+ξv(t) ≥
+� t0
+0
+gv(t) − gv(s)
+gv(t) + δ
+ds +
+� t1
+0
+k(t) − k(s)
+gv(t) + δ ds.
+Assume that for some such (t, v) we have that gv(t) + δ ≥ a. Since a ≥ 3δ, it follows that
+gv(t) ≥ 2δ and hence
+gv(t)
+gv(t) + δ ≥ 2
+3.
+Furthermore, from a ≥ 3 max|v|=1 gv(t0) we get for 0 ≤ s ≤ t0 that
+gv(s)
+gv(t) + δ ≤ gv(t0)
+a
+≤ 1
+3.
+These two inequalities imply that the first integral in (3.10) is bounded below by t0/3 ≥ c. If
+on the other hand gv(t) + δ < a then the denominator of the second integral in (3.10) is at most
+a, so the integral is ≥ c by the choice of t1. This shows that ξv(t) ≥ c for all |v| = 1 and
+t ≥ max{t0, t1}. Since c was arbitrary, condition (3.3) holds and hence Eψ has BCEH.
+□
+The following observation will be used in the proof of Theorem 1.3.
+Proposition 3.12. Denote by B the open unit ball in Rn. Let Eφ ⊂ Rn be a closed convex
+set of the form (3.1) with C 1 boundary having BCEH, where the function φ : Rn−1 → R is
+bounded from below and strictly convex near infinity. Then there is an r0 > 0 such that for every
+r ≥ r0 the convex hull Conv(Eφ ∪ rB) = {y ≥ ψ(x)} is a closed convex set with BCEH, and
+ψ : Rn−1 → R is a convex function of class C 1 such that ψ ≤ φ and these functions agree near
+infinity. Furthermore, if r ≥ r0 is large enough then the function φt : Rn−1 → R defined by
+(3.11)
+φt(x) = (1 − t)φ(x) + tψ(x),
+x ∈ Rn−1
+is strictly convex for every t ∈ (0, 1), and for any 0 < t0 < t1 < 1 the closure of the set
+{(x, y) ∈ Rn : φt1(x) < y < φt0(x)}
+is a strictly convex cap with the base in the strictly convex hypersurface {y = φt0(x)}.
+
+14
+B. Drinovec Drnovˇsek and F. Forstneriˇc
+Proof. Consider the function on Rn−1 given by
+˜φr(x) =
+�
+min{φ(x), −
+�
+r2 − |x|2},
+|x| < r,
+φ(x),
+|x| ≥ r.
+(Note that ˜φr may be discontinuous at the points of the sphere |x| = r.) The convex hull of its
+epigraph E˜φr equals Conv(E∪rB), which is closed by Theorem 3.1 (iv), and the set h(E, rB) =
+Conv(E ∪ rB) \ E is bounded since E has BCEH. By smoothing ˜φr we get a function ˜ψr of
+class C 1 which agrees with φ near infinity such that Conv(E ˜ψr) = Conv(E ∪ rB). By [28,
+Theorem 3.2] we conclude that Conv(E ∪ rB) has C 1 boundary, so it is the epigraph Eψr of a
+convex function ψr : Rn−1 → R of class C 1 which agrees with φ near infinity.
+Since φ grows at least linearly, there is a function τ(r) defined for r ∈ R+ large enough such
+that ψr(x) = −
+�
+r2 − |x|2 for |x| ≤ τ(r) and τ(r) → +∞ as r → +∞. By choosing r large
+enough, the compact set of points where the function φ fails to be strictly convex is contained
+in the ball |x| < τ(r). Since on this ball we have that ψr(x) = −
+�
+r2 − |x|2 which is strictly
+convex, the convex combinations φt in (3.11) of φ and ψ = ψr are strictly convex on Rn−1 for
+all 0 < t < 1. For such r, the last statement in the proposition is evident. (Note that the strictly
+convex functions ρt(x, y) = exp(ψt(x) − y) − 1 for t ∈ (0, 1) correspond to those used in
+Section 2.)
+□
+4. Proof of Theorem 1.3
+For the definition and the main theorem on Oka manifolds, see [20, Definition 5.4.1 and
+Theorem 5.4.4]. We shall use the following version of the Oka principle; see [22, Theorem 1.3].
+Theorem 4.1. Assume that X is a Stein manifold, K is a compact O(X)-convex set in X, X′
+is a closed complex subvariety of X, Ω is an Oka domain in a complex manifold Y , f : X → Y
+is a continuous map which is holomorphic on a neighbourhood of K, f|X′ : X′ → Y is
+holomorphic, and f(X \ ˚
+K) ⊂ Ω. Then there is a homotopy {ft}t∈[0,1] of continuous maps
+ft : X → Y connecting f = f0 to a holomorphic map f1 : X → Y such that for every t ∈ [0, 1]
+the map ft is holomorphic on a neighbourhood of K, it agrees with f on X′, it approximates f
+uniformly on K and uniformly in t ∈ [0, 1] as closely as desired, and ft(X \ ˚
+K) ⊂ Ω.
+Proof of Theorem 1.3. By Proposition 3.4 there are complex coordinates z = (z′, zn) on Cn
+such that the given set E is an epigraph of the form (1.2). We shall write z = (x, y) where
+x = (z′, ℜzn) ∈ Cn−1 × R ∼= R2n−1 and y = ℑzn ∈ R, so E = Eφ = {y ≥ φ(x)} where
+φ ≥ 0 is a convex function as in Proposition 3.4. Let the set K ⊂ X and the map f0 : K → Cn
+be as in the theorem; in particular, f0(bK) ⊂ Cn \ E. Thus, there are an open neighbourhood
+U ⊂ X of K and ǫ > 0 such that f0 is holomorphic in U and f0(U \ ˚
+K) ⊂ Cn \ Eφ−ǫ. By
+Azagra [6, Theorem 1.8] there is a a real analytic strictly convex function φ0 : R2n−1 → R such
+that φ − ǫ < φ0 < φ. Its epigraph E0 = {(x, y) ∈ Cn : y ≥ φ0(x)} is a closed strictly convex
+set with real analytic boundary which has BCEH by Lemma 3.7, and f0(U \ ˚
+K) ⊂ Cn \ E0.
+Let B denote the open unit ball in Cn centred at 0. Recall the notation h(E, K) in (1.1). Pick
+a number r0 > 0. We can find an increasing sequence rk > 0 diverging to infinity such that
+(4.1)
+h(E0, rkB) ⊂ rk+1B
+for k = 0, 1, 2, . . . .
+
+Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
+15
+Indeed, since E0 has BCEH, the set h(E0, rkB) is bounded for each k, and hence (4.1) holds if
+the number rk+1 is chosen large enough. Set
+Ek+1 = Conv(E0 ∪ rkB) = E0 ∪ h(E0, rkB)
+for k = 0, 1, 2, . . ..
+We clearly have that E0 ⊂ E1 ⊂ · · · ⊂ �∞
+k=0 Ek = Cn. Furthermore, (4.1) shows that for
+j = 0, 1, . . . , k + 1 we have that E0 ⊂ Ej ⊂ E0 ∪ rk+1B and hence
+(4.2)
+Ek+2 = Conv(Ej ∪ rk+1B) for j = 0, 1, . . . , k + 1.
+Proposition 3.12 shows that for each k = 1, 2, . . . we have Ek = {y ≥ φk(x)} where φk a
+convex function of class C 1 which agrees with φ0 near infinity, and Ek has BCEH. Hence,
+Ωk = Cn \ Ek = {(x, y) ∈ Cn : y < φk(x)}
+is an Oka domain for every k = 0, 1, . . . by Corollary 3.6. In view of Ek+2 = Conv(Ek∪rk+1B)
+(see (4.2)), Proposition 3.12 also shows that if rk+1 is chosen large enough then the function
+(4.3)
+ψt = (1 − t)φk + tφk+2 : Cn−1 × R → R
+is strictly convex for every t ∈ (0, 1), and for each 0 < t0 < t1 < 1 the closure of the set
+(4.4)
+C = {(x, y) : ψt1 < y < ψt0}
+is a strictly convex cap as described in Section 2. (Note that the strictly convex functions
+ρt(x, y) = exp(ψt(x) − y) − 1 for t ∈ (0, 1) correspond to those used in Section 2.)
+Choose an exhaustion D0 ⊂ D1 ⊂ · · · ⊂ �∞
+k=0 Dk = X by smoothly bounded, relatively
+compact, strongly pseudoconvex domains with O(X)-convex closures such that K ⊂ D0 ⊂
+¯D0 ⊂ U. For consistency of notation we set D−1 = K. We now construct a sequence of
+holomorphic maps fk : ¯Dk → Cn satisfying the following conditions for k = 0, 1, 2, . . .:
+(a) fk(Dk \ Dk−1) ⊂ Ωk = Cn \ Ek,
+(b) fk+1(Dk \ Dk−1) ⊂ Ωk, and
+(c) fk+1 approximates fk uniformly on ¯Dk−1 as closely as desired.
+For k = 0 the initial map f0 in Theorem 1.3 satisfies condition (a) while conditions (b) and (c)
+are void. Assuming inductively that we found maps f0, . . . , fk satisfying these conditions, the
+construction of the next map fk+1 is made in two steps as follows.
+By compactness of the set fk(bDk) ⊂ Ωk = {y < φk(x)} we can choose t0 ∈ (0, 1) small
+enough such that f(bDk) ⊂ {y < ψt0(x)}, where the function ψt (t ∈ [0, 1]) is given by (4.3).
+By (4.1) we can also choose t1 ∈ (t0, 1) sufficiently close to 1 such that
+Ek+1 ⊂ {(x, y) : y ≥ ψt1(x)}.
+Proposition 2.1 applied to the map fk : ¯Dk → Cn, the set Ek, and the strictly convex cap
+C (4.4) (which corresponds to C1 in Proposition 2.1) gives holomorphic map gk : ¯Dk → Cn
+approximating fk on Dk−1 and satisfying
+(4.5) gk(bDk) ⊂ {(x, y) : y < ψt1(x)} ⊂ Cn \ Ek+1 = Ωk+1 and gk(Dk \ Dk−1) ⊂ Ωk.
+In the second step, we use that Ωk+1 is an Oka domain. Since Ωk+1 is contractible and
+gk(bDk) ⊂ Ωk+1 by (4.5), gk extends from ¯Dk to a continuous map X → Cn sending
+X \ Dk to Ωk+1. Theorem 4.1 applied to gk gives a holomorphic map fk+1 : ¯Dk+1 → Cn
+approximating gk on ¯Dk and satisfying fk+1(Dk+1 \ Dk) ⊂ Ωk+1 (which is condition (a) for
+k + 1) and fk+1(Dk \ Dk−1) ⊂ Ωk (condition (b)). Since fk+1 approximates gk on ¯Dk and gk
+approximates fk on ¯Dk−1, fk+1 also satisfies condition (c). This completes the induction step.
+
+16
+B. Drinovec Drnovˇsek and F. Forstneriˇc
+If the approximations are close enough then the sequence fk converges uniformly on
+compacts in X to a holomorphic f : X → Cn. Conditions (a)–(c) and the fact that the sets
+Ek exhaust Cn imply that f is a proper holomorphic map satisfying f(X \ ˚
+K) ⊂ Ω0 = Cn \E0.
+To construct proper holomorphic immersions and embeddings in suitable dimensions given in
+the theorem, we use the general position argument at every step to ensure that every map fk in the
+sequence is an immersion or an embedding. (See e.g. [20, Corollary 8.9.3].) If the convergence
+is fast enough then the same holds for the limit map f by a standard argument.
+□
+Proof of Corollary 1.4. Given a holomorphic map f0 : K → Cn with f0(bK) ⊂ Cn \ Eφ as
+in Theorem 1.3, Proposition 3.11 furnishes a closed convex set Eψ ⊃ Eφ with BCEH such that
+f0(bK) ⊂ Cn \ Eψ. Applying Theorem 1.3 with Eψ gives the desired conclusion.
+□
+We have the following analogue of Theorem 1.3 with interpolation on a closed complex
+subvariety of X. Unlike in the above corollary, approximation of E from the outside by convex
+sets enjoying BCEH cannot be used since the subvariety f(X′) may have zero distance to bE.
+This results extends the case of [24, Theorem 15] when E is a compact convex set.
+Theorem 4.2. Let E be a closed convex set in Cn (n > 1) with C 1 boundary which is strictly
+convex near infinity and has bounded convex exhaustion hulls. Let X be a Stein manifold,
+K ⊂ X be a compact O(X)-convex set, U ⊂ X be an open set containing K, X′ be a
+closed complex subvariety of X, and f0 : U ∪ X′ → Cn be a holomorphic map such that
+f0|X′ : X′ → Cn is proper holomorphic and f0(bK ∪ (X′ \ K)) ∩ E = ∅. Given ǫ > 0 there
+exists a proper holomorphic map f : X → Cn satisfying the following conditions:
+(a) f(X \ ˚
+K) ⊂ Cn \ E,
+(b) ∥f − f0∥K < ǫ,
+(c) f|X′ = f0|X′.
+If 2 dim X ≤ n then f can be chosen an immersion (and an embedding if 2 dim X + 1 ≤ n)
+provided that f0|X′ is one.
+Proof. This is proved by a small modification of the proof of Theorem 1.3, similar to the one
+in [24, proof of Theorem 15]. The initial step in the proof, approximating E from the outside
+by a strictly convex set, is unnecessary since bE is strictly convex near infinity. The main (and
+essentially the only) change comes in the choice of the exhaustion Dk of the Stein manifold
+X. In the inductive step when constructing the map fk+1, we must assume in addition that
+fk(bDk ∩ X′) ⊂ Ωk+1 = Cn \Ek+1. Then, we push the image of bDk out of Ek+1 by the same
+method as before, using Proposition 2.1 but ensuring that the modifications are kept fixed on X′
+and small near bDk ∩ X′. This is possible since the method from [15] is applied locally near
+bDk (away from bDk ∩ X′), and these local modifications are glued together by preserving the
+value of the map on X′. We refer to [24, proof of Theorem 15] for a more precise description.
+This gives the next holomorphic map fk+1 : X → Cn satisfying fk+1(X \ Dk) ⊂ Ωk+1,
+fk+1|X′ = fk|X′, and conditions (b) and (c) in the proof of Theorem 1.3. We then choose the
+next domain Dk+1 ⊂ X big enough such that fk+1(bDk+1 ∩X′) ⊂ Ωk+2 = Cn \Ek+2. This is
+possible since the map fk+1|X′ = f0|X′ : X′ → Cn is proper, f0(X′ \ ˚
+K) ⊂ Ω = Cn \ E, and
+the domain Ωk+2 agrees with Ω near infinity by the construction. Clearly the induction step is
+now complete. Assuming that the approximations are close enough, the sequence fk converges
+to a limit holomorphic map f : X → Cn satisfying the stated conditions.
+□
+Acknowledgements. The first named author is supported by grants P1-0291, J1-3005, and N1-
+0137 from ARRS, Republic of Slovenia. The second named author is supported by the European
+Union (ERC Advanced grant HPDR, 101053085) and grants P1-0291, J1-3005, and N1-0237
+
+Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
+17
+from ARRS, Republic of Slovenia. The authors wish to thank Antonio Alarc´on for helpful
+discussions and information concerning the case pertaining to minimal surfaces.
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+[16] B. Drinovec Drnovˇsek and F. Forstneriˇc. Flexible domains for minimal surfaces in Euclidean spaces. J. Math.
+Anal. Appl., 517:126653, 2023.
+[17] Y. Eliashberg and M. Gromov. Embeddings of Stein manifolds of dimension n into the affine space of dimension
+3n/2 + 1. Ann. of Math. (2), 136(1):123–135, 1992.
+[18] J. E. Fornæss, F. Forstneriˇc, and E. Wold. Holomorphic approximation: the legacy of Weierstrass, Runge, Oka-
+Weil, and Mergelyan. In Advancements in complex analysis. From theory to practice, pages 133–192. Cham:
+Springer, 2020.
+[19] F. Forstneriˇc. Embedding strictly pseudoconvex domains into balls. Trans. Amer. Math. Soc., 295(1):347–368,
+1986.
+[20] F. Forstneriˇc. Stein manifolds and holomorphic mappings (The homotopy principle in complex analysis),
+volume 56 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Springer, Cham, second edition,
+2017.
+[21] F. Forstneriˇc. Holomorphic embeddings and immersions of Stein manifolds: a survey. In Geometric complex
+analysis, volume 246 of Springer Proc. Math. Stat., pages 145–169. Springer, Singapore, 2018.
+[22] F.
+Forstneriˇc.
+Recent
+developments
+on
+Oka
+manifolds.
+arXiv
+e-prints,
+2022.
+https://arxiv.org/abs/2006.07888.
+[23] F. Forstneriˇc and J. Globevnik. Proper holomorphic discs in C2. Math. Res. Lett., 8(3):257–274, 2001.
+[24] F. Forstneriˇc and T. Ritter. Oka properties of ball complements. Math. Z., 277(1-2):325–338, 2014.
+[25] F. Forstneriˇc and E. F. Wold. Oka domains in Euclidean spaces. Int. Math. Res. Not., to appear.
+https://arxiv.org/abs/2203.12883.
+[26] D. Gale and V. Klee. Continuous convex sets. Math. Scand., 7:379–391, 1959.
+[27] J. Globevnik. Boundary interpolation by proper holomorphic maps. Math. Z., 194(3):365–373, 1987.
+[28] A. Griewank and P. J. Rabier. On the smoothness of convex envelopes. Trans. Amer. Math. Soc., 322(2):691–
+709, 1990.
+[29] R. C. Gunning and H. Rossi. Analytic functions of several complex variables. AMS Chelsea Publishing,
+Providence, RI, 2009. Reprint of the 1965 original.
+
+18
+B. Drinovec Drnovˇsek and F. Forstneriˇc
+[30] M. Hakim. Applications holomorphes propres continues de domaines strictement pseudoconvexes de Cn dans
+la boule unit´e de Cn+1. (On the extension of proper holomorphic mappings from strictly pseudoconvex domains
+in Cn into the unit ball of Cn+1). Duke Math. J., 60(1):115–133, 1990.
+[31] M. Hakim and N. Sibony. Fonctions holomorphes born´ees sur la boule unite de Cn. Invent. Math., 67:213–222,
+1982.
+[32] D. Hoffman and W. H. Meeks, III. The strong halfspace theorem for minimal surfaces. Invent. Math.,
+101(2):373–377, 1990.
+[33] E. Løw. A construction of inner functions on the unit ball in Cp. Invent. Math., 67:223–229, 1982.
+[34] E. Løw. Embeddings and proper holomorphic maps of strictly pseudoconvex domains into polydiscs and balls.
+Math. Z., 190(3):401–410, 1985.
+[35] R. Narasimhan. Imbedding of holomorphically complete complex spaces. Amer. J. Math., 82:917–934, 1960.
+[36] R. Remmert. Sur les espaces analytiques holomorphiquement s´eparables et holomorphiquement convexes. C.
+R. Acad. Sci. Paris, 243:118–121, 1956.
+[37] R. Schoen and S. T. Yau. Lectures on harmonic maps. Conference Proceedings and Lecture Notes in Geometry
+and Topology, II. International Press, Cambridge, MA, 1997.
+[38] J. Sch¨urmann. Embeddings of Stein spaces into affine spaces of minimal dimension. Math. Ann., 307(3):381–
+399, 1997.
+[39] B. Stensønes. Proper holomorphic mappings from strongly pseudoconvex domains in C2 to the unit polydisc in
+C3. Math. Scand., 65(1):129–139, 1989.
+Barbara Drinovec Drnovˇsek
+Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI–1000 Ljubljana,
+Slovenia
+Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia.
+e-mail: barbara.drinovec@fmf.uni-lj.si
+Franc Forstneriˇc
+Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI–1000 Ljubljana,
+Slovenia
+Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia
+e-mail: franc.forstneric@fmf.uni-lj.si
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf,len=1068
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='01268v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='CV]  3 Jan 2023  Proper holomorphic maps in Euclidean spaces  avoiding unbounded convex sets  Barbara Drinovec Drnovˇsek and Franc Forstneriˇc  Abstract We show that if E is a closed convex set in Cn (n > 1) contained in a closed halfspace H such  that E ∩ bH is nonempty and bounded, then the concave domain Ω = Cn \\ E contains images of proper  holomorphic maps f : X → Cn from any Stein manifold X of dimension < n, with approximation of  a given map on closed compact subsets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' If in addition 2 dim X + 1 ≤ n then f can be chosen an  embedding, and if 2 dim X = n then it can be chosen an immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Under a stronger condition on E  we also obtain the interpolation property for such maps on closed complex subvarieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Keywords Stein manifold, holomorphic embedding, Oka manifold, minimal surface, convexity  MSC (2010): 32H02, 32Q56;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' 52A20, 53A10  Date: 3 January 2023  In memoriam Nessim Sibony  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Introduction  Let X be a Stein manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Denote by O(X, Cn) the Frechet space of holomorphic maps  X → Cn endowed with the compact-open topology and write O(X, C) = O(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' A theorem of  Remmert [36] (1956), Narasimhan [35] (1960), and Bishop [7] (1961) states that almost proper  maps are residual in O(X, Cn) if dim X = n, proper maps are dense if dim X < n, proper  immersions are dense if 2 dim X ≤ n, and proper embeddings are dense if 2 dim X < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' A  proof is also given in the monograph [29] by Gunning and Rossi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  It is natural to ask how much space proper maps need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We pose the following question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' For which domains Ω ⊂ Cn are proper holomorphic maps (immersions,  embeddings) X → Cn as above, with images contained in Ω, dense in O(X, Ω)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  It is evident that Ω cannot be contained in a halfspace of Cn since every holomorphic map  from C to a halfspace lies in a complex hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' In this paper we give an affirmative answer  for concave domains whose complement E = Cn \\ Ω satisfies the following condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' A closed convex set E in a real or complex Euclidean space V has bounded  convex exhaustion hulls (BCEH) if for every compact convex set K in V  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1)  the set h(E, K) = Conv(E ∪ K) \\ E is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Here, Conv denotes the convex hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The following is our first main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Let E be an unbounded closed convex set in Cn (n > 1) with bounded convex  exhaustion hulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Given a Stein manifold X with dim X < n, a compact O(X)-convex set K in  X, and a holomorphic map f0 : K → Cn with f0(bK) ⊂ Ω = Cn \\ E, we can approximate f0  uniformly on K by proper holomorphic maps f : X → Cn satisfying f(X \\ ˚  K) ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The map  f can be chosen an embedding if 2 dim X < n and an immersion if 2 dim X ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  2  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Drinovec Drnovˇsek and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Forstneriˇc  In this paper, a map f : K → Cn from a compact set K is said to be holomorphic if it is the  restriction to K of a holomorphic map on an open neighbourhood of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  In particular, if f0(K) ⊂ Ω then the theorem gives uniform approximation of f0 by proper  holomorphic maps f : X → Cn with f(X) ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' If bE is of class C 1 and strictly convex  near infinity, we obtain an analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3 with additional interpolation on a closed  complex subvariety X′ of X such that f0 : X′ → Cn is proper holomorphic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Without the condition on the range, interpolation of proper holomorphic embeddings X ֒→ Cn  on a closed complex subvariety was obtained by Acquistapace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' [1] in 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  The analogue of the BCEH condition for unbounded closed sets in Stein manifolds, with the  convex hull replaced by the holomorphically convex hull, is used in holomorphic approximation  theory of Arakelyan and Carleman type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' see the survey in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  It is evident that a closed convex set E ⊂ Rn has BCEH if and only if there is an increasing  sequence K1 ⊂ K2 ⊂ · · · of compact convex sets exhausting Rn such that the set h(E, Kj) (see  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1)) is bounded for every j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='. In particular, BCEH is a condition at infinity which is  invariant under perturbations supported on a compact subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' For compact convex sets E ⊂ Cn,  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3 was proved in [24];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' in this case BCEH trivially holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  We show in Section 3 that a closed convex set E in Rn has BCEH if and only if E is  continuous in the sense of Gale and Klee [26];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' If E has BCEH then  Conv(E ∪ K) is closed for any compact convex set K ⊂ Rn (see [26, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' If such E  is unbounded, which is the main case of interest, there are affine coordinates (x, y) ∈ Rn−1 × R  such that E = Eφ = {(x, y) ∈ Rn : y ≥ φ(x)} is the epigraph of a convex function  φ : Rn−1 → R+ = [0, +∞) growing at least linearly near infinity (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' In  particular, an unbounded closed convex set E ⊂ Cn with BCEH is of the form  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2)  E = Eφ = {z = (z′, zn) ∈ Cn : ℑzn ≥ φ(z′, ℜzn)}  in some affine complex coordinates z = (z′, zn) on Cn, with φ as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' (Here, ℜ and ℑ denote  the real and the imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=') For a convex function φ of class C 1 we give a differential  characterization of the BCEH condition on its epigraph Eφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The BCEH  property holds if the radial derivative of φ tends to infinity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' On the other hand,  there are convex functions of linear growth whose epigraphs have BCEH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' see Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' By  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='11, a convex function φ with at least linear growth at infinity can be approximated  uniformly on compacts by functions ψ ≤ φ of the same kind whose epigraphs Eψ have BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  This allows us to extend Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3 as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' see Section 4 for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The conclusion of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3 holds for any convex epigraph Eφ of the form  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2) such that φ ≥ 0 and the set {φ = 0} is nonempty and compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  A closed convex set E ⊂ Cn with BCEH does not contain any affine real line (see Proposition  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4), and for n > 1 its complement Ω = Cn \\ E is an Oka domain according to Wold and the  second named author;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' see [25, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' This fact plays an important role in our proof  of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3, given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' (The precise result from Oka theory which we shall use  is stated as Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=') Among closed convex epigraphs (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2), the class of sets with Oka  complement is strictly bigger than the class of sets with BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' In particular, the former class  contains many sets containing boundary lines, which is impossible for a set with BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Is there a (not necessarily convex) set Eφ ⊂ Cn of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2) with φ ≥ 0 of  sublinear growth for which Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3 holds?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Is there a set of this kind in C2 such that C2\\Eφ  contains the image of a proper holomorphic disc D = {z ∈ C : |z| < 1} → C2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets  3  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3 is the first general result in the literature providing proper holomorphic maps  X → Cn from any Stein manifold of dimension < n whose images avoid large convex sets in  Cn close to a halfspace, and with approximation of a given map on a compact holomorphically  convex set in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Without the approximation condition and assuming that dim X ≤ n − 2, there  are proper holomorphic maps of X into a complex hyperplane in Cn \\ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  On the other hand, there are many known results concerning proper holomorphic maps in  Euclidean spaces and in more general Stein manifolds whose images avoid certain small closed  subsets, such as compact or complete pluripolar ones, and results in which the source manifold  is the disc D = {z ∈ C : |z| < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Proper holomorphic discs in C2 avoiding closed complete  pluripolar sets of the form E = E′ × C, with E′ ⊂ C, were constructed by Alexander [5] in  1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The first named author showed in [13] (2004) that for every closed complete pluripolar  set E in a Stein manifold Y with dim Y > 1 and point p ∈ Y \\ E there is a proper holomorphic  disc f : D → Y with p ∈ f(D) ⊂ Y \\ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' If Y = C2 there also exist embedded holomorphic  discs with this property according to Borell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' [8] (2008), and for dim Y ≥ 3 this holds  by the general position argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Proper holomorphic discs in C2 with images contained in  certain concave cones were constructed by Globevnik and the second named author [23] in  2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' They also constructed proper holomorphic discs in C2 with images in (C \\ {0})2, and  hence proper harmonic discs D → R2, disproving a conjecture by Schoen and Yau [37, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  (Another construction of such maps was given by Boˇzin [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=') More generally, it was shown  by Alarc´on and L´opez [4, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1] in 2012 that every open Riemann surface X admits a  proper harmonic map to R2 which is the projection of a conformal minimal immersion X → R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  The aforementioned result from [23] was used by the first named author in [12] (2002) to classify  closed convex sets in C2 whose complement is filled by images of holomorphic discs which are  proper in C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' More recently, Forstneriˇc and Ritter [24] (2014) proved Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3 in the case  when E ⊂ Cn is a compact polynomially convex set and 2 dim X ≤ n (for immersions) or  2 dim X < n (for embeddings), and for proper holomorphic maps X → Cn when dim X < n  and E is a compact convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' A further development in this direction is the analogue of  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3 when Cn is replaced by a Stein manifold Y with the density property and E ⊂ Y  is a compact O(Y )-convex set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' see [22, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='5] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' However, in all  mentioned results except those in [23, 12], the avoided sets are thin or compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Without insisting on approximation, the theorem of Remmert, Bishop, and Narasimhan is not  optimal with respect to the dimension of the target space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Indeed, it was shown by Eliashberg and  Gromov [17] in 1992, with an improvement for odd dimensional Stein manifolds by Sch¨urmann  [38] in 1997, that a Stein manifold X of dimension m ≥ 2 embeds properly holomorphically  in Cn with n =  �3m  2  �  + 1, and for m ≥ 1 it immerses properly holomorphically in Cn with  n =  � 3m+1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' (See also [20, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=') However, the construction method in these papers,  which relies on the Oka principle for sections of certain stratified holomorphic fibre bundles,  does not give the density statement, and we do not know whether Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3 holds for maps to  these lower dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' It is also an open problem whether every open Riemann surface  embeds properly holomorphically in C2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' see [20, Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='10-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='11] and the survey [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3 is proved in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The proof relies on two main ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' One is the  result of Wold and the second named author [25, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='8] which shows in particular that the  complement Ω = Cn \\E of a closed convex set E having BCEH is an Oka domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The second  main technique comes from the work of Dor [10, 11] (1993-95), following earlier papers by  Stensønes [39] (1989) and Hakim [30] (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Dor constructed proper holomorphic immersions  and embeddings of any smoothly bounded, relatively compact, strongly pseudoconvex domain  D in a Stein manifold X into any pseudoconvex domain Ω in Cn under the dimension conditions  4  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Drinovec Drnovˇsek and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Forstneriˇc  in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Previously, Hakim [30] constructed proper holomorphic maps to balls in  codimension one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The main idea is to inductively lift the image of bD under a holomorphic  map f : ¯D → Ω to a given higher superlevel set of a strongly plurisubharmonic exhaustion  function ρ : Ω → R+ in a controlled way, taking care not to decrease the value of ρ ◦ f very  much anywhere on D during the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' When D is a finite bordered Riemann surface, this can  be achieved by using approximate solutions of a Riemann-Hilbert boundary value problem (see  [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' In higher dimensions the proof is more subtle and uses carefully controlled holomorphic  peak functions on ¯D to push a given map f : ¯D → Ω locally at a point z ∈ f(bD) in the direction  of the zero set Sz of the holomorphic (quadratic) Levi polynomial of the exhaustion function  ρ : Ω → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' At a noncritical point z ∈ Ω of ρ, Sz is a smooth local complex hypersurface and  the restricted function ρ|Sz increases quadratically as we move away from z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' If ρ is a strictly  convex function, this can be achieved by pushing the image of f(bD) in the direction of suitably  chosen affine complex hyperplanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Dor’s construction was extended by the authors to maps  from strongly pseudoconvex domains in Stein manifolds to an arbitrary Stein manifold Ω, and  also to q-convex complex manifolds for suitable values of q ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' see the papers [14, 15] from  2007 and 2010, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' In those papers we introduced the technique of gluing holomorphic  sprays of manifold-valued maps on a strongly pseudoconvex Cartan pair with control up to the  boundary (a nonlinear version of the Cousin-I problem) and a systematic approach for avoiding  critical points of a q-convex Morse exhaustion function on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Earlier constructions of this type, using simpler holomorphic peak functions and higher  codimension, were given in 1985 by Løw [34] and Forstneriˇc [19] who showed that every  relatively compact strongly pseudoconvex domain D in a Stein manifold embeds properly  holomorphically in a high dimensional Euclidean ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' A related result with interpolation on  a suitable subset of the boundary of D is due to Globevnik [27] (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' This peak function  technique was inspired by the construction of inner functions on the ball of Cn by Løw [33] in  1982, based on the work of Hakim and Sibony [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  We apply this technique to push the boundary f0(bD) ⊂ Ω = Cn \\ E of a holomorphic map  f0 : ¯D → Cn in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3 out of a certain compact convex cap C attached to E along a part  of bC contained in bE and such that the set E1 = E ∪ C is convex and has bounded convex  exhaustion hulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' At the same time, we ensure that the new map g : ¯D → Cn still sends D \\ K  to Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' For a precise result, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' In the next step, we use that Ω1 = Cn \\ E1 is  an Oka domain (see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since g(bD) ⊂ Ω1, we can apply the Oka principle (see  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1) to approximate g by a holomorphic map f1 : X → Cn with f1(X \\ D) ⊂ Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Continuing inductively, we obtain a sequence of holomorphic maps X → Cn converging to a  proper map satisfying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The details are given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  The analogues of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4 also hold for minimal surfaces in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Let n ≥ 3, and let φ : Rn−1 → R+ be a convex function such that the set {φ = 0}  is nonempty and compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Given an open Riemann surface X, a compact O(X)-convex set K  in X, and a conformal minimal immersion f0 : U → Rn from a neighbourhood of K with  f0(bK) ⊂ Ω = {y < φ(x)}, we can approximate f0 uniformly on K by proper conformal  minimal immersions f : X → Rn (embeddings if n ≥ 5) satisfying f(X \\ ˚  K) ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  If in addition φ is of class C 1, strictly convex at infinity, and the epigraph Eφ = {y ≥ φ(x)}  has BCEH then one can add to this statement the interpolation of the map on discrete sets, in  analogy to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='6 is obtained by following the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3, replacing Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1  by the analogous result obtained by the Riemann–Hilbert deformation method for conformal  Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets  5  minimal surfaces (see [2] or [3, Chapter 6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Furthermore, it has recently been shown by the  authors [16, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='5] that the complement of a closed convex set E ⊂ Rn (n ≥ 3) is  flexible for minimal surfaces (an analogue of the Oka property in complex geometry) if and only  if E is not a halfspace or a slab;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' clearly this includes all sets with BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Another method for constructing proper minimal surfaces, which yields the same result in  some examples not covered by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='6, was developed by Alarc´on and L´opez [4] in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  They showed that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='6 holds for any wedge domain Γ × R ⊂ R3, where Γ ⊂ R2 is an  open cone with angle > π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' see [4, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The complement of this set is convex but it  fails to satisfy the hypotheses of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='6 due to the presence of lines in the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' An  important difference between these two fields, which affects the possible construction methods,  is that every open Riemann surface admits a proper harmonic map to the plane R2 (see [4,  Theorem I]), while only few such surfaces admit proper holomorphic maps to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  The analogue of Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='5 for minimal surfaces asks whether there is a domain in R3 of  the form {x3 < φ(x1, x2)}, where φ : R2 → R+ is a function with sublinear growth, which  contains minimal surfaces of hyperbolic type that are proper in R3, or just a proper hyperbolic  end of a minimal surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' In particular, it would be interesting to know whether the domain  below the upper half of a vertical catenoid has this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' On the other hand, the strong  halfspace theorem of Hoffman and Meeks [32] says that the only proper minimal surfaces in R3  contained in a halfspace are planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Pushing a strongly pseudoconvex boundary out of a strictly convex cap  Let O be a convex domain in Cn for some n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Recall that a continuous function ρ : O → R  is said to be strictly convex if for any pair of points a, b ∈ O we have that  ρ(ta + (1 − t)b) < tρ(a) + (1 − t)ρ(b) for all 0 < t < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Assume now that ρt : O → R (t ∈ [0, 1]) is a continuous family of C 1 functions satisfying  the following conditions:  (a) For every t ∈ [0, 1] the function ρt is strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Note that dρt ̸= 0 on Mt := {ρt = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  (b) If 0 ≤ s < t ≤ 1 then ρt ≤ 0 on Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  (c) There is an open relatively compact subset ω0 of M0 such that for every pair of numbers  0 ≤ s < t ≤ 1 we have that Mt ∩ M0 = Mt ∩ Ms = M0 \\ ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  This means that the hypersurfaces Mt coincide on the subset M0 \\ ω0, and as t ∈ [0, 1]  increases the domains ωt = Mt \\ M0 ⊂ Mt are pairwise disjoint and move into the convex  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Each compact set of the form  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1)  Ct =  �  s∈[0,t]  ωs for t ∈ [0, 1]  is called a strictly convex cap with the base ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Note that bCt = ω0 ∪ ωt, Ct is strictly convex  along ωt, strictly concave along ω0, and it has corners along ω0 ∩ ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' As t ∈ [0, 1] increases to  1, the caps Ct monotonically increase to C1 and they share the same base ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Likewise, for any  0 ≤ s < t ≤ 1 the set Cs,t = �  u∈[s,t] ωu is a strictly convex cap with the base ωs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The sets  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2)  Et = {z ∈ O : ρt(z) ≤ 0} for t ∈ [0, 1]  are strictly convex along bEt = {ρt = 0}, they form a continuously increasing family in t, and  Et = E0 ∪ Ct for every t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  6  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Drinovec Drnovˇsek and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Forstneriˇc  Under these assumptions, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Let D be a smoothly bounded, relatively compact, strongly pseudoconvex  domain in a Stein manifold X with dim X < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Let the sets Et ⊂ O ⊂ Cn (t ∈ [0, 1]) be  given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2), and let f0 : ¯D → O be a map of class A ( ¯D) such that f0(bD) ∩ E0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Given  a compact set K ⊂ D such that f0(D \\ K) ∩ E0 = ∅ and a number ǫ > 0, there is a map  f : ¯D → O of class A ( ¯D) satisfying the following conditions:  (i) f(bD) ∩ E1 = ∅,  (ii) f(D \\ K) ∩ E0 = ∅, and  (iii) maxx∈K |f(x) − f0(x)| < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Recall that a map f : ¯D → O is said to be of class A ( ¯D) if it is continuous on ¯D and  holomorphic on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' In our application of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1 in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3, the set O  will be a ball (or the entire Euclidean space) and the hypersurfaces Mt = {ρt = 0} = bEt will  be convex graphs over the coordinate hyperplane Cn−1 × R ⊂ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  In the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1 we shall need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Assume that O is a convex open subset of Cn for n > 1, L is a compact subset of  O, and ρ : O → R is a C 1 smooth strictly convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Then there is a number δ > 0 with  the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' If D is a smoothly bounded strongly pseudoconvex domain in a Stein  manifold X of dimension dim X = m < n, K is a compact subset of D, and f : ¯D → O is a  map of class A ( ¯D) such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3)  ρ(f(z)) > −δ for all z ∈ bD  and  ρ(f(z)) > 0 if z ∈ bD and f(z) /∈ L,  then given η > 0 there is a map g : ¯D → O of class A ( ¯D) satisfying the following conditions:  (i) ρ(g(z)) > 0 for z ∈ bD,  (ii) ρ(g(z)) > δ for those z ∈ bD for which g(z) ∈ L,  (iii) ρ(g(z)) > ρ(f(z)) − η for z ∈ D \\ K, and  (iv) |f(z) − g(z)| < η for z ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  For m = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=', when D is a finite bordered Riemann surface, this is a simplified version of  [14, Lemmas 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3], which is proved by using approximate solutions of a Riemann–Hilbert  boundary value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' This method was employed in several earlier papers mentioned in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  When ρ is strictly convex, C 1 smoothness suffices since in the proof we may take a continuous  family of tangential linear discs to the sublevel set of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  For m ≥ 2, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2 is a simplified and slightly modified version of [15, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Besides the fact that we are considering single maps ¯D → O instead of sprays of maps, the only  difference is that the assumption in [15, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3] that the set {ρ = 0} is compact is replaced  by the assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3) saying that ρ(f(z)) for z ∈ bD may be negative only if f(z) lies in the  compact set L ⊂ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' This hypothesis ensures that the lifting for a relatively big amount (the role  of the constant δ) only needs to be made on a compact subset of O, while elsewhere it suffices to  pay attention not to decrease ρ ◦ f by more than a given amount and to approximate sufficiently  closely on K (the role of the constant η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The proof requires only a minor adaptation of [15,  proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3], using its local version [15, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2] in a finite induction with respect  to a covering of bD by small open sets on which there are good systems of local holomorphic  peak functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' In fact, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2 corresponds to a simplified version of [15, Sublemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4],  which explains how to lift the image of bD with respect to ρ for a sufficiently large amount at  those points in bD which the map f sends to a certain coordinate chart Ui in the target manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets  7  In our situation, the role of Ui is played by an open relatively compact neighbourhood of the set  L ∩ {ρ = 0} in O, and there is no need to use the rest of the proof of [15, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' For t ∈ [0, 1] let δt > 0 be a number for which the conclusion of  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2 holds for the function ρt and the compact set L = C1 (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The open sets  Ut = {z ∈ O : −δt < ρt(z) < δt} for t ∈ [0, 1]  form an open covering of C1, so there exists a finite subcovering {Utj} for 0 ≤ t1 < t2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' <  tk ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2 we inductively find maps f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' , fk ∈ A ( ¯D) such that for every  j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' , k we have that  (a) fj(bD) ∩ Etj = ∅ (where Et is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2)),  (b) fj(D \\ K) ∩ E0 = ∅, and  (c) |fj − fj−1| < ǫ/k on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Note that conditions (a) and (b) hold for f0 and (c) is void.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Assume inductively that for some  j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' , k} we have maps f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' , fj−1 satisfying these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2  with f = fj−1 and taking fj = g, condition (a) follows from part (i) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2, (b) follows  from (ii) provided that the number η > 0 in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2 is chosen small enough, and (c) follows  from (iii) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2 provided that η ≤ ǫ/k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' This gives the map fj satisfying conditions  (a)–(c) and the induction may continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The map f = fk then satisfies the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1 also holds, with the same proof, if ρt (t ∈ [0, 1]) are strongly  plurisubharmonic functions of class C 2 satisfying dρt ̸= 0 on Mt = {ρt = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Indeed, the  results from [15], which are used in the proof, pertain to this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' In the present paper we shall  only use the convex case under C 1 smoothness, which comes naturally in the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Closed convex sets with BCEH  In the context of convex analysis, closed unbounded convex sets that share several important  properties with compact convex sets were studied by Gale and Klee [26] in 1959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  They  introduced the class of continuous sets, and we show that this class coincides with the class  of sets having BCEH, introduced in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We then develop further  properties of these sets which are relevant to the proof of our main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  By a ray in Rn, we shall mean a closed affine halfline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Let E be a closed convex subset of  Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' A boundary ray of E is a ray contained in the boundary of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' An asymptote of E is a ray  L ⊂ Rn \\ E such that dist(L, E) = inf{|x − y| : x ∈ L, y ∈ E} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The function  σ : {u ∈ Rn : |u| = 1} → R ∪ {+∞},  σ(u) = sup{x · u : x ∈ E}  is called the the support function of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' (Here, x · u denotes the Euclidean inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=') A  closed convex set E is said to be continuous in the sense of Gale and Klee [26] if the support  function of E is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Note that every compact convex set is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  The following result is a part of [26, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3] due to Gale and Klee;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' we only list those  conditions that will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The last item (iv) uses also [26, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' For a closed convex subset E in Rn the following conditions are equivalent:  (i) E is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  (ii) E has no boundary ray nor asymptote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  (iii) For each point p ∈ Rn the convex hull Conv(E ∪ {p}) is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  (iv) For every compact convex set K ⊂ Rn the set Conv(E ∪ K) is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  8  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Drinovec Drnovˇsek and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Forstneriˇc  Condition (iii) implies that the closed convex hull Conv(E ∪ {p}) is the union of the line  segments connecting p to the points in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' It also shows that an unbounded continuous closed  convex subset E of Rn is not contained in any affine hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Let us record the following observation which will be used in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Let E ⊂ Rn be a closed convex set, p ∈ Rn\\E, and L ⊂ Rn be an affine subspace  containing p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Then, Conv(E ∪ {p}) ∩ L = Conv((E ∩ L) ∪ {p}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Set E′ = E ∩ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' It is obvious that Conv(E′ ∪ {p}) ⊂ Conv(E ∪ {p}) ∩ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Conversely,  since E is convex, every point q ∈ Conv(E ∪ {p}) belongs to a line segment from p to a point  q′ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' If in addition q ∈ L and q ̸= p then q′ ∈ E′, and hence q ∈ Conv(E′ ∪ {p}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  □  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' A closed convex set E ⊂ Rn has BCEH if and only if it is continuous in the  sense of Gale and Klee [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since all closed bounded convex sets have BCEH and are continuous, it suffices to  consider the case when the set E is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  If E is not continuous then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1 it has a boundary ray or an asymptote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Denote it  by L, and let ℓ be the affine line containing L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Pick any affine 2-plane H ⊂ Rn containing ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  There is a point p ∈ H \\(ℓ∪E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' By considering rays from p to points q ∈ E approaching L and  going to infinity (if L is a boundary ray, we can choose points q ∈ L), we see that the closure  of the set h(E, p) = Conv(E ∪ {p}) \\ E contains the parallel translate L′ ⊂ H+ of L passing  through p, so h(E, p) is unbounded and hence E does not have BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Assume now that E is a continuous and let us prove that it has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We need to show that  for any closed ball B ⊂ Rn the set h(E, B) = Conv(E ∪ B) \\ E is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Assume to the  contrary that there is a sequence xm ∈ h(E, B) with |xm| → ∞ as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since the sets E  and B are convex, we have that  xm = tmbm + (1 − tm)em for tm ∈ [0, 1], bm ∈ B, em ∈ E, and m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Note that (1 − tm)|em| → ∞ as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' By compactness of the respective sets we may  assume, passing to a subsequence, that em ̸= 0 for all m and the sequences tm, bm, and  1  |em|em  are convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Denote their respective limits by t, b, and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We have that  xm = tmbm + (1 − tm)em = bm + (1 − tm)|em|  � em  |em| − bm  |em|  �  = bm + (1 − tm)|em|fm  where fm =  � em  |em|− bm  |em|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Note that limm→∞ fm = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Pick a number α ≥ 0 and set p = b+αf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  If m is large enough then (1−tm)|em| > α, so the point ym = bm+αfm lies on the line segment  connecting bm and xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since xm ∈ Conv(E ∪ {bm}), it follows that ym ∈ Conv(E ∪ {bm}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Note that the sequence ym converges to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since E is continuous, Conv(E ∪ {b}) is closed by  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1, so p = limm→∞ ym ∈ Conv(E ∪ {b}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since this holds for every α ≥ 0, the ray  L = {b + αf : α ∈ [0, ∞)} lies in Conv(E ∪ {b}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2 there is α0 ∈ [0, ∞) such  that the ray L′ = {b + αf : α ≥ α0} lies in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since E is continuous, L is not a boundary ray  of E by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1, thus L contains a point q = b+α1f ∈ E \\bE for some α1 ≥ α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Choose  a neighbourhood Uq ⊂ E of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' For any large enough m we then have pm := bm + α1fm ∈ Uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Let Lm = {bm + αfm : α ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Note that Lm ∩ Conv(E ∪ {bm}) = Conv((Lm ∩ E) ∪ {bm})  by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' However, for m large enough the point xm ∈ Lm lies on the opposite side of pm  than bm, so xm belongs to Lm ∩ Conv(E ∪ {bm}) but not to Conv((Lm ∩ E) ∪ {bm}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' This  contradiction proves that E has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  □  Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets  9  Given a function φ : Rn−1 → R, the epigraph of φ is the set  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1)  E = Eφ = {(x, y) ∈ Rn−1 × R : y ≥ φ(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Note that a function is convex if and only if its epigraph is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' If E ⊊ Rn is a closed unbounded convex set with BCEH then  (i) E does not contain any affine real line, and  (ii) for every affine line ℓ intersecting E in a ray and any hyperplane H transverse to ℓ, E is  the epigraph of a convex function on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' In particular, there are affine coordinates (x, y)  on Rn in which E is of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1) for a convex function φ : Rn−1 → R+ satisfying  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2)  lim inf  |x|→+∞  φ(x)  |x|  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  The condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2) says that φ grows at least linearly at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We show in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='10  that linear growth is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' (i) Assume that ℓ ⊂ E is an affine line and let us prove that E does not have BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Since E is a proper subset of Rn, there is a parallel translate ℓ′ of ℓ which is not contained in E,  and hence ℓ′ \\E contains a ray L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Let p be the endpoint of L, and let p′ ∈ L be an arbitrary other  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since E ∩L = ∅, there is a ball B around p′ such that Conv(B ∪{p})∩E = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Clearly,  there is a point q ∈ B such that the ray Lq with the endpoint p and containing q intersects the  line ℓ, so the line segment from p to q belongs to Conv(E ∪ {p}) \\ E = h(E, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' By moving  p′ ∈ L to infinity we see that h(E, p) is unbounded, so E does not have BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  (ii) Since E is unbounded, it contains a ray L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Denote by ℓ the affine line containing L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Let  ℓ′ be any parallel translate of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since E contains no affine lines by part (i), there is a point  p ∈ ℓ′ \\E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The closed convex hull of the union of L and p contains the parallel translate L′ ⊂ ℓ′  of L passing through p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since E has BCEH, we conclude that L′ ⊂ Conv(E ∪ {p}) and L′ \\ E  is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since E ∩ L′ is convex, L′ ∩ E is a closed ray with the endpoint on bE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' This shows  that E is a union of closed rays contained in parallel translates of the line ℓ, so it is an epigraph  of a convex function defined on any hyperplane H ⊂ Rn transverse to ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Choosing H such that  H ∩ E = ∅ there are affine coordinates (x, y) on Rn with H = {y = 0} and ℓ = {x = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' In  these coordinates, E is of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1) for a positive convex function φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Finally, if condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2) fails then there is a sequence (xk, yk) ∈ E with |xk| → +∞ and  yk/|xk| → 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The union of the line segments Lk connecting p = (0, −1) ∈ Rn−1×R  to (xk, yk), intersected with the lower halfspace y ≤ 0, is then an unbounded subset of  h(E, p) = Conv(E ∪ {p}) \\ E, contradicting the assumption that E has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The growth condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2) for an epigraph can always be achieved in suitable  linear coordinates (even without the BCEH property) if there is a supporting hyperplane H ⊂ Rn  for E such that the set E∩H is nonempty and compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Indeed, we may then choose coordinates  (x, y) on Rn such that H = {y = 0}, E ⊂ {y ≥ 0}, and 0 ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' If the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2) fails,  there is a sequence (xk, yk) ∈ E with |xk| → +∞ and yk/|xk| → 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' After passing  to a subsequence, a ray in E ∩ H lies in the closure of the union of the line segments Lk ⊂ E  connecting the origin to (xk, yk), contradicting the assumption that the latter set is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' If E is a closed convex set in Cn (n > 1) having BCEH then Cn \\ E is Oka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4 the set E does not contain any affine real line, and hence Cn \\ E is  Oka by [25, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  □  10  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Drinovec Drnovˇsek and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Forstneriˇc  The following lemma shows that the BCEH condition is stable under uniform approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Assume that φ : Rn−1 → R is a convex function whose epigraph Eφ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1) has  BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Then for any ǫ > 0 and convex function ψ : Rn−1 → R satisfying |φ − ψ| < ǫ the  epigraph Eψ also has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' If Eψ fails to have BCEH then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3 it has a boundary  ray or an asymptote, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since dist(L, Eψ) = 0 and Eψ is convex, dist(x, Eψ) converges  to zero as x ∈ L goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Thus, by making L shorter if necessary, we have that  L ⊂ Eφ−2ǫ \\ Eφ+2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Hence, L lies out of Eφ+2ǫ but the vertical translation of L for 4ǫ pushes it  in Eφ+2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since Eφ+2ǫ, being a translate of Eφ, has BCEH, this contradicts Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  The contradiction shows that Eψ has BCEH as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  □  We now give a differential characterization of the BCEH property of an epigraph (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' If φ : Rn−1 → R is a convex function of class C 1 satisfying condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2),  then the epigraph E = {(x, y) ∈ Rn : y ≥ φ(x)} has BCEH if and only if  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3)  lim  |x|→∞ |x|  �  1 −  φ(x)  x · ∇φ(x)  �  = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We first consider the case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Then, x is a single variable and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3) is equivalent to  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4)  lim  x→+∞  �  x − φ(x)  φ′(x)  �  = +∞  and  lim  x→−∞  �  x − φ(x)  φ′(x)  �  = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  For every x ∈ R such that φ′(x) ̸= 0 the number  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='5)  ξ(x) = x − φ(x)  φ′(x)  is the first coordinate of the intersection of the tangent line to the graph of φ at the point (x, φ(x))  with the first coordinate axis y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2) and convexity of φ we have that |φ′(x)| is bounded  away from zero for all sufficiently big |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' This shows that conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4) are invariant under  translations, so we may assume that φ ≥ 0 and φ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' It is easily seen that the function ξ is  increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' If φ is of class C 2, we have that ξ′(x) = φ(x)φ′′(x)/φ′(x)2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Assume now that conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Pick a pair of sequences aj < bj in R with  limj→∞ aj = −∞ and limj→∞ bj = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The intervals Ij = [ξ(aj), ξ(bj)] then increase  to R as j → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We identify Ij with Ij × {0} ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since φ is convex, its epigraph lies above  the tangent line at any point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' It follows that the set h(E, Ij) (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1)) is the bounded region in  R×R+ whose boundary consists of Ij, the two line segments Lj and L′  j connecting the endpoints  (ξ(aj), 0) and (ξ(bj), 0) of Ij to the respective points Aj = (aj, φ(aj)) and Bj = (bj, φ(bj))  on bE, and the graph of φ over [aj, bj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The supporting lines of Lj and L′  j intersect at a point  Cj in the lower halfspace y < 0, and we obtain a closed triangle ∆j with the endpoints Aj, Bj,  and Cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Note that ∆j ∩ (R × {0}) = Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since φ grows at least linearly (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2)), the triangles  ∆j ⊂ R2 exhaust R2 as j → ∞, and the set h(E, ∆j) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1) is bounded for every j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Hence,  E has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' This argument furthermore shows that for any point p = (0, −c) /∈ E there is  a unique pair of tangent lines to bE passing through p such that, denoting by q1, q2 ∈ bE the  respective points where these lines intersect bE, the convex hull Conv(E ∪ {p}) is the union of  E and the triangle with vertices p, q1, q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Conversely, if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3) fails then it is easily seen that E has a boundary ray or an asymptote, so  it does not have BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We leave the details to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets  11  The case with n ≥ 3 now follows easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Pick a unit vector v ∈ Rn−1, |v| = 1, and let Lv  denote the 2-plane in Rn passing through the origin and spanned by v and en = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' , 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Then, Ev := E ∩ Lv = {(t, y) ∈ R2 : y ≥ φ(tv)} and the first condition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4) reads  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='6)  lim  t→+∞  �  t −  φ(tv)  �n−1  j=1 vj  ∂φ  ∂xj (tv)  �  = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Writing x = tv with t ≥ 0 and v = x/|x|, this is clearly equivalent to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' As before, let  p = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' , 0, −c) /∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' If (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3) holds then Conv(Ev ∪ {p}) ⊂ Lv is obtained by adding to  Ev the triangle in Lv obtained by the two tangent lines to bEv passing through p as described in  the case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The sizes of these triangles are uniformly bounded with respect to the direction  vector |v| = 1, and condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2) implies that these triangles increase to Lv as c → +∞,  uniformly with respect to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since �  |v|=1 Lv = Rn, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2 shows that  Conv(E ∪ {p}) =  �  |v|=1  Conv(Ev ∪ {p}),  and hence E has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The converse is seen as in the special case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  □  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' If φ : Rn−1 → R+ is a convex function of class C 1 such that  lim  |x|→+∞  x · ∇φ(x)  |x|  = +∞,  then the epigraph E = {(x, y) ∈ Rn : y ≥ φ(x)} has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' By restricting to planes as in the above proof, it suffices to consider the case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We  may assume that φ ≥ 0 and φ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since φ is convex, g(x) = φ′(x) is an increasing function  and the above condition reads limx→±∞ g(x) = ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' For any x0 > 0 and x ≥ x0 we have that  ξ(x) := x −  1  g(x)  � x  0  g(t)dt =  � x  0  �  1 − g(t)  g(x)  �  dt ≥  � x0  0  �  1 − g(t)  g(x)  �  dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Letting x → +∞ we have that g(t)  g(x) → 0 uniformly on t ∈ [0, x0], and hence the last integral  converges to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Letting x0 → ∞ we see that limx→+∞ ξ(x) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The analogous argument  applies when x → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Hence, conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3) hold and therefore E has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  □  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' There exist convex epigraphs (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1) having BCEH where the function φ grows  linearly, although it cannot be too close to linear near infinity in the absence of boundary rays  and asymptotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We give such an example in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Let g : R → (−1, 1) be an odd, continuous,  increasing function with limx→+∞ g(x) = 1 and  � ∞  0 (1−g(x))dx = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' (An explicit example  is g(x) = 2  πArctan x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=') Its integral φ(x) =  � x  0 g(t)dt for x ∈ R then clearly satisfies φ(x) ≥ 0,  φ′(x) = g(x) ∈ (−1, +1) (hence φ grows linearly), and φ is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We now show that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Let x > 0 be large enough so that g(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We have that  ξ(x) = x −  1  g(x)  � x  0  g(t)dt =  � x  0  �  1 − g(t)  g(x)  �  dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Fix x0 > 0 and let x ≥ x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Then, ξ(x) ≥  � x0  0 (1 − g(t)/g(x))dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since limx→+∞ g(x) = 1  and ξ is increasing for large enough |x|, it follows that limx→+∞ ξ(x) ≥  � x0  0 (1 − g(t))dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Sending x0 → +∞ gives limx→+∞ ξ(x) ≥  � ∞  0 (1 − g(t))dt = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Similarly we see that  limx→−∞ ξ(x) = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Thus, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3) holds, and hence the epigraph of φ has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  By using the idea in the above example we now prove the following approximation result,  which extends Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3 to a much bigger class of convex epigraphs (see Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  12  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Drinovec Drnovˇsek and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Forstneriˇc  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Assume that φ : Rn−1 → R+ is a convex function such that the set {φ = 0}  is nonempty and compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Given numbers ǫ > 0 (small) and R > 0 (big) there is a smooth  convex function ψ : Rn−1 → R such that ψ < φ on Rn−1, φ(x) − ψ(x) < ǫ for all |x| ≤ R,  and the epigraph Eψ = {y ≥ ψ} has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' By Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='5 the function φ grows at least linearly near infinity (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Set  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='7)  A = lim inf  |x|→∞  φ(x)  |x| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Since the set φ = 0 does not contain any affine line, Azagra’s result [6, Theorem 1 and  Proposition 1] implies that for every ǫ > 0 there is a smooth strictly convex function ψ on  Rn−1 satisfying φ − ǫ < ψ < φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Replacing φ by ψ − minx ψ(x) ≥ 0 we may therefore assume  that φ is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' By increasing the number R > 0 if necessary, we may assume that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='8)  φ(x)  |x|  ≥ A  2  for all |x| ≥ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Pick a number r ∈ (0, 1) close to 1 such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='9)  (1 − r) sup  |x|≤R  φ(x) < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Choose a smooth increasing function h : R → R+ such that  h(t) = 0 for t ≤ R,  lim  t→+∞ h(t) = 1,  and  � ∞  0  (1 − h(t))dt = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  (We can take a smoothing of the Arctan function used in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=') Set  H(x) =  � |x|  0  h(s)ds  for x ∈ Rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Clearly, H ≥ 0 is a radially symmetric smooth convex function that vanishes on |x| ≤ R and  satisfies H(x) ≤ |x| for all x ∈ Rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' With A and r as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='9) we set  δ = A(1 − r)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  We claim that the function  ψ(x) = rφ(x) + δH(x)  for x ∈ Rn−1  satisfies the conditions in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Clearly, ψ ≥ rφ is a smooth convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' For  |x| ≤ R we have H(x) = 0, so ψ(x) = rφ(x) ≤ φ(x) and φ(x) − ψ(x) = (1 − r)φ(x) < ǫ by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' If |x| > R then φ(x)/|x| ≥ A/2 by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='8) and H(x) < |x|, which implies  ψ(x)  |x|  ≤ rφ(x)  |x| + δ ≤ φ(x)  |x| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Indeed, we have that φ(x)  |x| − r φ(x)  |x| = (1 − r)φ(x)  |x| ≥ A(1−r)  2  = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Hence, ψ ≤ φ on Rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  It remains to show that the epigraph Eψ satisfies BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We shall verify (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3), which is  equivalent to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='6) with uniform convergence with respect to the vector v = x/|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Write  gv(t) = r∂φ(tv)  ∂t  ,  k(t) = δh(t),  ˜gv(t) = ∂ψ(tv)  ∂t  = gv(t) + k(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets  13  The quantity in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='6) associated to the function ψ is given by  ξv(t)  =  t − ψ(tv)  ˜gv(t) =  � t  0  �  1 − gv(s) + k(s)  gv(t) + k(t)  �  ds  =  � t  0  gv(t) − gv(s)  gv(t) + k(t) ds +  � t  0  k(t) − k(s)  gv(t) + k(t)ds  ≥  � t  0  gv(t) − gv(s)  gv(t) + δ  ds +  � t  0  k(t) − k(s)  gv(t) + δ ds,  where the last inequality holds since the functions gv and k are nonnegative and increasing and  k < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Pick c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We will show that for large enough t > 0 and any unit vector v ∈ Rn−1 the  above expression is bigger than or equal to c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Choose positive numbers t0, a, t1 as follows:  t0 = 3c,  a = max{3 max  |v|=1 gv(t0), 3δ},  � t1  0  (k(t1) − k(s))ds > ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Such t1 exists since limt→+∞  � t  0(k(t) − k(s))ds = δ  � ∞  0 (1 − h(s))ds = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since the  integrands in the bound for ξv(t) are nonnegative, we have for t ≥ max{t0, t1} and |v| = 1 that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='10)  ξv(t) ≥  � t0  0  gv(t) − gv(s)  gv(t) + δ  ds +  � t1  0  k(t) − k(s)  gv(t) + δ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Assume that for some such (t, v) we have that gv(t) + δ ≥ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since a ≥ 3δ, it follows that  gv(t) ≥ 2δ and hence  gv(t)  gv(t) + δ ≥ 2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Furthermore, from a ≥ 3 max|v|=1 gv(t0) we get for 0 ≤ s ≤ t0 that  gv(s)  gv(t) + δ ≤ gv(t0)  a  ≤ 1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  These two inequalities imply that the first integral in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='10) is bounded below by t0/3 ≥ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' If  on the other hand gv(t) + δ < a then the denominator of the second integral in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='10) is at most  a, so the integral is ≥ c by the choice of t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' This shows that ξv(t) ≥ c for all |v| = 1 and  t ≥ max{t0, t1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since c was arbitrary, condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3) holds and hence Eψ has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  □  The following observation will be used in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Denote by B the open unit ball in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Let Eφ ⊂ Rn be a closed convex  set of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1) with C 1 boundary having BCEH, where the function φ : Rn−1 → R is  bounded from below and strictly convex near infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Then there is an r0 > 0 such that for every  r ≥ r0 the convex hull Conv(Eφ ∪ rB) = {y ≥ ψ(x)} is a closed convex set with BCEH, and  ψ : Rn−1 → R is a convex function of class C 1 such that ψ ≤ φ and these functions agree near  infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Furthermore, if r ≥ r0 is large enough then the function φt : Rn−1 → R defined by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='11)  φt(x) = (1 − t)φ(x) + tψ(x),  x ∈ Rn−1  is strictly convex for every t ∈ (0, 1), and for any 0 < t0 < t1 < 1 the closure of the set  {(x, y) ∈ Rn : φt1(x) < y < φt0(x)}  is a strictly convex cap with the base in the strictly convex hypersurface {y = φt0(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  14  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Drinovec Drnovˇsek and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Forstneriˇc  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Consider the function on Rn−1 given by  ˜φr(x) =  �  min{φ(x), −  �  r2 − |x|2},  |x| < r,  φ(x),  |x| ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  (Note that ˜φr may be discontinuous at the points of the sphere |x| = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=') The convex hull of its  epigraph E˜φr equals Conv(E∪rB), which is closed by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1 (iv), and the set h(E, rB) =  Conv(E ∪ rB) \\ E is bounded since E has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' By smoothing ˜φr we get a function ˜ψr of  class C 1 which agrees with φ near infinity such that Conv(E ˜ψr) = Conv(E ∪ rB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' By [28,  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2] we conclude that Conv(E ∪ rB) has C 1 boundary, so it is the epigraph Eψr of a  convex function ψr : Rn−1 → R of class C 1 which agrees with φ near infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Since φ grows at least linearly, there is a function τ(r) defined for r ∈ R+ large enough such  that ψr(x) = −  �  r2 − |x|2 for |x| ≤ τ(r) and τ(r) → +∞ as r → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' By choosing r large  enough, the compact set of points where the function φ fails to be strictly convex is contained  in the ball |x| < τ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since on this ball we have that ψr(x) = −  �  r2 − |x|2 which is strictly  convex, the convex combinations φt in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='11) of φ and ψ = ψr are strictly convex on Rn−1 for  all 0 < t < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' For such r, the last statement in the proposition is evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' (Note that the strictly  convex functions ρt(x, y) = exp(ψt(x) − y) − 1 for t ∈ (0, 1) correspond to those used in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=')  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3  For the definition and the main theorem on Oka manifolds, see [20, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1 and  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We shall use the following version of the Oka principle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' see [22, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Assume that X is a Stein manifold, K is a compact O(X)-convex set in X, X′  is a closed complex subvariety of X, Ω is an Oka domain in a complex manifold Y , f : X → Y  is a continuous map which is holomorphic on a neighbourhood of K, f|X′ : X′ → Y is  holomorphic, and f(X \\ ˚  K) ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Then there is a homotopy {ft}t∈[0,1] of continuous maps  ft : X → Y connecting f = f0 to a holomorphic map f1 : X → Y such that for every t ∈ [0, 1]  the map ft is holomorphic on a neighbourhood of K, it agrees with f on X′, it approximates f  uniformly on K and uniformly in t ∈ [0, 1] as closely as desired, and ft(X \\ ˚  K) ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4 there are complex coordinates z = (z′, zn) on Cn  such that the given set E is an epigraph of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We shall write z = (x, y) where  x = (z′, ℜzn) ∈ Cn−1 × R ∼= R2n−1 and y = ℑzn ∈ R, so E = Eφ = {y ≥ φ(x)} where  φ ≥ 0 is a convex function as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Let the set K ⊂ X and the map f0 : K → Cn  be as in the theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' in particular, f0(bK) ⊂ Cn \\ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Thus, there are an open neighbourhood  U ⊂ X of K and ǫ > 0 such that f0 is holomorphic in U and f0(U \\ ˚  K) ⊂ Cn \\ Eφ−ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' By  Azagra [6, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='8] there is a a real analytic strictly convex function φ0 : R2n−1 → R such  that φ − ǫ < φ0 < φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Its epigraph E0 = {(x, y) ∈ Cn : y ≥ φ0(x)} is a closed strictly convex  set with real analytic boundary which has BCEH by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='7, and f0(U \\ ˚  K) ⊂ Cn \\ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Let B denote the open unit ball in Cn centred at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Recall the notation h(E, K) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Pick  a number r0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We can find an increasing sequence rk > 0 diverging to infinity such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1)  h(E0, rkB) ⊂ rk+1B  for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets  15  Indeed, since E0 has BCEH, the set h(E0, rkB) is bounded for each k, and hence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1) holds if  the number rk+1 is chosen large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Set  Ek+1 = Conv(E0 ∪ rkB) = E0 ∪ h(E0, rkB)  for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='.  We clearly have that E0 ⊂ E1 ⊂ · · · ⊂ �∞  k=0 Ek = Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Furthermore, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1) shows that for  j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' , k + 1 we have that E0 ⊂ Ej ⊂ E0 ∪ rk+1B and hence  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2)  Ek+2 = Conv(Ej ∪ rk+1B) for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' , k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='12 shows that for each k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' we have Ek = {y ≥ φk(x)} where φk a  convex function of class C 1 which agrees with φ0 near infinity, and Ek has BCEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Hence,  Ωk = Cn \\ Ek = {(x, y) ∈ Cn : y < φk(x)}  is an Oka domain for every k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' In view of Ek+2 = Conv(Ek∪rk+1B)  (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2)), Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='12 also shows that if rk+1 is chosen large enough then the function  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3)  ψt = (1 − t)φk + tφk+2 : Cn−1 × R → R  is strictly convex for every t ∈ (0, 1), and for each 0 < t0 < t1 < 1 the closure of the set  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4)  C = {(x, y) : ψt1 < y < ψt0}  is a strictly convex cap as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' (Note that the strictly convex functions  ρt(x, y) = exp(ψt(x) − y) − 1 for t ∈ (0, 1) correspond to those used in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=')  Choose an exhaustion D0 ⊂ D1 ⊂ · · · ⊂ �∞  k=0 Dk = X by smoothly bounded, relatively  compact, strongly pseudoconvex domains with O(X)-convex closures such that K ⊂ D0 ⊂  ¯D0 ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' For consistency of notation we set D−1 = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We now construct a sequence of  holomorphic maps fk : ¯Dk → Cn satisfying the following conditions for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' :  (a) fk(Dk \\ Dk−1) ⊂ Ωk = Cn \\ Ek,  (b) fk+1(Dk \\ Dk−1) ⊂ Ωk, and  (c) fk+1 approximates fk uniformly on ¯Dk−1 as closely as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  For k = 0 the initial map f0 in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3 satisfies condition (a) while conditions (b) and (c)  are void.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Assuming inductively that we found maps f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' , fk satisfying these conditions, the  construction of the next map fk+1 is made in two steps as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  By compactness of the set fk(bDk) ⊂ Ωk = {y < φk(x)} we can choose t0 ∈ (0, 1) small  enough such that f(bDk) ⊂ {y < ψt0(x)}, where the function ψt (t ∈ [0, 1]) is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1) we can also choose t1 ∈ (t0, 1) sufficiently close to 1 such that  Ek+1 ⊂ {(x, y) : y ≥ ψt1(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1 applied to the map fk : ¯Dk → Cn, the set Ek, and the strictly convex cap  C (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4) (which corresponds to C1 in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1) gives holomorphic map gk : ¯Dk → Cn  approximating fk on Dk−1 and satisfying  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='5) gk(bDk) ⊂ {(x, y) : y < ψt1(x)} ⊂ Cn \\ Ek+1 = Ωk+1 and gk(Dk \\ Dk−1) ⊂ Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  In the second step, we use that Ωk+1 is an Oka domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since Ωk+1 is contractible and  gk(bDk) ⊂ Ωk+1 by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='5), gk extends from ¯Dk to a continuous map X → Cn sending  X \\ Dk to Ωk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1 applied to gk gives a holomorphic map fk+1 : ¯Dk+1 → Cn  approximating gk on ¯Dk and satisfying fk+1(Dk+1 \\ Dk) ⊂ Ωk+1 (which is condition (a) for  k + 1) and fk+1(Dk \\ Dk−1) ⊂ Ωk (condition (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Since fk+1 approximates gk on ¯Dk and gk  approximates fk on ¯Dk−1, fk+1 also satisfies condition (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' This completes the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  16  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Drinovec Drnovˇsek and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Forstneriˇc  If the approximations are close enough then the sequence fk converges uniformly on  compacts in X to a holomorphic f : X → Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Conditions (a)–(c) and the fact that the sets  Ek exhaust Cn imply that f is a proper holomorphic map satisfying f(X \\ ˚  K) ⊂ Ω0 = Cn \\E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  To construct proper holomorphic immersions and embeddings in suitable dimensions given in  the theorem, we use the general position argument at every step to ensure that every map fk in the  sequence is an immersion or an embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' (See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' [20, Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=') If the convergence  is fast enough then the same holds for the limit map f by a standard argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  □  Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Given a holomorphic map f0 : K → Cn with f0(bK) ⊂ Cn \\ Eφ as  in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='11 furnishes a closed convex set Eψ ⊃ Eφ with BCEH such that  f0(bK) ⊂ Cn \\ Eψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Applying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3 with Eψ gives the desired conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  □  We have the following analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3 with interpolation on a closed complex  subvariety of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Unlike in the above corollary, approximation of E from the outside by convex  sets enjoying BCEH cannot be used since the subvariety f(X′) may have zero distance to bE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  This results extends the case of [24, Theorem 15] when E is a compact convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Let E be a closed convex set in Cn (n > 1) with C 1 boundary which is strictly  convex near infinity and has bounded convex exhaustion hulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Let X be a Stein manifold,  K ⊂ X be a compact O(X)-convex set, U ⊂ X be an open set containing K, X′ be a  closed complex subvariety of X, and f0 : U ∪ X′ → Cn be a holomorphic map such that  f0|X′ : X′ → Cn is proper holomorphic and f0(bK ∪ (X′ \\ K)) ∩ E = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Given ǫ > 0 there  exists a proper holomorphic map f : X → Cn satisfying the following conditions:  (a) f(X \\ ˚  K) ⊂ Cn \\ E,  (b) ∥f − f0∥K < ǫ,  (c) f|X′ = f0|X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  If 2 dim X ≤ n then f can be chosen an immersion (and an embedding if 2 dim X + 1 ≤ n)  provided that f0|X′ is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' This is proved by a small modification of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3, similar to the one  in [24, proof of Theorem 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The initial step in the proof, approximating E from the outside  by a strictly convex set, is unnecessary since bE is strictly convex near infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The main (and  essentially the only) change comes in the choice of the exhaustion Dk of the Stein manifold  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' In the inductive step when constructing the map fk+1, we must assume in addition that  fk(bDk ∩ X′) ⊂ Ωk+1 = Cn \\Ek+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Then, we push the image of bDk out of Ek+1 by the same  method as before, using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='1 but ensuring that the modifications are kept fixed on X′  and small near bDk ∩ X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' This is possible since the method from [15] is applied locally near  bDk (away from bDk ∩ X′), and these local modifications are glued together by preserving the  value of the map on X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We refer to [24, proof of Theorem 15] for a more precise description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  This gives the next holomorphic map fk+1 : X → Cn satisfying fk+1(X \\ Dk) ⊂ Ωk+1,  fk+1|X′ = fk|X′, and conditions (b) and (c) in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' We then choose the  next domain Dk+1 ⊂ X big enough such that fk+1(bDk+1 ∩X′) ⊂ Ωk+2 = Cn \\Ek+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' This is  possible since the map fk+1|X′ = f0|X′ : X′ → Cn is proper, f0(X′ \\ ˚  K) ⊂ Ω = Cn \\ E, and  the domain Ωk+2 agrees with Ω near infinity by the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Clearly the induction step is  now complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Assuming that the approximations are close enough, the sequence fk converges  to a limit holomorphic map f : X → Cn satisfying the stated conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  □  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The first named author is supported by grants P1-0291, J1-3005, and N1-  0137 from ARRS, Republic of Slovenia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The second named author is supported by the European  Union (ERC Advanced grant HPDR, 101053085) and grants P1-0291, J1-3005, and N1-0237  Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets  17  from ARRS, Republic of Slovenia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The authors wish to thank Antonio Alarc´on for helpful  discussions and information concerning the case pertaining to minimal surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  References  [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Every bordered Riemann surface is a complete  conformal minimal surface bounded by Jordan curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Alarc´on, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Forstneriˇc, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Minimal surfaces from a complex analytic viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Springer  Monographs in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Springer, Cham, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Alarc´on and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Minimal surfaces in R3 properly projecting into R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Kutzschebauch, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Wold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Proper holomorphic disks in the complement of varieties in C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Note on harmonic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Notices, 1999(19):1081–1085, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Approximation by proper holomorphic maps into convex domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Scuola Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Sup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Pisa Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Drinovec Drnovˇsek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Proper holomorphic discs avoiding closed convex sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Rossi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Analytic functions of several complex variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' AMS Chelsea Publishing,  Providence, RI, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Reprint of the 1965 original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Drinovec Drnovˇsek and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Applications holomorphes propres continues de domaines strictement pseudoconvexes de Cn dans  la boule unit´e de Cn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' (On the extension of proper holomorphic mappings from strictly pseudoconvex domains  in Cn into the unit ball of Cn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Meeks, III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' The strong halfspace theorem for minimal surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Paris, 243:118–121, 1956.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+page_content=' Schoen and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Yau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Lectures on harmonic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Conference Proceedings and Lecture Notes in Geometry  and Topology, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' International Press, Cambridge, MA, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  [38] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Sch¨urmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Embeddings of Stein spaces into affine spaces of minimal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=', 307(3):381–  399, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  [39] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Stensønes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Proper holomorphic mappings from strongly pseudoconvex domains in C2 to the unit polydisc in  C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=' Scand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content=', 65(1):129–139, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  Barbara Drinovec Drnovˇsek  Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI–1000 Ljubljana,  Slovenia  Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='  e-mail: barbara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='drinovec@fmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='uni-lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='si  Franc Forstneriˇc  Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI–1000 Ljubljana,  Slovenia  Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia  e-mail: franc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='forstneric@fmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='uni-lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
+page_content='si' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfUfyk/content/2301.01268v1.pdf'}
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+Symmetric Kondo Lattice States in Doped Strained Twisted Bilayer Graphene
+H. Hu,1 G. Rai,2 L. Crippa,3 J. Herzog-Arbeitman,4 D. C˘alug˘aru,4 T. Wehling,2, 5
+G. Sangiovanni,3 R. Valent´ı,6 A. M. Tsvelik,7 and B. A. Bernevig4, 1, 8, ∗
+1Donostia International Physics Center, P. Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain
+2I. Institute of Theoretical Physics, University of Hamburg, Notkestrasse 9, 22607 Hamburg, Germany
+3Institut f¨ur Theoretische Physik und Astrophysik and W¨urzburg-Dresden Cluster
+of Excellence ct.qmat, Universit¨at W¨urzburg, 97074 W¨urzburg, Germany
+4Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
+5The Hamburg Centre for Ultrafast Imaging, 22761 Hamburg, Germany
+6Institut f¨ur Theoretische Physik, Goethe Universit¨at Frankfurt,
+Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany
+7Division of Condensed Matter Physics and Materials Science,
+Brookhaven National Laboratory, Upton, NY 11973-5000, USA
+8IKERBASQUE, Basque Foundation for Science, Bilbao, Spain
+We use the topological heavy fermion (THF) model [1] and its Kondo Lattice (KL) formulation [2] to study
+the possibility of a symmetric Kondo state in twisted bilayer graphene. Via a large-N approximation, we find
+a symmetric Kondo state in the KL model at fillings ν = 0, ±1, ±2 where a KL model can be constructed [2].
+In the symmetric Kondo state, all symmetries are preserved and the local moments are Kondo screened by the
+conduction electrons. At the mean-field level of the THF model at ν = 0, ±1, ±2, ±3 we also find a similar
+symmetric state that is adiabatically connected to the symmetric Kondo state [3]. We study the stability of the
+symmetric state by comparing its energy with the ordered (symmetry-breaking) states found in Ref. [1] and
+find the ordered states to have lower energy at ν = 0, ±1, ±2. However, moving away from integer fillings
+by doping holes to the light bands, our mean-field calculations find the energy difference between the ordered
+state and the symmetric state to be reduced, which suggests the loss of ordering and a tendency towards Kondo
+screening. We expect that including the Gutzwiller projection in our mean-field state will further reduce the
+energy of the symmetric state. In order to include many-body effects beyond the mean-field approximation, we
+also performed dynamical mean-field theory (DMFT) calculations on the THF model in the non-ordered phase.
+The spin susceptibility follows a Curie behavior at ν = 0, ±1, ±2 down to ∼ 2K where the onset of screening
+of the local moment becomes visible. This hints to very low Kondo temperatures at these fillings, in agreement
+with the outcome of our mean-field calculations. At non-integer filling ν = ±0.5, ±0.8, ±1.2 DMFT shows
+deviations from a 1/T-susceptibility at much higher temperatures, suggesting a more effective screening of local
+moments with doping. Finally, we study the effect of a C3z-rotational-symmetry-breaking strain via mean-field
+approaches and find that a symmetric phase (that only breaks C3z symmetry) can be stabilized at sufficiently
+large strain at ν = 0, ±1, ±2. Our results suggest that a symmetric Kondo phase is strongly suppressed at
+integer fillings, but could be stabilized either at non-integer fillings or by applying strain.
+Introduction— The experiments on magic-angle (θ
+=
+1.05◦) twisted bilayer graphene (MATBLG) [4–6] have es-
+tablished the existence of a variety of interesting phases [7–
+28], including correlated insulating phases [29–39] and super-
+conductivity [40–44]. Their discovery has been followed by
+considerable theoretical efforts [45–69] aimed at understand-
+ing their origin. An extended Hubbard model has been con-
+structed to analyze the interacting physics [60, 70–82], how-
+ever, due to the non-trivial topology of the flat bands [83–
+91], certain symmetries become non-local. Alternatively, an
+approach based on a momentum space model has been con-
+sidered [92–100], in which correlated insulators [101–108],
+superconductivity [109–114], and other correlated quantum
+phases [115–119] have been identified and studied. Besides,
+various numerical calculations [120–127] have also been per-
+formed to investigate the correlated nature of the phenom-
+ena. However, the active phase diagram including the states
+at non-integer fillings is not well understood. The exact map-
+ping between the MATBLG and topological heavy-fermion
+∗ bernevig@princeton.edu
+model constructed in Ref. [1] could be used for develop-
+ments in this direction. This mapping establishes a bridge
+between heavy-fermions [3, 128–131] and moir´e systems [1,
+2, 132]. The presence of localized moments in MATBG is
+supported by recent entropy measurements which have found
+a Pomeranchuk-type transition [19, 133]. A large entropy ob-
+served at high-temperatures, originates from weakly interact-
+ing local moments whose fluctuations are quenched at low
+temperatures [19, 133]. Since a similar behavior is observed
+in heavy fermion systems [3, 128], where the fluctuating lo-
+cal moments are screened by conduction electrons (Kondo
+effect), this observation is suggestive of a Kondo state with
+screened local moments in MATBLG [128, 134].
+In this paper we use the KL model [2], to describe and study
+the symmetric Kondo (SK) state. We focus on integer fillings
+ν = 0, ±1, ±2, where a KL model can be constructed [2] (a
+KL description fails at ν = ±3 as demonstrated in Ref. [2]).
+The SK phase preserves all symmetries; the local moments
+are screened. We discuss the topology and the band struc-
+ture of the SK state and extend the study to the THF model
+where we identify the symmetric state that is adiabatically
+connected to the SK state [3]. In order to address integer
+arXiv:2301.04673v1  [cond-mat.str-el]  11 Jan 2023
+
+2
+and fractional fillings on equal footing, we perform both a
+mean-field and a dynamical mean-field theory (DMFT) calcu-
+lations of the THF defining a “periodic Anderson model” with
+a momentum-dependent hybridization between the correlated
+f- and the dispersive c-electrons in the non-ordered state.
+Our mean-field calculations indicate that the energy of the
+symmetric state is higher than that of the ordered (symmetry-
+breaking) states found in Ref. [1] at integer filling. We thus
+conclude that ordered states are more energetically favored
+at integer fillings. DMFT supports this picture as we obtain
+a Curie behavior of the local spin susceptibility at integer
+fillings, down to very low temperatures ∼ 2K, hinting to a
+very small Kondo scale (lower than ∼ 2K). Together with the
+mean-field results we would then expect an ordered state to be
+favored at low temperatures for these fillings.
+Turning to the effect of doping, instead, from our mean-
+field analysis, we find that the energy difference between the
+symmetric phase and the ordered phase can be sizeably re-
+duced. Doping hence suppresses the ordering and enhances
+the Kondo screening. This conclusion is further supported
+by the DMFT results at non-integer fillings. Here, we find
+clear deviations from the Curie behavior in the entire range
+from 10K down to ∼1K. Even though it is computationally
+too demanding to go further down in temperature, we point
+out that our evidence of a clear-cut difference in the screening
+properties between integer and fractional fillings is reliable.
+DMFT treats indeed local quantum fluctuations exactly [135]
+and hence takes into account the many-body processes that
+can potentially lead to the screening of local moments at any
+filling.
+Since realistic samples have intrinsic strains, we finally
+study the effect of a C3z-breaking strain on the symmetric
+phase. Our mean-field calculations show that the order is sup-
+pressed by the strain effect and a symmetric state can be sta-
+bilized at a sufficiently large strain at ν = 0, ±1, ±2.
+In summary, we conclude that a symmetric Kondo phase
+is absent at integer fillings of MATBLG, but could in princi-
+ple be stabilized either at non-integer fillings or by applying
+strain.
+Topological Heavy Fermion model and the Kondo lattice
+model— The THF model [1] contains two types of electrons:
+topological conduction c-electrons (ck,aηs) and localized f-
+electrons (fR,αηs). The operator ck,aηs annihilates conduc-
+tion c-electron with momentum k, orbital a ∈ {1, 2, 3, 4},
+valley η ∈ {+, −} and spin s ∈ {↑, ↓}. At the ΓM-point for
+each valley and each spin projection, c-electrons in the orbital
+1 and 2 transform according to the Γ3 irreducible represen-
+tation (of magnetic space group P6′2′2) [1]. The remaining
+c-electrons (a = 3, 4) at the same valley with the same spin
+projection transform in the Γ1 ⊕ Γ2 reducible representation
+(of magnetic space group P6′2′2) [1]. We will call them Γ3 c-
+electrons (a = 1, 2) and Γ1⊕Γ2 c-electrons (a = 3, 4) respec-
+tively. fR,αηs is the annihilation operator of the f-electron at
+the moir´e unit cell R with orbital α ∈ {1, 2}, valley η and
+spin s. The Hamiltonian of the THF model [1, 136] is
+ˆHT HF = ˆHc + ˆHfc + ˆHU + ˆHW + ˆHV + ˆHJ
+(1)
+where ˆHc describes the kinetic term of conduction elec-
+trons,
+ˆHfc describes the hybridization between f-c elec-
+trons [1, 136]. The interactions include an on-site Hubbard
+interaction of f-electrons ( ˆHU with U = 57.95meV), a re-
+pulsion between f- and c-electrons ( ˆHW with W = 48meV),
+a Coulomb interaction between c-electrons ( ˆHV with V (q =
+0)/Ω0 = 48.33meV and Ω0 the area of moir´e unit cell), and a
+ferromagnetic exchange coupling between f-and c-electrons
+( ˆHJ with J = 16.38meV) [1, 136].
+Based on the THF model [1], a KL model of MATBLG
+has been constructed via a generalized Schrieffer–Wolff (SW)
+transformation as shown in Ref. [2]. The KL model is de-
+scribed by the following Hamiltonian
+ˆHKondo = ˆHc + ˆHcc + ˆHK + ˆHJ .
+(2)
+where
+ˆHc, ˆHJ come from the original THF model and
+ˆHcc, ˆHK emerge from the SW transformation. ˆHcc is the one-
+body scattering term of Γ3 c-electrons with the form of
+ˆHcc =
+�
+|k|<Λc
+�
+a,a′∈{1,2}
+η,s
+e−|k|2λ2 : c†
+k,aηsck,a′ηs :
+�
+−1
+Dνc,νf
++
+−1
+Dνc,νf
+� �
+γ2/2
+γv′
+⋆(ηkx − iky)
+γv′
+⋆(ηkx + iky)
+γ2/2
+�
+a,a′ .
+(3)
+λ is the damping factor of the f-c hybridization in the THF
+model. γ, v′
+⋆ characterize the zeroth order and linear order
+f-c hybridization of the THF model with v′
+⋆ characterizing
+a k-dependent hybridization matrix [1, 136]. D1,νc,νf and
+D2,νc,νf are defined as
+D1,νc,νf = (U − W)νf − U
+2 + (−V (0)
+Ω0
++ W)νc
+D2,νc,νf = (U − W)νf + U
+2 + (−V (0)
+Ω0
++ W)νc ,
+(4)
+where νf, νc are the filling of f- and c-electrons deter-
+mined from the calculations of the THF model at the zero-
+hybridization limit [2]. We point out that in the single-orbital
+Kondo model, the one-body scattering term merely introduces
+a chemical potential shift [3, 137] of the c-electrons and is
+usually omitted. However, in our model, ˆHcc cannot be ig-
+nored for two reasons. First, ˆHcc is k-dependent and thus
+introduces additional kinetic energy to the conduction elec-
+trons. From Eq. S24, we observe the k-dependency mainly
+comes from the linear k term that is proportional to v′
+⋆ and
+can be traced back to the k-dependency of the hybridization
+matrix in the THF model. Secondly, even if we drop the v′
+⋆
+term in Eq. S24 (v′
+⋆ = 0 corresponding to the chiral limit [1]),
+ˆHcc still produces an energy shift for the Γ3 c-electrons. Thus
+ˆHcc leads to the energy splitting between Γ3 and Γ1 ⊕ Γ2 c-
+electrons and cannot be simply treated as a shift of the chem-
+ical potential.
+ˆHK is the Kondo interaction between f- and Γ3 c-electrons
+whose explicit form is given in Refs. [2, 136]. The Kondo
+interaction consists of two parts: the zeroth order Kondo in-
+teraction proportional to γ2/Dνc,νf and the first order Kondo
+
+3
+interaction proportional to γv′
+⋆/Dνc,νf , where D−1
+νc,νf
+=
+−D−1
+1,νc,νf + D−1
+2,νc,νf . The zeroth order Kondo interaction
+term describes the antiferromagnetic interaction between the
+U(8) moments of the f- and the Γ3 c-electrons and has a U(8)
+symmetry. The linear-order Kondo interaction only has a flat
+U(4) symmetry and is k-dependent [1, 136].
+ˆHJ is the fer-
+romagnetic exchange interaction between Γ1 ⊕ Γ2 c- and f-
+electrons that already exists in the TFH model [1, 136]. We
+also note that, for both the THF model and the KL model,
+ground states at filling ν and −ν are connected by a charge-
+conjugation transformation [1]. This can be broken by other
+one-body terms which we did not consider here. Therefore, in
+what follows, we only focus on ν ≤ 0.
+Mean-field Hamiltonian of the Kondo model— We next per-
+form a mean-field study of the KL model [3]. This MF sup-
+presses the RKKY interaction and essentially restores the hy-
+bridization term ˆHfc of the original periodic Anderson model,
+but in a renormalized form. It becomes exact in the N → ∞
+limit (we have N = 4 here which corresponds to the approxi-
+mate flat U(4) symmetry of the KL Hamiltonian in Eq. 2). At
+the mean-field level, the Kondo interaction ˆHK can be written
+as (see Supplementary Materials (SM))
+ˆHMF
+K
+=
+�
+R,|k|<Λc
+�
+αηs
+eik·R−|k|2λ2/2
+√NMDνc,νf
+�
+− f †
+R,αηsck,aηs
+�
+γ2V ∗
+1 + γv′
+⋆V ∗
+2
+V ∗
+1 (ηkx − iky)
+V ∗
+1 (ηkx + iky) γ2V ∗
+1 + γv′
+⋆V ∗
+2
+�
+α,a
++ h.c.
+�
++ NM
+�
+γ2|V ∗
+1 |2 + γv′
+⋆(V ∗
+1 V ∗
+2 + V ∗
+2 V ∗
+1 )
+�
++ H.T.
+(5)
+where we have introduced the following mean fields
+V ∗
+1 =
+�
+R,|k|<Λc
+�
+αηs
+eik·R−|k|2λ2/2
+√NMNM
+⟨Ψ|f †
+R,αηsck,αηs|Ψ⟩
+V ∗
+2 =
+�
+R,|k|<Λc
+�
+αaηs
+eik·R−|k|2λ2/2
+√NMNM
+(ηkxσx + kyσy)αa
+⟨Ψ|f †
+r,αηsck,αηs|Ψ⟩
+(6)
+with |Ψ⟩ being the mean-field ground state, and H.T. denotes
+the Hartree term (⟨f †f⟩, ⟨c†c⟩) whose explicit formula is in
+the Supplementay Materials (SM) [136]. Several points are
+in order. First, as we have mentioned above, the mean field
+restores the hybridization of the original Anderson model, but
+in a renormalized form. V ∗
+1 , V ∗
+2 describe the renormalized
+hybridization between the f- and Γ3 c-electrons driven by the
+Kondo interactions between two types of electrons(f and Γ3
+c) [3, 138]. Second, it is necessary to keep the Hartree con-
+tributions. In the canonical Kondo model, the Hartree term
+merely produces a chemical potential shift (in the case without
+symmetry breaking) and hence can be omitted. Here, Hartree
+contributions (see SM [136]) are k-dependent because of the
+k-dependency of the Kondo interactions, and thus contribute
+to the dispersion of the conduction c-electrons. Furthermore,
+since only Γ3 c-electrons contribute to the Kondo interaction,
+the Hartree term also produces an energy splitting between the
+Γ3 and the Γ1 ⊕ Γ2 c-electrons.
+As for ˆHJ, we perform a similar mean-field decoupling
+ˆHMF
+J
+=J
+�
+R,|k|<Λc,αηs
+eik·R
+√NM
+�
+V3δ1,η(−1)α+1f †
+R,αηsck,α+2ηs
++ V4δ−1,η(−1)α+1ηf †
+R,αηsck,α+2ηs + h.c.
+�
+− JNM
+�
+|V3|2 + |V4|2
+�
++ H.T.
+(7)
+where we have introduced the following two mean-field aver-
+ages that describe the f-c hybridization:
+V3 =
+�
+R,|k|<Λc
+�
+αη,s
+eik·Rδ1,η(−1)α+1
+√NMNM
+⟨Ψ|f †
+R,αηsck,α+2ηs|Ψ⟩
+V4 =
+�
+R,|k|<Λc
+�
+αη,s
+eik·Rδ−1,η(−1)α+1
+√NMNM
+⟨Ψ|ηf †
+R,αηsck,α+2ηs|Ψ⟩ ,
+(8)
+To impose the filling of the f-electrons to be νf, we intro-
+duce the Lagrange multiplier [134, 136, 138]:
+ˆHλf =
+�
+R,αηs
+λf
+�
+: f †
+R,αηsfR,αηs : −νf
+�
+(9)
+with λf to be determined self-consistently [136]. Finally, we
+introduce a chemical potential µc to the c-electrons
+ˆHµc = −µc
+�
+|k|<Λc,aηs
+: c†
+k,aηsck,aηs : .
+(10)
+In the calculation, we tune µc and λc together to fix both the
+total filling ν = νf + νc and the filling of f-electrons [136].
+The final mean-field Hamiltonian of the KL model now is
+ˆHMF
+Kondo = ˆHc + ˆHcc + ˆHMF
+K
++ ˆHMF
+J
++ ˆHλf + ˆHµc .
+(11)
+We then self-consistently solve the mean-field equations
+(see SM [136]). At ν = νf = 0, −1, −2 (where a KL model
+can be constructed), we identify a SK state that preserves
+all the symmetries and is characterized by V ∗
+1
+̸= 0, V ∗
+2
+̸=
+0, V ∗
+3 = 0, V ∗
+4 = 0 [136]. We comment that the exchange
+interaction ˆHJ [1] between f- and Γ1 ⊕ Γ2 c-electrons is fer-
+romagnetic, and hence disfavors the singlet formation or hy-
+bridization (V3, V4) between f- and Γ1 ⊕ Γ2 c-electrons. We
+find that V3, V4 vanish (their numerical amplitudes are smaller
+than 10−5). In fact, ˆHJ favors the triplet formation or pair-
+ing formation (f †c†), where both lead to a symmetry-breaking
+state at the mean-field level and are beyond our current con-
+sideration of SK state.
+Properties of the symmetric Kondo phase— In Fig. 1, we
+plot the band structure of the SK phase and compare it with
+the non-interacting THF model. We find the hybridization in
+
+4
+(a)
+(b)
+(c)
+(d)
+FIG. 1.
+(a) Band structure of the non-interaction THF model at ν = 0. (b), (c), (d) Band structure of the SK phase at ν = 0, −1, −2
+respectively.
+the SK state defined in Eq. 5 to be enhanced compared to the
+non-interacting limit of THF model, which is clear from the
+increase of the gap of the Γ3 states at the Γ point [1] from
+its non-interacting value 24.75meV at ν = 0, to 168meV,
+190meV, 213meV at ν = 0, −1, −2 respectively. We also
+find that in the SK phase the bandwidths of the flat bands at
+ν = −1, −2 become 16meV, 53meV, which are (much) larger
+than the non-interacting flat-band bandwidth (= 7.4meV) of
+the THF model (Fig. 1). However, at ν = 0, the flat-band
+bandwidth is the same as the non-interacting flat-band band-
+width. This is because, at ν = 0, the one-body scattering
+term and the Hartree contributions from ˆHK, ˆHJ all van-
+ish [136], and the enhanced hybridization pushes the remote
+bands away from the Fermi energy and does not change much
+the band structures of the flat bands. In addition, unlike the
+non-interacting case, here we found the flat bands are mostly
+formed by Γ1⊕Γ2 c-electrons with orbital weights larger than
+70% at ν = 0, −1, −2. This is because the large f-c hy-
+bridization induced by V1, V2 (Eq. 5) pushes the energy of Γ3
+c- and f-electrons away from the Fermi energy and reduces
+their orbital weights [136].
+The flat bands in the SK phase form Γ1 ⊕ Γ2, M1 ⊕ M2
+and K2K3 representations at ΓM, MM, KM respectively, and
+have the same topology as the flat bands in the non-interacting
+THF model [1]. More explicitly, the flat bands for each val-
+ley and each spin projection belong to a fragile topology [1]
+at ν = −1, −2. At ν = 0, due to the additional particle-hole
+symmetry, flat bands have a stable topology [1, 85, 91, 136],
+which is characterized by the odd winding number of the Wil-
+son loop as shown in supplementary material [136]. We men-
+tion that the interplay between Kondo effect and the topologi-
+cal bands has also been studied in various other systems [139–
+144].
+Symmetric phase in the topological heavy-fermion model—
+We next investigate the similar symmetric phase in the THF
+model Eq. 1.
+We first focus on integer fillings ν
+=
+0, −1, −2, −3 and perform the mean-field calculations of
+THF as introduced in Ref. [1, 136]. By enforcing the mean-
+field Hamiltonian to preserve all the symmetries, we are able
+to identify a symmetric state that preserves all the symmetries
+at ν = 0, −1, −2, −3. To observe the stability of the sym-
+metric phase, we compare its energy (Esym) with the energy
+(Eorder) of the ordered (symmetry-breaking) ground states
+derived in Ref. [1]. The ordered ground states in Ref. [1]
+are a Kramers inter-valley-coherent (KIVC) state at ν = 0,
+0.5
+0.0
+0.5
+0
+10
+20
+30
+40
+50
+E/meV
+= -3
+0.5
+0.0
+0.5
+0
+10
+20
+30
+40
+50
+= -2
+0.5
+0.0
+0.5
+0
+10
+20
+30
+40
+50
+= -1
+0.5
+0.0
+0.5
+0
+10
+20
+30
+40
+50
+= 0
+FIG. 2. Doping dependence of the ground state energy difference
+∆E = Esym − Eorder near integer fillings νt = 0, −1, −2, −3.
+a KIVC+valley polarized (VP) state at ν = −1, a KIVC state
+at ν = −2 and a VP state at ν = −3. We point out that at
+ν = −3 other states with lower energy exist [145]. In our
+numerical calculations, we find ∆E = Esym − Eorder =
+47meV, 40meV, 33meV, 23meV at ν = 0, −1, −2, −3 re-
+spectively. In all integer filling cases, the symmetric states
+have higher energy, and the ground states cannot be the sym-
+metric state, which is consistent with the previous calcula-
+tions of Ref. [1, 2, 103]. Note that our mean-field calcula-
+tion does not include a Gutzwiller projection to fix the fill-
+ing of f-electrons at each site, and hence we expect the en-
+ergy of projected symmetric states will be lower. However, as
+we show later, after including the effect of local correlations
+via the DMFT approach, we confirm that the Kondo phase,
+which is adiabatically connected to the symmetric phase in
+the mean-field calculations, is strongly suppressed at integer
+fillings (down to temperatures ∼ 1-2K). This further supports
+the development of ordering at integer fillings.
+Effects of doping— We next investigate the effects of dop-
+ing, first at the level of mean-field theory. We stick to a nar-
+row region ν ∈ [νint−0.5, νint+0.5] near each integer filling
+νint = 0, −1, −2, −3 and compare the energies of the ordered
+states Eord and the symmetric states Esym in the THF model.
+To find the ordered state solutions, we first initialize the cal-
+culations with the mean-field solutions at integer filling νint
+and fill the mean-field bands up to current filling ν. We then
+self-consistently solve the mean-field equations and calculate
+the energy of the resulting states. We obtain the symmetric-
+state solution in a similar manner but take the symmetric so-
+lution at νint as initialization and enforce the symmetry of
+the mean-field Hamiltonian during the calculations. Fig. 2
+displays a plot of the difference of the ground state energies
+
+5
+∆E = Esym −Eorder as a function of doping ∆ν = ν −νint
+near νint = 0, −1, −2, −3. We observe that hole doping at
+ν = 0, −1, −2, −3 and electron doping at ν = 0 decreases
+the ∆E. Doping holes at ν = 0, −1, −2, −3 and doping elec-
+trons at ν = 0 to the ordered states is equivalent to doping the
+light (dispersive) bands which are mostly formed by conduc-
+tion c-electrons. After doping, the conduction electrons will
+stay close to the Fermi energy, and then enhance the tendency
+towards the Kondo effect.
+However, doping electrons at ν = −1, −2 to the ordered
+states is equivalent to doping heavy (flat) bands which mostly
+come from the f-electrons. Because of the flatness of the
+band, we find the nature of the ordered states will change with
+doping (see SM [136]). For example at ν = 2, the KIVC
+order is suppressed by the electron doping (see SM [136]).
+Thus, ∆E will be affected by both, changes of ordering and
+doping. However, a sizeable change of the order parameters
+is not observed for hole doping at ν = 0, −1, −2, −3 and also
+electron doping at ν = 0 (see SM [136]), because we are dop-
+ing conduction c-electrons in both cases. We also point out
+the complexity of ν = −3. First, other states that break trans-
+lational invariant [145] could have lower energy than the VP
+state we currently considered. Second, even for the VP state,
+doping electrons is equivalent to doping both heavy and light
+bands [1], since both light and heavy bands appear in the elec-
+tron doping case [1]. In practice, as we increase ∆ν, we find
+that, at ν = −1, −2, ∆E will first decrease and then increase
+and, at ν = −3, ∆E will always increase.
+In summary, we conclude that hole doping can suppress the
+long-range order and enhance the tendency towards the Kondo
+effect near ν = 0, −1, −2. Electron doping, depending on the
+fillings, could also enhance the tendency toward the Kondo ef-
+fect. However, on the electron doping side, the change of or-
+der moments indicates the importance of the correlation effect
+which could be underestimated in the mean-field approach. In
+the next section, we provide a more comprehensive study of
+the doping effect using the DMFT calculation.
+Dynamical mean-field theory results of the THF model—
+In the following, we present the dynamical mean-field the-
+ory resultss of the THF model (Eq. 1), where we describe the
+local quantum many-body effects of the density-density Hub-
+bard term ˆHU within the f-subspace. The ˆHW and ˆHV in-
+teractions involving density fluctuations on the c orbitals are
+accounted for at the static mean-field level. We neglect or-
+dered phases and perform calculations in the non-ordered one.
+There, we focus in particular on lifetime effects, quasiparticle
+weights and exploit the ability of DMFT to take local vertex
+corrections to the spin-spin correlation function into account.
+First, DMFT finds a qualitative difference between the
+strong quasiparticle renormalization when the f+c manifold
+is occupied with an integer number of electrons and a lighter
+Fermi liquid at fractional fillings: this can be seen from the
+scattering rate Γf = −ImΣf(ω = 0) which is shown as a
+function of the total filling ν at T = 11.6K (light blue empty
+circles) in Fig.3(a). The largest scattering rates are found close
+to ν = 0.0, -1.0 and -2.0, progressively decreasing as one
+moves away from the charge neutrality point. Correspond-
+ingly, the spectral weight at the Fermi level (black and grey
+solid circles) is suppressed at these fillings, with a residual
+nonzero value due to the finite temperature on the one hand
+and the resilient f/c hybridization on the other.
+Fig.3(b) illustrates the temperature-dependent screening of
+the local magnetic moment on the f orbitals at different fill-
+ings. A flat T · χloc
+spin(ω = 0) indicates Curie behavior and a
+well-defined effective local moment, while deviations signal
+the onset of screening and a crossover towards a renormalized
+Pauli-like behavior, in agreement with the general expecta-
+tion of zero-temperature Fermi-liquid in the periodic Ander-
+son model [146]. While at ν = 0.0, -1.0 and -2.0 the 1/T
+local spin susceptibility persists down to 1-2 K, the fractional
+fillings deviate from Curie at much higher temperatures, in
+line with the better Fermi-liquid nature signaled by the single-
+particle quantities in Fig.3(a).
+As in the Hartree-Fock treatment of the THF model, DMFT
+confirms the difference between electron doping and hole dop-
+ing (particle-hole asymmetry) near integer filling ν = −1 and
+-2. Here, DMFT reveals particle-hole asymmetric scattering
+rates (Fig. 3(a)) and also in the difference of effective local
+moments at ν = −0.8 and −1.2(Fig. 3(b)).
+In summary, our DMFT calculations confirm that the
+Kondo phase is strongly suppressed at integer fillings ν =
+0, −1, −2, increasing the propensity towards long-range order
+at these fillings. However, by doping the system, the develop-
+ment of Kondo screening (starting from ∼ 10K) is observed,
+which suggests that doping could enhance the Kondo effect.
+This picture is consistent with our mean-field calculations.
+Effects of strain— Since twisted bilayer graphene samples
+exhibit intrinsic strain [147] and the ordered states are disfa-
+vored by strain, we investigate the effect of strain on the sym-
+metric state of THF model via mean-field approach. We focus
+on ν = 0, −1, −2, −3 and introduce the following Hamilto-
+nian [136] that qualitatively characterizes the effect of strain
+ˆHstrain = α
+�
+R,ηs
+(f †
+R,1ηsfR,2ηs + h.c.)
+where α is proportional to the strain amplitude (we leave the
+construction of a realistic strain Hamiltonian [148–150] for
+a future study). A non-zero α breaks the C3z symmetry but
+preserves all other symmetries [136]. We compare the ground
+state energies of the symmetric states (Estrain
+sym
+) and the or-
+dered states (Estrain
+ord
+) at non-zero strain. To obtain the sym-
+metric state solution, we solve the mean-field equations by
+requiring the mean-field Hamiltonian to satisfy all symme-
+tries except for the C3z. We obtain the solution of the or-
+dered states by initializing the mean-field calculations with
+the ordered ground states at zero strain and then perform self-
+consistent calculations. In Fig. 4, we plot the difference be-
+tween the ground state energies of the symmetric and the or-
+dered states ∆Estrain = Estrain
+sym
+− Estrain
+order as a function of
+the effective strain amplitude α with 0meV < α < 20meV at
+ν = 0, −1, −2, −3. We observe that ∆E at ν = 0, −1 van-
+ishes at sufficiently large strain. A detailed analysis [136] of
+the wavefunction shows that the ordered state cannot be sta-
+bilized and converged to a C3z broken symmetric solution at
+large strain. By further increasing strain, we find a symmet-
+ric state at ν = −2 can also be stabilized at α ∼ 45meV
+
+6
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+A(
+= 0)
+a
+Atot
+Af
+Ac
+f
+0
+2
+4
+6
+8
+10
+T [K]
+0.5
+1.0
+1.5
+2.0
+T
+loc
+spin(
+= 0)
+b
+total
+eff
+= 1.39
+total
+eff
+= 1.34
+total
+eff
+= 1.19
+= 0.00
+=
+0.50
+=
+0.80
+=
+1.00
+=
+1.20
+=
+2.00
+2
+1
+0
+2
+0
+f/c
+f
+c
+0
+20
+40
+f [meV]
+FIG. 3. DMFT solution of the THF model. (a) Doping ν dependent low-energy spectral function at the Fermi level (A(ω = 0)) for the
+full system Atot, the c- (Ac) and the f-electrons (Af) at 11.6 K. Also shown is the scattering rate Γf as extracted from the local f-electron
+self-energy. (b) Effective local moment T · χloc
+spin(ω = 0) as a function of temperature T for different doping levels ν.
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+/meV
+0
+2
+4
+6
+8
+10
+12
+14
+16
+Estrain/meV
+=
+0
+=
+1
+=
+2
+=
+3
+FIG. 4. Energy difference ∆Estrain = Estrain
+sym
+− Estrain
+ord
+between
+the symmetric state that only breaks C3z symmetry (Estrain
+sym
+) and
+the ordered state (Estrain
+ord
+) as a function of α - a parameter charac-
+terizing the strain amplitude. We note that even at zero strain α = 0,
+a symmetric state that only breaks C3z symmetry has lower energy
+than the fully symmetric state. Thus ∆Estrain at α = 0 is smaller
+than the corresponding ∆E in Fig. 2.
+(see SM [136]). We conclude that a symmetric phase can be
+stabilized by sufficiently large strain at ν = 0, −1, −2. As
+for ν = −3, we mention that other ordered states, that break
+translational symmetry and have lower energy than the VP
+state, exist even at zero strain. We leave a systematical analy-
+sis of ν = −3 for future study. Finally, we comment that even
+at zero strain, a symmetric state that breaks C3z symmetry has
+lower energy than the fully symmetric state that preserves all
+the symmetries (including C3z). Therefore, ∆Estrain (energy
+difference between a symmetric state that only breaks C3z and
+an ordered state) at zero strain α = 0 in Fig. 4 is smaller
+than the corresponding ∆E (energy difference between a fully
+symmetric state and an ordered state) in Fig. 2.
+Discussion and summary— Our main result is that an or-
+dered state, instead of a SK state, will be the ground state of
+the system at integer filling ν = 0, −1, −2, −3. Our mean-
+field calculations of THF model indicate ground state energy
+of the symmetric state is higher than the one of the ordered
+states at these fillings. Via DMFT calculations, we find the
+Kondo temperature to be substantially smaller than 2K. Thus,
+we conclude the Kondo effect is suppressed at integer filling
+ν = 0, −1, −2, −3, and the ground state is likely to be an
+ordered state. However, our mean-field calculations suggest
+doping can reduce the energy difference between the symmet-
+ric state and the ordered state enhancing the tendency towards
+the SK state. This has also been confirmed by the DMFT
+calculations which show a strong deviation from the Curie
+Weiss law at non-integer fillings ν = −0.5, −0.8, −1.2 al-
+ready around 10K. Furthermore, we show that a sufficiently
+large C3z breaking strain could also stabilize a symmetric
+state that only breaks the C3z symmetry at ν = 0, −1, −2.
+Therefore, we conclude both doping and strain enhance the
+Kondo effect and could, in principle, stabilize a SK state. Our
+results may explain the recent entropy experiments [18, 19]
+which reveal a high-temperature phase with fluctuating mo-
+ments and a low-temperature Fermi liquid phase with unpolar-
+ized isospins. This could be understood as a sign of screening
+of the local moments and the development of the SK phase at
+low temperatures.
+As far as the SK state is concerned, we have performed a
+systematic study of its band structure and topology. Via the
+mean-field approach, we successfully identified the SK state
+in the KL model, and a symmetric state, that is adiabatically
+connected to the SK state, in the THF model. For the SK
+state in the KL model, we find that the Γ3 states near the ΓM
+point have been pushed away, and the bandwidth of the flat
+bands is enlarged at ν = −1, −2. The hybridization between
+f-electrons and Γ3 c-electons is enhanced by the Kondo in-
+teractions. Consequently, the flat bands are mostly formed by
+Γ1 ⊕ Γ2 c-electrons. The topology of the flat bands remains
+the same as in the non-interacting case. However, for the sym-
+metric state in the THF model, the enhanced f-c hybridization
+does not appear. We mention that the mean-field solution of
+
+7
+the symmetric state in the THF model underestimates the cor-
+relation effect, which could be the origin of the weak f-c hy-
+bridization. We expect introducing a Gutzwiller projector will
+give a more precise description of the symmetric state in the
+THF model.
+Note added— After finishing this work, we learned that re-
+lated, but not identical, results have also recently been ob-
+tained by the S. Das Sarma’s [151], P. Coleman’s [134], and
+Z. Song’s groups [152]. We also mention that results from Z.
+Song’s group are compatible with our DMFT results.
+Acknowledgements— B. A. B.’s work was primarily sup-
+ported by the DOE Grant No.
+DE-SC0016239, the Si-
+mons Investigator Grant No. 404513. H. H. was supported
+by the European Research Council (ERC) under the Euro-
+pean Union’s Horizon 2020 research and innovation program
+(Grant Agreement No. 101020833). J. H. A. was supported
+by a Hertz Fellowship.
+D. C. was supported by the DOE
+Grant No. DE-SC0016239 and the Simons Investigator Grant
+No. 404513. A. M. T. was supported by the Office of Ba-
+sic Energy Sciences, Material Sciences and Engineering Divi-
+sion, U.S. Department of Energy (DOE) under Contract No.
+DE-SC0012704. G. R, L. K, T. W., G. S. and R. V. thank
+the Deutsche Forschungsgemeinschaft (DFG, German Re-
+search Foundation) for funding through QUAST FOR 5249-
+449872909 (Projects P4 and P5). G.R. acknowledges funding
+from the European Commission via the Graphene Flagship
+Core Project 3 (grant agreement ID: 881603). T.W. acknowl-
+edges support by the Cluster of Excellence “CUI: Advanced
+Imaging of Matter” of the Deutsche Forschungsgemeinschaft
+(DFG)–EXC 2056–Project ID No. 390715994.
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+
+13
+Supplementary Materials
+CONTENTS
+References
+7
+S1. Toplogical heavy-fermion model
+14
+S2. Kondo lattice model
+15
+S3. Symmetry
+16
+S4. Mean-field solutions of the Kondo lattice model
+16
+A. Mean-field decoupling of ˆHK
+17
+1. Fock term
+17
+2. Hartree term
+18
+3. Fock and Hartree terms
+19
+B. Mean-field decoupling of ˆHJ
+20
+1. Fock term
+20
+2. Hartree term
+21
+3. Fock and Hartree terms
+21
+C. Filling constraints and mean-field equations
+21
+D. Mean-field equations of the symmetric Kondo state
+22
+E. Properties of the symmetric Kondo state
+24
+S5. Mean-field solutions of the topological heavy-fermion model
+27
+A. Mean-field equations of fully symmetric state
+27
+B. Mean-field equations of the symmetric state in the presence of strain
+29
+C. Effect of doping
+30
+D. Effect of strain
+31
+1. ν = −1
+32
+2. ν = −2
+34
+3. Discussions about ν = −3
+36
+E. Strain
+36
+S6. Dynamical mean field theory: implementation
+39
+
+14
+S1.
+TOPLOGICAL HEAVY-FERMION MODEL
+The topological heavy-fermion (THF) model introduced in Ref. [1] takes the following Hamiltonian
+ˆH = ˆHc + ˆHfc + ˆHU + ˆHJ + ˆHW + ˆHV + ˆHµ
+(S12)
+The single-particle Hamiltonian of conduction c-electrons has the form of
+ˆHc =
+�
+η,s,a,a′,|k|<Λc
+H(c,η)
+a,a′ (k)c†
+kaηscka′ηs
+,
+H(c,η)(k) =
+�
+02×2
+v⋆(ηkxσ0 + ikyσz)
+v⋆(ηkxσ0 − ikyσz)
+Mσx.
+�
+(S13)
+where σ0,x,y,z are identity and Pauli matrices. ckaηs represents the annihilation operator of the a(= 1, 2, 3, 4)-th conduction
+band basis of the valley η(= ±) and spin s(=↑, ↓) at the moir´e momentum k. At ΓM point (k = 0) of the moir´e Brillouin zone
+(MBZ), ck1ηs, ck2ηs form a Γ3 irreducible representation (of P6′2′2 group), ck3ηs, ck4ηs form a Γ1 ⊕ Γ2 reducible (into Γ1 and
+Γ2 - as they are written, the ck3ηs, ck4ηs are just the σx linear combinations of Γ1 ± Γ2 ) representation (of P6′2′2 group). Λc is
+the momentum cutoff for the c-electrons. N is the total number of moir´e unit cells. The parameter values are v⋆ = −4.303eV·˚A,
+M = 3.697meV.
+The hybridization between f and c electrons has the form of
+ˆHfc =
+1
+√NM
+�
+|k|<Λc
+R
+�
+αaηs
+�
+eik·R− |k|2λ2
+2
+H(fc,η)
+αa
+(k)f †
+Rαηsckaηs + h.c.
+�
+,
+(S14)
+where fRαηs represents the annihilation operators of the f electrons with orbital index α(= 1, 2), valley index η(= ±) and spin
+s(=↑, ↓) at the moir´e unit cell R. NM is the number of moir´e unit cells and λ = 0.3376aM is the damping factor, where aM is
+the moir´e lattice constant. The hybridization matrix H(fc,η) has the form of
+H(fc,η)(k) =
+�γσ0 + v′
+⋆(ηkxσx + kyσy), 02×2
+�
+(S15)
+which describe the hybridization between f electrons and Γ3 c electrons (a = 1, 2). The parameter values are γ = −24.75meV,
+v′
+⋆ = 1.622eV · ˚A.
+ˆHU (U = 57.89meV) describes the on-site interactions of f-electrons.
+ˆHU = U
+2
+�
+R
+: nf
+R :: nf
+R :,
+(S16)
+where nf
+R = �
+αηs f †
+RαηsfRαηs is the f-electrons density and the colon symbols represent the normal ordered operator with
+respect to the normal state: : f †
+Rα1η1s1fRα2η2s2 := f †
+Rα1η1s1fRα2η2s2 − 1
+2δα1η1s1;α2η2s2.
+The ferromagnetic exchange interaction between f and c electrons ˆHJ is defined as
+HJ = −
+J
+2NM
+�
+Rs1s2
+�
+αα′ηη′
+�
+|k1|,|k2|<Λc
+ei(k1−k2)·R(ηη′ + (−1)α+α′) : f †
+Rαηs1fRα′η′s2 :: c†
+k2,α′+2,η′s2ck1,α+2,ηs1 :
+(S17)
+where J = 16.38meV and : c†
+k2,α′+2,η′s2ck1,α+2,ηs1 := c†
+k2,α′+2,η′s2ck1,α+2,ηs1 − 1
+2δk1,k2δα,α′δη,η′δs1,s2
+The repulsion between f and c electrons ˆHW has the form of
+ˆHW =
+�
+η,s,η′,s′,a,α
+�
+|k|<Λc,|k+q|<Λc
+Wae−iq·R : f †
+R,aηsfR,aηs :: c†
+k+q,aη′s′ck,aη′s′ :
+(S18)
+where we take W1 = W2 = 44.03meV and W3 = W4 = 50.20meV.
+The Coulomb interaction between c electrons has the form of
+ˆHV =
+1
+2Ω0N
+�
+η1s1a1
+�
+η2s2a2
+�
+|k1|,|k2|<Λc
+�
+q
+|k1+q|,|k2+q|<Λc
+V (q) : c†
+k1a1η1s1ck1+qa1η1s1 :: c†
+k2+qa2η2s2ck2a2η2s2 :
+(S19)
+
+15
+where Ω0 is the area of the moir´e unit cell and V (q = 0)/Ω0 = 48.33meV. We will always treat ˆHV at mean-field level (int
+both the THF model and the Kondo lattice (KL) model) [1]
+ˆHV ≈ V (0)
+Ω0
+νc
+�
+|k|<Λc,a,η,s
+c†
+k,aηsck,aηs − V (0)
+2Ω0
+NMν2
+c + V (0)
+Ω0
+�
+|k|<Λc
+8νc
+(S20)
+where νc is the filling of c electrons νc =
+1
+NM
+�
+|k|<Λc,a,η,s⟨Ψ| : c†
+k,aηsck,aηs : |Ψ⟩ with |ψ⟩ the ground state.
+Finally, we introduce a chemical potential term
+ˆHµ = −µ
+�
+|k|<Λc,aηs
+c†
+k,aηsck,aηs − µ
+�
+R,αηs
+f †
+R,αηsfR,αηs .
+(S21)
+S2.
+KONDO LATTICE MODEL
+The Kondo lattice model is derived by performing a generalized Schrieffer-Wolff (SW) transformation on the topological
+heavy fermion model (detailed derivation in Ref. [2]). The Hamiltonian has the form of
+ˆHKondo = ˆHc + ˆHV + ˆHW + ˆHJ + ˆHK + ˆHcc − ˆHµc
+(S22)
+where ˆHc (Eq. S13), ˆHV (Eq. S19) and ˆHW (Eq. S18) and ˆHJ (Eq. S17) come from the original TFH model. The Kondo
+interactions and the one-body scattering term are
+ˆHK =
+�
+R,|k|<Λc,|k′|<Λc
+�
+α,α′,a,a′,η,η′,s,s′
+ei(k−k′)R−|k|2λ2/2−|k′|2λ2/2
+NMDνc,νf
+: f †
+R,αηsfR,α′η′s′ :: c†
+k′,a′η′s′ck,aηs :
+�
+γ2δα′,a′δα,a + γv′
+⋆δα,a[η′k′
+xσx − k′
+yσy]α′a′ + γv′
+⋆δα′,a′[ηkxσx + kyσy]αa
+�
+(S23)
+and
+ˆHcc = −
+�
+|k|<Λc,η,s
+�
+a,a′∈{1,2}
+e−|k|2λ2�
+1
+D1,νc,νf
++
+1
+D2,νc,νf
+� �
+γ2/2
+γv′
+⋆(ηkx − iky)
+γv′
+⋆(ηkx + iky)
+γ2/2
+�
+a,a′ : c†
+k,aηsck,a′ηs : (S24)
+where
+D1,νc,νf = (U − W)νf − U
+2 + (−V0
+Ω0
++ W)νc
+,
+D2,νc,νf = (U − W)νf + U
+2 + (−V0
+Ω0
++ W)νc
+Dνc,νf =
+�
+−
+1
+D1,νc,νf
++
+1
+D2,νc,νf
+�−1
+.
+(S25)
+We point out that, at ν = νf = νc = 0, D1,νc,νf = −D2,νc,νf and the on-body term ˆHcc(= 0) vanishes.
+We note that in the Kondo model the filling of f electron at each site is fixed to be νf. Then we can replace �
+αηs :
+f †
+R,αηsfR,αηs : with νf and ˆHW becomes
+ˆHW =
+�
+|k|<Λc,|k′|<Λc,aηs
+�
+Q
+Wνf : c†
+k,aη′s′ck′,aηsδk,k′+Q
+(S26)
+where Q ∈ {mbM1 + mbM2|m, n ∈ Z} and bM1, bM2 are the reciprocal lattice vectors. If we focus on the conduction
+electrons within the first MBZ, we can replace δk,k′+Q by δk,k′ and
+ˆHW =
+�
+|k|<Λc,aηs
+�
+Q
+Wνf : c†
+k,aη′s′ck,aηs
+(S27)
+which is a chemical shift of conduction electrons. We also set W1 = W2 = W3 = W4 = W = 47.12meV in ˆHW to simplify
+the SW transformation. The realistic values of W1,2,3,4 are not identical but the difference is about 15%.
+Finally, we introduce a chemical potential µc to tune the filling of the system
+ˆHµc = −µc
+�
+|k|<Λc,aηs
+: c†
+k,aηsck,aηs :
+(S28)
+
+16
+S3.
+SYMMETRY
+We now provide the symmetry transformation of electron operators.
+For a given symmetry operation g, we let
+Df(g), Dc′(g), Dc′′(g) denote the representation matrix of f-, Γ3 c- and Γ1 ⊕ Γ2 c-electrons:
+gf †
+R,αηsg−1 =
+�
+α′η′s′
+f †
+gR,α′η′s′Df(g)α′η′s′,αηs
+gc†
+k,aηsg−1 =
+�
+a′∈{1,2},η′s′
+c†
+gk,a′η′s′Dc′(g)a′η′s′,aηs,
+a ∈ {1, 2}
+gc†
+k,aηsg−1 =
+�
+a′∈{3,4},η′s′
+c†
+gk,a′η′s′Dc′′(g)a′+2η′s′,a+2ηs,
+a ∈ {3, 4}
+(S29)
+We consider the following symmetry operations as given in Ref. [1].
+T, C3z, C2x, C2zT
+(S30)
+with the following representation matrices
+T :
+Df(T) = σ0τxς0,
+Dc′(T) = σ0τxς0,
+Dc′′(T) = σ0τxς0
+C3z :
+Df(C3z) = ei 2π
+3 σzτzς0,
+Dc′(C3z) = ei 2π
+3 σzτzς0,
+Dc′′(C3z) = σ0τ0ς0
+C2x :
+Df(C2x) = σxτ0ς0,
+Dc′(C2x) = σxτ0ς0,
+Dc′′(C2x) = σxτ0ς0
+C2zT :
+Df(C2xT) = σxτ0ς0,
+Dc′(C2xT) = σxτ0ς0,
+Dc′′(C2zT) = σxτ0ς0
+(S31)
+where σx,y,z,0, τx,y,z,0, ςx,y,z,0 are Pauli or identity matrices of orbital, valley and spin degrees of freedom respectively.
+At M ̸= 0, v′
+⋆ ̸= 0, we also have U(1)c charge symmetry, U(1)v valley symmetry and SU(2)η spin symmetry for each
+valley η. We also mention that at M = 0, we have an enlarged flat U(4) symmetry and at v′
+⋆ = 0 we have an enlarged
+chiral U(4) symmetry [1, 2]. At M = 0, v′
+⋆ = 0, we have a U(4) × U(4) symmetry [1, 2]. Here, we consider the case of
+M ̸= 0, v′
+⋆ ̸= 0, where we only have a U(1)c ×U(1)v ×SU(2)η=+ ×SU(2)η=− symmetry. We comment that M = 3.698meV
+is relatively small and we have an approximate flat U(4) symmetry. Under U(1)c transformation gU(1)c(θc) (characterized
+by a real number θc), U(1)v transformation gU(1)v(θv) (characterized by a real number θv) and SU(2)η spin transformation
+gSU(2)η(θµ
+η ) (characterized by three real numbers θµ
+η , µ ∈ {x, y, z} ), we have
+U(1)c :
+Df(gU(1)c((θc)) = e−iθcσ0τ0ς0,
+Dc′(gU(1)c((θc)) = e−iθcσ0τ0ς0,
+Dc′′(gU(1)c((θc)) = e−iθcσ0τ0ς0
+U(1)v :
+Df(gU(1)v((θv)) = σ0e−iθvτzς0,
+Dc′(gU(1)v((θv)) = σ0e−iθvτzς0,
+Dc′′(gU(1)v((θv)) = σ0e−iθvτzς0
+SU(2)η :
+Df(gSU(2)η(θµ
+η )) = σ0e−i �
+µ θη
+µ
+τ0+ητz
+4
+ςµ,
+Dc′(gSU(2)η(θµ
+η )) = σ0e−i �
+µ θη
+µ
+τ0+ητz
+4
+ςµ,
+Dc′′(gSU(2)η(θµ
+η )) = σ0e−i �
+µ θη
+µ
+τ0+ητz
+4
+ςµ
+(S32)
+S4.
+MEAN-FIELD SOLUTIONS OF THE KONDO LATTICE MODEL
+The Kondo Hamiltonian in Eq. S22 contains two single-particle term ˆHc and ˆHcc and two interaction terms ˆHK + ˆHJ. We
+now discuss the mean-field decoupling of ˆHK, ˆHJ.
+
+17
+A.
+Mean-field decoupling of ˆHK
+We treat the interaction terms via mean-field decoupling
+ˆHK ≈ ˆHMF
+K
+=
+�
+R,|k|<Λc,|k′|<Λc
+�
+α,α′,a,a′,η,η′,s,s′
+ei(k−k′)R−|k|2λ2/2−|k′|2λ2/2
+NMDνc,νf
+�
+γ2δα′,a′δα,a + γv′
+⋆δα,a[η′k′
+xσx − k′
+yσy]α′a′ + γv′
+⋆δα′,a′[ηkxσx + kyσy]αa
+�
+�
+⟨f †
+R,αηsck,aηs⟩⟨c†
+k′,a′η′s′fR,α′η′s′⟩ − ⟨f †
+R,αηsck,aηs⟩c†
+k′,a′η′s′fR,α′η′s′ − f †
+R,αηsck,aηs⟨c†
+k′,a′η′s′fR,α′η′s′⟩
+− ⟨: f †
+R,αηsfR,α′η′s′ :⟩⟨: c†
+k′,a′η′s′ck,aηs :⟩ + ⟨: f †
+R,αηsfR,α′η′s′ :⟩ : c†
+k′,a′η′s′ck,aηs : + : f †
+R,αηsfR,α′η′s′ : ⟨: c†
+k′,a′η′s′ck,aηs :⟩
+�
+(S33)
+where for an operator O, ⟨O⟩ = ⟨Ψ|O|Ψ⟩ with |Ψ⟩ the mean-field ground state.
+1.
+Fock term
+We first consider the Fock term (F.T.), which takes the form of
+F.T. =
+�
+R,|k|<Λc,|k′|<Λc
+�
+α,α′,a,a′,η,η′,s,s′
+ei(k−k′)R−|k|2λ2/2−|k′|2λ2/2
+NMDνc,νf
+�
+γ2δα′,a′δα,a + γv′
+⋆δα,a[η′k′
+xσx − k′
+yσy]α′a′ + γv′
+⋆δα′,a′[ηkxσx + kyσy]αa
+�
+�
+⟨f †
+R,αηsck,aηs⟩⟨c†
+k′,a′η′s′fR,α′η′s′⟩ − ⟨f †
+R,αηsck,aηs⟩c†
+k′,a′η′s′fR,α′η′s′ − f †
+R,αηsck,aηs⟨c†
+k′,a′η′s′fR,α′η′s′⟩
+�
+=
+�
+R
+1
+Dνc,νf
+�
+γ2⟨
+�
+|k|<Λc
+�
+αηs
+eik·R−|k|2λ2/2
+√NM
+f †
+R,αηsck,aηs⟩⟨
+�
+|k′|<Λc
+�
+α′η′s′
+e−ik′·R−|k′|2λ2/2
+√NM
+c†
+k′,α′η′s′fR,α′η′s′⟩
+−
+�
+γ2 �
+|k|<Λc
+�
+αηs
+eik·R−|k|2λ2/2
+√NM
+f †
+R,αηsck,aηs⟨
+�
+|k′|<Λc
+�
+α′η′s′
+e−ik′·R−|k′|2λ2/2
+√NM
+c†
+k′,α′η′s′fR,α′η′s′⟩ + h.c.
+�
++ γv′
+⋆⟨
+�
+|k|<Λc
+�
+αηs
+eik·R−|k|2λ2/2
+√NM
+f †
+R,αηsck,αηs⟩⟨
+�
+|k′|<Λc
+�
+a′α′η′s′
+e−ik′·R−|k′|2λ2/2[η′k′
+xσx − k′
+yσy]α′a′
+√NM
+c†
+k′,a′η′s′fR,α′η′s′⟩
++ γv′
+⋆⟨
+�
+|k|<Λc
+�
+αaηs
+eik·R−|k|2λ2/2[ηkxσx + kyσy]αa
+√NM
+f †
+R,αηsck,αηs⟩⟨
+�
+|k′|<Λc
+�
+α′η′s′
+e−ik′·R−|k′|2λ2/2
+√NM
+c†
+k′,α′η′s′fR,α′η′s′⟩
+− γv′
+⋆
+� �
+|k|<Λc
+�
+αaηs
+eik·R−|k|2λ2/2[ηkxσx + kyσy]αa
+√NM
+f †
+R,αηsck,αηs⟨
+�
+|k′|<Λc
+�
+α′η′s′
+e−ik′·R−|k′|2λ2/2
+√NM
+c†
+k′,α′η′s′fR,α′η′s′⟩
++
+�
+|k|<Λc
+⟨
+�
+αaηs
+eik·R−|k|2λ2/2[ηkxσx + kyσy]αa
+√NM
+f †
+R,αηsck,αηs⟩
+�
+|k′|<Λc
+�
+α′η′s′
+e−ik′·R−|k′|2λ2/2
+√NM
+c†
+k′,α′η′s′fR,α′η′s′ + h.c.
+��
+(S34)
+
+18
+We introduce the following mean-field expectation values
+V1 =
+�
+R,|k|<Λc
+�
+αηs
+eik·R−|k|2λ2/2
+NM
+√NM
+⟨Ψ|f †
+R,αηsck,αηs|Ψ⟩
+V2 =
+�
+R,|k|<Λc
+�
+αaηs
+eik·R−|k|2λ2/2
+NM
+√NM
+(ηkxσx + kyσy)αa⟨Ψ|f †
+R,αηsck,aηs|Ψ⟩
+(S35)
+and assume the ground state is translational invariant such that
+�
+|k|<Λc
+�
+αηs
+eik·R−|k|2λ2/2
+√NM
+⟨Ψ|f †
+R,αηsck,αηs|Ψ⟩ =
+1
+NM
+�
+R,|k|<Λc
+�
+αηs
+eik·R−|k|2λ2/2
+√NM
+⟨Ψ|f †
+R,αηsck,αηs|Ψ⟩ = V1
+�
+|k|<Λc
+�
+αaηs
+eik·R−|k|2λ2/2
+√NM
+(ηkxσx + kyσy)αa⟨Ψ|f †
+r,αηsck,αηs|Ψ⟩
+= 1
+NM
+�
+R,|k|<Λc
+�
+αaηs
+eik·R−|k|2λ2/2
+√NM
+(ηkxσx + kyσy)αa⟨Ψ|f †
+r,αηsck,αηs|Ψ⟩ = V2
+(S36)
+Then the Fock term (Eq. S34) becomes
+F.T. = −
+γ2
+Dνc,νf
+�
+R,|k|<Λc
+�
+αη,s
+eik·R−|k|2λ2/2
+√NM
+�
+V ∗
+1 f †
+R,αηsck,αηs + h.c.
+�
++ NMγ2|V1|2
+Dνc,νf
+−
+γv′
+⋆
+Dνc,νf
+�
+R,|k|<Λc
+�
+α,a,η,s
+eik·R−|k|2λ2/2
+√NM
+�
+V ∗
+1 (ηkxσx + kyσy)αaf †
+R,αηsck,αηs + h.c.
+�
++ NMγv′
+⋆V ∗
+1 V2
+Dνc,νf
+−
+γv′
+⋆
+Dνc,νf
+�
+R,|k|<Λc
+�
+α,η,s
+eik·R−|k|2λ2/2
+√NM
+�
+V ∗
+2 f †
+R,αηsck,αηs + h.c.
+�
++ NMγv′
+⋆V ∗
+2 V1
+Dνc,νf
+(S37)
+2.
+Hartree term
+For the Hartree term (H.T.), we introduce the following density matrices Of, Oc′,1, Oc′,2, where Of have also been used in
+the mean-field calculations of the THF model as shown in Ref. [1] (however, Oc′,1, Oc′,2, V1, V2 are absent in the THF model)
+Of
+αηs,α′η′s′ =
+1
+NM
+�
+R
+⟨Ψ| : f †
+R,αηsfR,α′η′s′ : |Ψ⟩
+Oc′,1
+aηs,a′η′s′ =
+1
+NM
+�
+|k|<Λc
+e−|k|2λ2⟨Ψ| : c†
+k,aηsck,a′η′s′ : |Ψ⟩,
+a, a′ ∈ {1, 2}
+Oc′,2
+a′η′s′,αηs =
+1
+NM
+�
+|k|<Λc
+�
+a=1,2
+e−|k|2λ2(ηkxσx + kyσy)αa⟨Ψ| : c†
+k,a′η′s′ck,aηs : |Ψ⟩,
+a′, α ∈ {1, 2} .
+(S38)
+We then assume the ground state is translational invariance such that
+⟨Ψ| : f †
+R,αηsfR,α′η′s′ : |Ψ⟩ =
+1
+NM
+�
+R
+⟨Ψ| : f †
+R,αηsfR,α′η′s′ : |Ψ⟩ = Of
+αηs,α′η′s′ .
+(S39)
+
+19
+Using Eq. S38 and Eq. S39, the Hartree term can be written as
+H.T.
+=
+�
+R,
+|k|<Λc,|k′|<Λc
+�
+α,α′,a,a′,
+η,η′,s,s′
+ei(k−k′)R−|k|2λ2/2−|k′|2λ2/2
+NMDνc,νf
+�
+γ2δα′,a′δα,a + γv′
+⋆δα,a[η′k′
+xσx − k′
+yσy]α′a′ + γv′
+⋆δα′,a′[ηkxσx + kyσy]αa
+�
+�
+− ⟨: f †
+R,αηsfR,α′η′s′ :⟩⟨: c†
+k′,a′η′s′ck,aηs :⟩ + ⟨: f †
+R,αηsfR,α′η′s′ :⟩ : c†
+k′,a′η′s′ck,aηs : + : f †
+R,αηsfR,α′η′s′ : ⟨: c†
+k′,a′η′s′ck,aηs :⟩
+�
+=
+�
+α,α′,
+η,η′,s,s′
+NM
+Dνc,νf
+�
+− γ2Of
+αηs,αη′s′Oc′1
+α′η′s′,αηs −
+�
+γv′
+⋆Of
+αηs,α′η′s′Oc′,2
+α′η′s′,αηs + h.c.
+��
++
+�
+|k|<Λc
+�
+α,α′,
+η,η′,s,s′
+�
+Of
+αηs,α′η′s′e−|k|2λ2 : c†
+k,a′η′s′ck,aηs : δα,aδα′,a′ +
+�
+γv′
+⋆Of
+αηs,α′η′s′δα,a[η′kxσx − kyσy]α′a′e−|k|2λ2
+: c†
+k,a′η′sck,aηs : +h.c.
+��
++
+�
+R
+�
+α,α′,
+η,η′,s,s′
+�
+: f †
+R,αηsfR,α′η′s′ : Oc′,1
+α′η′s′,αηs +
+�
+γv′
+⋆ : f †
+R,αηsfR′,α′η′s′ : Oc′,2
+α′η′s′,αηs + h.c.
+��
+(S40)
+3.
+Fock and Hartree terms
+Combining Fock and Hartree (Eq. S37 and Eq. S37) terms, we have
+ˆHK ≈ ˆHMF
+K
+=F.T. + H.T.
+= −
+γ2
+Dνc,νf
+�
+R,|k|<Λc
+�
+αη,s
+eik·R−|k|2λ2/2
+√NM
+��
+V ∗
+1 f †
+R,αηsck,αηs + h.c.
+��
++ NMγ2|V1|2
+Dνc,νf
+−
+γv′
+⋆
+Dνc,νf
+�
+R,|k|<Λc
+�
+α,a,η,s
+eik·R−|k|2λ2/2
+√NM
+��
+V ∗
+1 (ηkxσx + kyσy)αaf †
+R,αηsck,αηs + h.c.
+��
++ NMγv′
+⋆V ∗
+1 V2
+Dνc,νf
+−
+γv′
+⋆
+Dνc,νf
+�
+R,|k|<Λc
+�
+α,η,s
+eik·R−|k|2λ2/2
+√NM
+��
+V ∗
+2 f †
+R,αηsck,αηs + h.c.
+�
+− V ∗
+2 V1
+�
++ NMγv′
+⋆V ∗
+2 V1
+Dνc,νf
++
+�
+α,α′,
+η,η′,s,s′
+NM
+Dνc,νf
+�
+− γ2Of
+αηs,αη′s′Oc′1
+α′η′s′,αηs −
+�
+γv′
+⋆Of
+αηs,α′η′s′Oc′,2
+α′η′s′,αηs + h.c.
+��
++
+�
+|k|<Λc
+�
+α,α′,
+η,η′,s,s′
+�
+Of
+αηs,α′η′s′e−|k|2λ2 : c†
+k,a′η′s′ck,aηs : δα,aδα′,a′ +
+�
+γv′
+⋆Of
+αηs,α′η′s′δα,a[η′kxσx − kyσy]α′a′e−|k|2λ2
+: c†
+k,a′η′sck,aηs : +h.c.
+��
++
+�
+R
+�
+α,α′,
+η,η′,s,s′
+�
+: f †
+R,αηsfR,α′η′s′ : Oc′,1
+α′η′s′,αηs +
+�
+γv′
+⋆ : f †
+R,αηsfR′,α′η′s′ : Oc′,2
+α′η′s′,αηs + h.c.
+��
+(S41)
+V1, V2 describes the Fock contribution that characterize the hybridization between f- and Γ3 c-electrons. Of, Oc′,1, Oc′,2 are
+the mean fields taking the form of ⟨f †f⟩, ⟨c†c⟩ which represent the Fock contribution.
+
+20
+B.
+Mean-field decoupling of ˆHJ
+We now perform a mean-field decoupling of the ferromagnetic exchange coupling term [1]
+ˆHJ ≈ ˆHMF
+J
+= −
+J
+2NM
+�
+R
+�
+αα′ηη′,ss′
+�
+|k|,|k′|<Λc
+ei(k−k′)·R(ηη′ + (−1)α+α′)
+�
+⟨f †
+R,αηsck′,α+2,ηs⟩⟨c†
+k′,α′+2,η′s′fR,α′η′s′⟩
+− ⟨f †
+R,αηsck,α+2,ηs1⟩c†
+k′,α′+2,η′s′fR,α′η′s′ − f †
+R,αηs1ck,α+2,ηs⟨c†
+k′,α′+2,η′s′fR,α′η′s′⟩
+− ⟨: f †
+R,αηsfR,α′η′s′ :⟩⟨: c†
+k′,α′+2,η′s′ck,α+2,ηs :⟩+ : f †
+R,αηsfR,α′η′s′ : ⟨: c†
+k′,α′+2,η′s′ck,α+2,ηs :⟩
++ ⟨: f †
+R,αηsfR,α′η′s′ :⟩ : c†
+k′,α′+2,η′s′ck,α+2,ηs :
+�
+(S42)
+1.
+Fock term
+The Fock term takes the form of
+F.T. = −
+J
+2NM
+�
+R
+�
+αα′ηη′,ss′
+�
+|k|,|k′|<Λc
+ei(k−k′)·R(ηη′ + (−1)α+α′)
+�
+⟨f †
+R,αηsck′,α+2,ηs⟩⟨c†
+k′,α′+2,η′s′fR,α′η′s′⟩
+− ⟨f †
+R,αηsck,α+2,ηs1⟩c†
+k′,α′+2,η′s′fR,α′η′s′ − f †
+R,αηs1ck,α+2,ηs⟨c†
+k′,α′+2,η′s′fR,α′η′s′⟩
+�
+= − J
+�
+R
+�
+ξ=±
+�
+⟨
+�
+|k′|<Λc,αηs
+ei(−k′)·R
+√NM
+δξ,η(−1)α+1f †
+R,αηsck′,α+2,ηs⟩⟨
+�
+|k|<Λc,α′η′s′
+δξ,η′(−1)α′+1 eik·R
+√NM
+c†
+k′,α′+2,η′s′fR,α′η′s′⟩
+−
+�
+|k|<Λc,αηs
+ei(−k′)·R
+√NM
+δξ,η(−1)α+1f †
+R,αηsck′,α+2,ηs⟨
+�
+|k′|<Λc,αηs
+δξ,η′(−1)α′+1 eik·R
+√NM
+�
+α′η′s′
+c†
+k′,α′+2,η′s′fR,α′η′s′⟩
+− ⟨
+�
+k′,αηs
+ei(−k′)·R
+√NM
+δξ,η(−1)α+1f †
+R,αηsck′,α+2,ηs⟩
+�
+|k|<Λc,α′η′s′
+eik·R
+√NM
+δξ,η′(−1)α′+1c†
+k′,α′+2,η′s′fR,α′η′s′
+�
+(S43)
+We then introduce the following mean-fields
+V3 =
+�
+R,|k|<Λc
+�
+αη,s
+eik·Rδ1,η(−1)α+1
+NM
+√NM
+⟨Ψ|f †
+R,αηsck,α+2ηs|Ψ⟩
+V4 =
+�
+R,|k|<Λc
+�
+αη,s
+eik·Rδ−1,η(−1)α+1
+NM
+√NM
+⟨Ψ|ηf †
+R,αηsck,α+2ηs|Ψ⟩
+(S44)
+and assume the ground state is translational invariant such that
+�
+|k|<Λc
+�
+αη,s
+eik·Rδ1,η(−1)α+1
+√NM
+⟨Ψ|f †
+R,αηsck,α+2ηs|Ψ⟩ =
+1
+NM
+�
+R
+�
+|k|<Λc
+�
+αη,s
+eik·Rδ1,η(−1)α+1
+√NM
+⟨Ψ|f †
+R,αηsck,α+2ηs|Ψ⟩ = V3
+�
+|k|<Λc
+�
+αη,s
+eik·Rδ−1,η(−1)α+1
+√NM
+⟨Ψ|f †
+R,αηsck,α+2ηs|Ψ⟩ =
+1
+NM
+�
+R
+�
+|k|<Λc
+�
+αη,s
+eik·Rδ−1,η(−1)α+1
+√NM
+⟨Ψ|f †
+R,αηsck,α+2ηs|Ψ⟩ = V4
+(S45)
+Then the Fock term can be written as
+F.T. = − JNM[|V3|2 + |V4|2]
++ J
+�
+R,|k|<Λc,αηs
+�ei(−k′)·R
+√NM
+�
+δ1,η(−1)α+1f †
+R,αηsck′,α+2,ηsV ∗
+3 + δ−1,η(−1)α+1f †
+R,αηsck′,α+2,ηsV ∗
+4 +
+�
++ h.c.
+�
+(S46)
+
+21
+2.
+Hartree term
+The Hartree term takes the form of
+H.T. = −
+J
+2NM
+�
+R
+�
+αα′ηη′,ss′
+�
+|k|,|k′|<Λc
+ei(k−k′)·R(ηη′ + (−1)α+α′)
+�
+− ⟨: f †
+R,αηsfR,α′η′s′ :⟩⟨: c†
+k′,α′+2,η′s′ck,α+2,ηs :⟩
++ : f †
+R,αηsfR,α′η′s′ : ⟨: c†
+k′,α′+2,η′s′ck,α+2,ηs :⟩ + ⟨: f †
+R,αηsfR,α′η′s′ :⟩ : c†
+k′,α′+2,η′s′ck,α+2,ηs :
+�
+(S47)
+We introduce the following density matric which has also been used in Ref. [1]
+Oc′′
+aηs,a′η′s′ =
+1
+NM
+�
+|k|<Λc
+⟨Ψ| : c†
+k,a+2ηsck,a′+2η′s′ : |Ψ⟩,
+a, a′ ∈ {1, 2} .
+(S48)
+Using Eq. S38 and Eq. S48, the Hartree term becomes
+H.T. = − JNM
+2
+�
+αα′ηη′ss′
+(ηη′ + (−1)α+α′)Of
+αηs,α′η′s′Oc′′
+α′η′s′,αηs
++ J
+2
+�
+Rc,αα′ηη′ss′
+: f †
+R,αηsfR,α′η′s′ : Oc′′
+α′η′s′,αηs + J
+2
+�
+|k|<Λc,αα′ηη′ss′
+Of
+αηs,α′η′s′ : c†
+k′,α′+2,η′s′ck,α+2,ηs :
+(S49)
+3.
+Fock and Hartree terms
+Combing Hartree and Fock terms (Eq. S46 and Eq. S49), we have
+ˆHJ ≈ ˆHMF
+J
+= − JNM
+�
+ξ=±
+|V3|2 + |V4|2
++ J
+�
+R,|k|<Λc,αηs
+�ei(−k′)·R
+√NM
+�
+δ1,η(−1)α+1f †
+R,αηsck′,α+2,ηsV ∗
+3 + δ−1,η(−1)α+1f †
+R,αηsck′,α+2,ηsV ∗
+4
+�
++ h.c.
+�
+− JNM
+2
+�
+αα′ηη′ss′
+(ηη′ + (−1)α+α′)Of
+αηs,α′η′s′Oc′′
+α′η′s′,αηs
++ J
+2
+�
+Rc,αα′ηη′ss′
+: f †
+R,αηsfR,α′η′s′ : Oc′′
+α′η′s′,αηs + J
+2
+�
+|k|<Λc,αα′ηη′ss′
+Of
+αηs,α′η′s′ : c†
+k′,α′+2,η′s′ck,α+2,ηs :
+(S50)
+V3, V4 describes the Fock contribution that characterize the hybridization between f- and Γ1 ⊕ Γ2 c-electrons. Of, Oc′′ are the
+mean fields taking the form of ⟨f †f⟩, ⟨c†c⟩ which represent the Fock contribution and have also been used in Ref. [1].
+C.
+Filling constraints and mean-field equations
+We note that in the Kondo model the filling of f electrons is fixed to be νf at each site. To simplify the calculation, we take
+a common approximation that only requires the average filling of f-electron to be νf [3, 138]. In other words, we only require
+1
+NM
+�
+R,αηs⟨Ψ| : f †
+R,αηsfR,αηs : |Ψ⟩ = νf. We then add the following term to the Hamiltonian
+ˆHλf =
+�
+R,αηs
+λf
+�
+: f †
+R,αηsfR,αηs : −νf
+�
+(S51)
+and determine the Langrangian multiplier λf from the following equation
+1
+NM
+�
+R,αηs
+⟨Ψ| : f †
+R,αηsfR,αηs : |Ψ⟩ = νf
+(S52)
+
+22
+In practice, we perform calculations at fixed total filling ν = νf + νc, where νf and νc are the average fillings of f and c
+electrons respectively. Since νf is also fixed in the Kondo model, we will self-consistently determine the chemical potential µc
+(in Eq. S28) by requiring
+1
+NM
+�
+|k|<Λc,aηs
+⟨Ψ| : c†
+k,aηsck,aηs : |Ψ⟩ = νc = ν − νf
+(S53)
+Finally, our mean-field Hamiltonian takes the form of
+ˆHMF = ˆHc + ˆHcc + ˆHMF
+K
++ ˆHMF
+J
++ ˆHλf + ˆHµc
+(S54)
+and we determine V1, V2, V3, V4, Of, Oc′,1, Oc′,2, Oc′′, λf, µc from the self-consistent equations (Eq. S35, Eq. S38, Eq. S44,
+Eq. S48, Eq. S52, Eq. S53). During the self-consistent solution, at each step, we will adjust λf, µc according to the current
+filling of f- and c-electrons. We use νi
+f and νi
+c denote the filling of f and c at i-th step. For the i + 1-th step, we will update
+λf, µc as λf → λf + r(νi
+f − νf), µc → µc − r(νi
+c − νc), where r(> 0) will be manually adjusted to improve the convergence
+(in practice, we take r ∼ 0.001).
+D.
+Mean-field equations of the symmetric Kondo state
+We focus on the symmetric Kondo phase without any symmetry breaking. Therefore, we require our density matrix of f-
+and c- electrons (Eq. S38, Eq. S48) to be symmetric. We can then utilize symmetry to simplify the self-consistent equations
+(Eq. S38, Eq. S48). We first consider the U(1)v symmetry. From Eq. S32, a U(1)v symmetric solution satisfies
+Of
+αηs,α′η′s′ = Of
+αηs,α′η′s′e−iθν(η−η′) ⇒ Of
+αηs,α−ηs′ = 0
+Oc′,1
+aηs,a′η′s′ = Oc′,1
+aηs,a′η′s′e−iθν(η−η′) ⇒ Oc′,1
+aηs,a′−η′s′ = 0
+Oc′,2
+aηs,α′η′s′ = Oc′,2
+aηs,α′η′s′e−iθν(η−η′) ⇒ Oc′,2
+aηs,α′−η′s′ = 0
+Oc′′
+aηs,a′η′s′ = Oc′′
+aηs,a′η′s′e−iθν(η−η′) ⇒ Oc′′
+aηs,a′−ηs′ = 0
+(S55)
+and V1, V2, V3, V4 are invariant under U(1)v transformation. From Eq. S55, Of, Oc′,1, Oc′,2, Oc′′ are block diagonalized in
+valley index. We next consider a SU(2)η transformation acting on the valley η. We find
+�
+s,s′
+[ei �
+µ θη
+µσµ]s2,sOf
+αηs,α′ηs′[ei �
+µ θη
+µσµ]s′,s′
+2 = Of
+αηs2,α′ηs′
+2 ⇒ Of
+aηs,a′ηs′ ∝ Is,s′
+�
+s,s′
+[ei �
+µ θη
+µσµ]s2,sOc′,1
+aηs,a′ηs′[ei �
+µ θη
+µσµ]s′,s′
+2 = Oc′,1
+aηs,a′ηs′ ⇒ Oc′,1
+aηs,a′ηs′ ∝ Is,s′
+�
+s,s′
+[ei �
+µ θη
+µσµ]s2,sOc′,2
+aηs,a′ηs′[ei �
+µ θη
+µσµ]s′,s′
+2 = Oc′,2
+aηs,a′ηs′ ⇒ Oc′,2
+aηs,a′ηs′ ∝ Is,s′
+�
+s,s′
+[ei �
+µ θη
+µσµ]s2,sOc′′
+aηs,a′ηs′[ei �
+µ θη
+µσµ]s′,s′
+2 = Oc′′
+aηs,a′ηs′ ⇒ Oc′′
+aηs,a′ηs′ ∝ Is,s′
+(S56)
+where I is an 2 × 2 identical matrix. In addition, V1, V2, V3, V4 are invariant under SU(2)η transformation.
+From Eq. S55 and Eq. S56, the density matrices Of, Oc′,1, Oc′′ are diagonalized in valley and spin incdies. We then introduce
+2 × 2 matrices, of,η, oc′,1,η, oc′,2,η, oc′′,η, such that
+Of
+αηs,α′η′s′ = of,η
+α,α′δη,η′δs,s′,
+Oc′,1
+aηs,a′η′s′ = oc′,1,η
+a,a′ δη,η′δs,s′,
+Oc′,2
+aηs,a′η′s′ = oc′,2,η
+a,a′ δη,η′δs,s′,
+Oc′′
+aηs,a′η′s′ = oc′′,η
+a,a′ δη,η′δs,s′
+(S57)
+We now consider the effect of discrete symmetries in Eq. S30. Using Eq. S31 and Eq. S57, we find
+T :
+(of,η)∗ = of,−η,
+(oc′,1,η)∗ = oc′,1,−η,
+(oc′,2,η)∗ = oc′,2,−η,
+(oc′′,η)∗ = oc′′,−η
+C3z :
+ei 2πη
+3 σzof,ηe−i 2πη
+3 σz = of,η,
+ei 2πη
+3 σzoc′,1,ηe−i 2πη
+3 σz = oc′,1,η,
+ei η2π
+3 oc′,2,ηe−i η2π
+2 σz = oc′,2,η,
+oc′′,η = oc′′,η
+C2x :
+σxof,ησx = of,η,
+σxoc′,1,ησx = oc′,1,η,
+σxoc′,2,ησx = oc′,2,η,
+σxoc′′,ησx = oc′′,η
+C2zT :
+(σxof,ησx)∗ = of,η,
+(σxoc′,1,ησx)∗ = oc′,1,η,
+(σxoc′,2,ησx)∗ = −oc′,2,−η,
+(σxoc′′,ησx)∗ = oc′′,η
+(S58)
+
+23
+From the definition (Eq. S38), Of, Oc′,1, Oc′′ are Hermitian matrices and then of, oc′,1, oc′′ are also Hermitian matrices. Com-
+bining Eq. S58 and the Hermitian properties, we can introduce real numbers χf
+0, χc′,1
+0
+, χc′′
+0 , χc′′
+1
+and then of, oc′, oc′′ take the
+following structure
+of,η = of,−η = χf
+0σ0,
+oc′,1,η = oc′,−η = χc′,1
+0
+σ0,
+oc′,2,η = 0,
+oc′′,η = χc′′
+0 σ0 + χc′′
+1 σx
+(S59)
+where σ0, σx,y,z are identity and Pauli matrices respectively with row and column indices α = 1, 2. Since the filling of f- and
+Γ1 ⊕ Γ2 c- electrons are νf = Tr[Of], νc′′ = Tr[Oc′′] respectively, we find χf
+0 = νf/8, χc′′
+0
+= νc′′/8. Using Eq. S59 and
+Eq. S57, for the symmetric solution, we finally have
+Of
+αηs,α′η′s′ = δα,α′δη,η′δs,s′νf/8
+Oc′,1
+aηs,a′η′s′ = δa,a′δη,η′δs,s′χc′,1
+0
+,
+Oc′,2
+aηs,a′η′s′ = 0,
+a, a′ ∈ {1, 2}
+Oc′′
+aηs,a′η′s′ = δη,η′δs,s′(δa,a′νc′′/8 + δa,3−a′χc′′
+x ),
+a, a′ ∈ {1.2}
+(S60)
+As for the hybridization fields, we find discrete symmetries will not impose constraints on V1, V2. As for V3, V4, we have
+T : V3 = V ∗
+4 ;
+C3z : V3 = ei2π/3V3,
+V4 = ei2π/3V4
+C2x : V3 = V4;
+C2zT : V3 = V ∗
+4
+(S61)
+therefore
+V3 = V4 = 0
+(S62)
+In summary, instead of solving self-consistent equations of Of, Oc′,1, Oc′′, V3, V4 (Eq. S38, Eq. S44, Eq. S48), we can use
+νf =
+1
+NM
+�
+R,αηs
+⟨Ψ| : f †
+R,αηsfR,αηs : |Ψ⟩
+χc′,1
+0
+=
+1
+8NM
+�
+|k|<Λc,a=1,2,ηs
+⟨Ψ|e−|k|2λ2 : c†
+k,aηsck,aηs : |Ψ⟩
+νc′′ =
+1
+NM
+�
+|k|<Λc,a=3,4,ηs
+⟨Ψ| : c†
+k,aηsck,aηs : |Ψ⟩
+χc′′
+1 =
+1
+8NM
+�
+|k|<Λc,ηs
+⟨Ψ|c†
+k,3ηsck,4ηs + c†
+k,4ηsck,3ηs|Ψ⟩
+V3 = V4 = 0
+(S63)
+and obtain Of, Oc, Oc′′ via Eq. S60. We note that the first equation in Eq. S63 is equivalent to Eq. S52. In summary, combining
+Eq. S35, Eq. S53 and Eq. S63, we have a complete set of mean-field self-consistent equations for the symmetric Kondo state.
+We comment that Eq. S63 are the same mean-field equations as we derived in Sec. S4 A, Sec. S4 B and Sec. S4 C, but with
+additional symmetry requirement, that is the ground states satisfy all symmetries.
+We mention that, at ν = νf = νc = 0, we have Of
+αηs,α′η′s′ = 0, Oc′
+αηs,α′η′s′ = 0 and the Hartree term in Eq. S41 vanishes.
+We now prove the Hartree term in Eq. S50 also vanishes. We note the only non-zero components of Oc′′ are Oc′′
+1ηs,2ηs, Oc′′
+2ηs,1ηs.
+From Eq. S49, the Hartree term takes the form of (with Of = 0)
+−
+J
+2NM
+�
+Rs1s2
+�
+αα′ηη′
+�
+|k1|,|k2|<Λc
+ei(k1−k2)·R(ηη′ + (−1)α+α′)
+�
+: f †
+Rαηs1fRα′η′s2 : Oc′′
+α′η′s2,αηs1
+�
+= −
+J
+2NM
+�
+Rs
+�
+αη
+�
+|k1|,|k2|<Λc
+ei(k1−k2)·R(ηη + (−1)α+3−α)
+�
+: f †
+RαηsfRα′ηs : Oc′′
+3−αηs,αηs
+�
+= −
+J
+2NM
+�
+Rs
+�
+αη
+�
+|k1|,|k2|<Λc
+ei(k1−k2)·R(0)
+�
+: f †
+RαηsfRα′ηs : Oc′′
+3−αηs,αηs
+�
+=0
+(S64)
+and hence vanishes. In summary, at ν = 0, we only need to consider V1, V2, and other mean fields vanish.
+
+24
+E.
+Properties of the symmetric Kondo state
+We solve the self-consistent equations Eq. S35, Eq. S44, Eq. S52, Eq. S53 and Eq. S63 at integer filling ν = 0, −1, −2
+with νf
+= ν, νc = 0.
+We identify the symmetric Kondo (SK) states at ν = 0, −1, −2 which are characterized by
+V1 ̸= 0 (|γ2V1/Dνf ,νc| = 95meV, 111meV, 209meV at ν = 0, −1, −2 respectively), V2 ̸= 0 (|v′
+⋆γV2/Dνf ,νc| =
+80meV, 97meV, 197meV at ν = 0, −1, −2 respectively) and V3 = 0, V4 = 0. Even if we allow non-zero V3, V4 and ini-
+tialize the mean-field calculations with non-zero V3, V4, V3, V4, we still find V3 = V4 = 0 after self-consistent calculations
+(amplitudes smaller than 10−5). This is because ˆHJ describes ferromagnetic interactions and disfavors the development of non-
+zero V3, V4. We also comment that the non-zero V1, V2, introduce an effective f-c hybridization (Eq. S37) and characterize the
+Kondo physics.
+We next discuss the topological feature of the bands. Since the SK states preserve all the symmetries, it is sufficient to only
+consider the bands in valley + and spin ↑. We find at ν = 0, −1, −2, the representations formed by flat bands at Γ, K, M are
+Γ1 ⊕ Γ2, K2K3, M1 ⊕ M2 respectively. We note that the representations formed by flat bands here are equivalent to that of the
+non-interacting THF model. Thus, the flat bands form a fragile topology at ν = −1, −2 and a stable topology at ν = 0 due to
+the additional particle-hole symmetry at ν = 0 [1, 85].
+In addition, we also calculate the Wilson loop of the flat bands (valley + spin ↑). In the calculation of Wilson loop, we
+let k1 ∈ { i
+N }i=1,...,N−1, k2 = { j
+N }j=1,...,N−1 and k1 ∈ { i
+N }i=1,...,N−1, k2 = { j
+N }j=1,...,N−1 and k = k1bM,1 + k2bM,2,
+where bM,1 =
+4π
+3aM (
+√
+3, 0), bM,2 =
+4π
+3aM (
+√
+3
+2 , 3
+2) are two moir´e reciprocal lattice vectors and aM is the moir´e lattice constant.
+We then let |un,k⟩ denote the n-th eigenvectors of the single-particle Hamiltonian H(k) (of valley + spin ↑). We focus on
+the subset of the bands, which we denote with band indices n = 1, .., nband. Here, we take the flat bands as the subset of
+the bands that we are interested in. We then define the matrix Uk as a matrix formed by the eigenvectors of the flat bands
+Uk = [|u1,k⟩, |u2,k⟩, ..., |unband,k⟩]. The Wilson loop [85] along the k2 direction is defined as
+W(k1) = U †
+k1,k2=0
+N−1
+�
+j=1
+�
+Uk1,k2= 2πj
+N U †
+k1,k2= 2π(j+1)
+N
+�
+V (k1=0,k2=2π)Uk1,k2=2π .
+(S65)
+where V G is defined as H(k+G) = V GH(k)V G,†, G = nbM,1 +mbM,2, n, m ∈ Z. We mention that c-electrons are defined
+in the momentum space that can be larger than the first MBZ (depending on the momentum cutoff Λc). Thus we introduce
+V G that maps ck to ck+G to restore the periodic condition H(k + G) = V GH(k)V G,†. The corresponding Wilson loop
+Hamiltonian [85] is
+H(k1) = −i ln(W(k1))
+(S66)
+We plot the Wilson loop spectrum (eigenvalues of H(k1)) in Fig. S5, where we observe the Wilson loop has winding number 1.
+As shown in Ref. [85], in the presence of additional particle-hole symmetry at ν = 0 [1, 85], (−1)n with n the winding number
+of Wilson loop is a stable topological index. We conclude that at ν = 0, the symmetric Kondo state has a stable topology that is
+characterized by the odd winding number of the Wilson loop.
+From Fig. S5, we observe the behaviors of the Wilson loop are similar at different fillings. We check the overlapping of the
+flat-band wavefunctions between different bands. We let {|uν
+i,k⟩}i=1,...,nband denote the wavefunction of flat bands at filling ν.
+We define the overlapping between wavefunctions at fillings ν and ν′ as
+Overlap(ν, ν′) = 1
+N
+�
+k
+�
+i,j∈{1,...,nband}
+⟨uν
+i,k|uν′
+j,k⟩⟨uν′
+j,k|uν
+i,k⟩
+(S67)
+We find Overlap(0, −1) = 99.1%, Overlap(−1, −2) = 91.6%. The large overlapping of wavefunctions between different
+fillings indicates similar behaviors of the Wilson loop at different fillings as we showed in Fig. S5.
+Finally, we analyze the mean-field Hamiltonian of the symmetric Kondo state. The mean-field single-particle Hamiltonian of
+valley η and spin ↑ (spin ↑ and spin ↓ are equivalent) of the Kondo symmetric state can be approximately written as
+˜H(η)(k) =
+� ˜H(f,η)(k)
+˜H(fc,η)(k)
+˜H(fc,η),†(k)
+˜H(c,η)(k)
+�
+(S68)
+˜H(f,η)(k) = EfI2×2
+˜H(c,η)(k) =
+�
+EcI2×2
+v⋆(ηkxσ0 + ikyσz)
+v⋆(ηkxσ0 − ikyσz)
+Ec′′I2×2
+�
+˜H(fc,η)(k) =
+�
+˜γσ0 + ˜v′⋆(ηkxσx + kyσy) 02×2
+�
+(S69)
+
+25
+FIG. S5. Wilson loop spectrum of flat bands of SK states at ν = 0, −1, −2.
+where ˜H(f,η), ˜H(c,η), ˜H(fc,η) denote the single-particle Hamiltonian of the f-block, c-block and fc-block respectively.
+Ef, Ec, Ec′′ denote the energy shifting induced by the Hartree term, one-body scattering term ˆHcc and chemical potential.
+Ef, Ec can be k dependent and we only keep its k-independent part which makes dominant contributions. Ec′′ comes from
+the Hartree contribution of ˆHJ (Eq. S49, which is relatively small and we set Ec′′ = 0. We also set M = 0, since it is small
+compared to the other parameters. ˜γ, ˜v⋆
+′ denote the renormalized f-c hybridization emerged from Kondo interactions (Eq. S37).
+We also drop the damping factor e−|k|2λ2/2 to simplify the analysis. In practice, we find |˜γ| =
+1
+Dνc,νf |γ2V ∗
+1 + γv′
+⋆V ∗
+2 | ≈
+175meV, 209meV, 406meV. In the chiral limit v′
+⋆ = 0, we also have ˜v′⋆ = 0 ( ˜v′⋆ = v′
+⋆ = γv′
+⋆V ∗
+1 /Dνc,νf ). We note that
+|˜v′
+⋆||k| can reach a similar amplitude as the k-independent hybridization |˜γ|. However, we expect in most regions of MBZ, |˜γ|
+makes the dominant contribution. We, therefore, drop the set ˜v′⋆ = 0 or equivalently v′
+⋆ = 0 as an approximation. By setting
+v′
+⋆ = 0, we can further separate ˜H(η)(k) into two blocks. The first block corresponds to the row and column indices 1, 3, 5
+with electron operators,fk,1ηs, ck,1ηs, ck,3ηs. The second block corresponds to the row and column indices 2, 4, 6 with electron
+operators,fk,2ηs, ck,2ηs, ck,4ηs. We focus on the first block whose single-particle Hamiltonian is
+˜h(η)(k) =
+�
+�
+Ef
+˜γ
+0
+˜γ
+Ec
+v⋆(ηkx + iky)
+0
+v⋆(ηkx − iky)
+0
+�
+�
+(S70)
+We next analyze the eigensystems of ˜h(η)(k). We note that ˜γ provides the largest energy scales near ΓM point and will gap
+out f- and Γ3 c-electrons. To observe this, we first consider the first 2 × 2 block of ˜h(η)(k) which describes the single-particle
+Hamiltonian of f- and Γ3 c-electrons
+�
+Ef
+˜γ
+˜γ
+Ec
+�
+(S71)
+The eigenvalues and eigenvectors are
+E1 = Ec + Ef
+2
+−
+�
+˜γ2 + (Ec − Ef)2
+4
+,
+E2 = Ec + Ef
+2
++
+�
+˜γ2 + (Ec − Ef)2
+4
+v1 =
+1
+�
+2Efc(Efc − E3)
+�E3 − Efc ˜γ�T ,
+v2 =
+1
+�
+2Efc(Efc + E3)
+�E3 + Efc ˜γ�T
+(S72)
+where E3 = Ef −Ec
+2
+, Efc =
+�
+˜γ2 + E2
+3. Since ˜γ is larger than Ec, Ef, Γ3 c-electrons and f-electrons are gapped out by the
+hybridization. Consequently, the flat bands are mostly formed by Γ1 ⊕ Γ2 c-electrons. Numerically, we indeed find the orbital
+weights of Γ1 ⊕ Γ2 c-electrons are large (71%, 77%, 89% at ν = 0, −1, −2 respectively).
+We next treat v⋆ perturbatively. We find the dispersion of the flat band becomes
+Eflat
+k
+≈ −Ef|k|2(v⋆)2
+E1E2
+= Ef|k|2(v⋆)2
+˜γ2 − EcEf
+(S73)
+At ν = 0 with particle-hole symmetry, Ef = Ec = 0 and Eflat
+k
+≈ 0. However, at ν = −1, −2, where Ef ̸= 0, Ec ̸= 0, flat
+bands become dispersive. We observe that |Ec| is much smaller than |˜γ| at ν = −1, −2. Ef increases as we change from ν = 0
+to ν = −2, because we are doping more holes to the f-orbitals. At ν = −2, Ef can reach ∼ 0.5|˜γ|, but at ν = −1, Ef ∼ 0.1|˜γ|.
+Approximately, the dispersion of the flat band is Eflat
+k
+≈ (v⋆)2Ef/˜γ2. At ν = −1, we have Ek ≈ 13meV · ˚A2|k|2 and, at
+
+26
+ν = −2, we have Ek ≈ 45meV · ˚A2|k|2. This indicates a larger dispersion at ν = −2, which is consistent with our numerical
+result shown in the main text Fig.1.
+We next analyze the wavefunctions of the flat bands. The corresponding electron operator of the flat band d†
+flat,k is
+d†
+flat,k ≈ 1
+Ak
+�
+c†
+k,3ηs + v⋆(ηkx − iky)
+EcEf − ˜γ2
+�
+− Efc†
+k,1ηs + ˜γf †
+k,1ηs
+��
+(S74)
+where the normalization factor
+Ak =
+�
+1 +
+|v⋆|2|k|2(E2
+f + ˜γ2)
+(EcEf − ˜γ2)2
+(S75)
+We observe that in the large |˜γ| limit, the flat bands are mostly formed by Γ1 ⊕ Γ2 c-electrons (c†
+k,3ηs). We also provide the
+Berry curvature derived from the wavefunction in Eq. S74
+Ω(k) =
+−2(EcEf − ˜γ2)2(E2
+f + ˜γ2)v2
+⋆
+�
+(EcEf − ˜γ2)2 + (E2
+f + ˜γ2)v2⋆|k|2
+�2
+(S76)
+We next calculate the Wilson loop from the wavefunction in Eq. S74. The wavefunctions of d†
+flat,k is
+u(k) = 1
+Ak
+�
+1
+v⋆ηkx−iky
+EcEf −˜γ2 (−Ef)
+v⋆ηkx−iky
+EcEf −˜γ2 ˜γ
+�T
+(S77)
+where the first, second and third rows denote c†
+k,3ηs, f †
+k,1ηs, c†
+k,1ηs respectively. We then parametrize the momentum as
+k = x1aMbM,1 + x2aMbM,2,
+bM,1 =
+4π
+3aM
+(
+√
+3, 0),
+bM,2 =
+4π
+3aM
+(
+√
+3
+2 , 3
+2)
+(S78)
+x1, x2 ∈ [−1
+2, 1
+2] 1
+aM
+(S79)
+and define |u(x1, x2)⟩ as |u(k)⟩ with k = x1aMbM,1 + x2aMbM,2. The Wilson loop can be written as
+W(x1) =
+N−1
+�
+j=0
+⟨u(x1, x2 = x2,i)|u(x1, x2 = x2,j+1⟩⟨u(x1, x2,N)|u(x1, x2,0)⟩,
+x2,i = −
+1
+2aM
++
+1
+aM
+i
+N
+(S80)
+The spectrum of the Wilson loop is
+N(x1) = −i ln(W(x1)) = −i
+�
+1
+2aM
+−
+1
+2aM
+⟨u(x1, x2)|∂x2|u(x1, x2)⟩dx2 − i ln(⟨u(x1, 1/(2aM))|u(x1, −1/(2aM)⟩)
+(S81)
+In the continuous limit with aM → 0, we find
+N(x1) = −i
+� ∞
+−∞
+⟨u(x1, x2)|∂x2|u(x1, x2)⟩dx2 − i ln(⟨u(x1, ∞)|u(x1, −∞⟩)
+(S82)
+Combining Eq. S77 and Eq. S82, we find
+N(x1) =π
+�
+1 +
+v2
+⋆x1
+�
+x2
+1v2⋆ + (˜γ2−EcEf )2
+E2
+f +˜γ2
+�
+(S83)
+Even though Eq. S82 is calculated from the perturbative wavefunction in Eq. S77, it qualitatively captures the behaviors of the
+Wilson loop shown in Fig. S5. We observe that N(−∞) = 0 and N(∞) = 2π, which indicates a 2π winding at ν = 0, −1, −2.
+We also mention that current calculations correspond to one of the two flat bands for each valley and each spin, because we only
+pick one block of the single-particle Hamiltonian as we discussed near Eq. S70. The other flat band can be derived in the same
+manner and has similar behaviors, since it has a similar single-particle Hamiltonian.
+
+27
+S5.
+MEAN-FIELD SOLUTIONS OF THE TOPOLOGICAL HEAVY-FERMION MODEL
+We now discuss the mean-field equations of topological heavy-fermion mode in Eq. S12. We use a similar Hartree-Fock ap-
+proximation as introduced in Ref. [1]. However, we decouple ˆHJ via Eq. S50. The mean-field expectation values we considered
+are
+Of
+αηs,α′η′s′ =
+1
+NM
+�
+R
+⟨Ψ| : f †
+R,αηsfR,α′η′s′ : |Ψ⟩
+Oc′
+aηs,a′η′s′ =
+1
+NM
+�
+|k|<Λc
+⟨Ψ| : c†
+k,aηsck,a′η′s′ : |Ψ⟩,
+a ∈ {1, 2}
+Oc′′
+aηs,a′η′s′ =
+1
+NM
+�
+|k|<Λc
+⟨Ψ| : c†
+k,a+2ηsck,a′+2η′s′ : |Ψ⟩,
+a, a′ ∈ {1, 2}
+Oc′f
+aηs,α′η′s′ =
+1
+√
+NN
+�
+|k|<Λc,R
+e−ik·R⟨Ψ|c†
+k,aηsfR,α′η′s′|Ψ⟩,
+a ∈ {1, 2}
+V3 =
+�
+R,|k|<Λc
+�
+αη,s
+eik·Rδ1,η(−1)α+1
+NM
+√NM
+⟨Ψ|f †
+R,αηsck,α+2ηs|Ψ⟩
+V4 =
+�
+R,|k|<Λc
+�
+αη,s
+eik·Rδ−1,η(−1)α+1
+NM
+√NM
+⟨Ψ|ηf †
+R,αηsck,α+2ηs|Ψ⟩
+(S84)
+where Of, Oc′′, V3, V4 have also been used in the Kondo lattice mean-field calculations (Eq. S38, Eq. S48 and Eq. S44).
+In addition, THF model also has a chemical potential term ˆHµ (Eq. S21) and we determine µ by requiring the total filling of
+f- and c-electrons to be ν:
+ν = Tr[Of] + Tr[Oc′] + Tr[Oc′′]
+(S85)
+where we note that the filling of f-, Γ3 c- and Γ1 ⊕ Γ2 c-electrons are
+νf = Tr[Of],
+νc′ = Tr[Oc′],
+νc′′ = Tr[Oc′′] ,
+(S86)
+respectively.
+We discuss the difference and similarities between the mean-field equations of the KL model and that of the THF model.
+For the THF model, we introduce mean fields Of, Oc′, Oc′′, Oc′f, V3, V4 (Eq. S84) (for a generic state without enforcing any
+symmetries). As for KL model, we introduce mean fields V1, V2, Of, Oc′,1, Oc′,2, Oc′′, V3, V4 (Eq. S38, Eq. S35, Eq. S48,
+Eq. S44) (for a generic state without enforcing any symmetry).
+• For both models, Oc′′, Of, V3, V4 are part of mean fields and contribute the mean-field decoupling of ˆHJ (Eq. S50).
+• In the THF model, we introduce a chemical potential term µ that couples to both the f-electron density operators and c-
+electron density operator (Eq. S21. We enforce the total filling of f- and c-electrons to be ν by tuning µ. In the KL model,
+we introduce a Lagrangian multiplier λf (Eq. S51) that couples to f-electron density operators, and a chemical potential
+µc (Eq. S28) that couples to the c-electron density operators. We enforce the fillings of f-electrons and c-electrons to be
+νf and νc respectively by tuning λf and µc in the KL model.
+• We also mention that Oc′ in THF model (Eq. S84) and Oc′,1 in the KL model (Eq. S38) are different, where the latter one
+has included an additional damping factor e−|k|2λ2.
+• In the THF model, we do not need hybridization fields V1, V2 (Eq. S35), since both come from the decoupling of Kondo
+interactions that only appear in the KL model.
+A.
+Mean-field equations of fully symmetric state
+We next discuss the solution of the symmetric state The fully symmetric state is characterized by density matrices
+Of, Oc′, Oc′′, Oc′f and hybridization fields V3, V4 that satisfy all symmetries. The structures of Of, Oc′′ in the fully sym-
+metric state are given in Eq. S60. We also prove that V3 = V4 = 0 in a fully symmetric state (near Eq. S62). We now discuss
+
+28
+the symmetry properties of Oc′, Oc′f. From Eq. S32, a U(1)v symmetric solution satisfies
+Oc′
+aηs,α′η′s′ = Oc′
+aηs,α′η′s′e−iθν(η−η′) ⇒ Oc′
+aηs,α−ηs′ = 0
+Oc′f
+aηs,α′η′s′ = Oc′f
+aηs,α′η′s′e−iθν(η−η′) ⇒ Oc′f
+aηs,α−ηs′ = 0
+(S87)
+Then, Oc′, Oc′f is block diagonalized in valley indices. We next consider a SU(2)η transformation acting on the valley η. It
+indicates
+�
+s,s′
+[ei �
+µ θη
+µσµ]s2,sOc′
+αηs,α′ηs′[ei �
+µ θη
+µσµ]s′,s′
+2 = Oc′
+αηs2,α′ηs′
+2 ⇒ Oc′
+αηs,α′ηs′
+�
+s,s′
+[ei �
+µ θη
+µσµ]s2,sOc′f
+αηs,α′ηs′[ei �
+µ θη
+µσµ]s′,s′
+2 = Oc′f
+αηs2,α′ηs′
+2 ⇒ Oc′f
+αηs,α′ηs′
+(S88)
+Combining Eq. S87 and Eq. S88, we can introduce 2 × 2 matrices oc′,η, oc′f,η, such that
+Oc′
+αηs,α′η′s′ = oc′
+α,α′δs,s′δη,η′,
+Oc′f
+αηs,α′η′s′ = oc′f
+α,α′δs,s′δη,η′
+(S89)
+We now consider the effect of discrete symmetries in Eq. S30. Using Eq. S31 and Eq. S89, we find
+T :
+(oc′,η)∗ = oc′,−η,
+(oc′f,η)∗ = oc′f,−η
+C3z :
+ei2π/3ησzoc′,ηe−i2πη/3σz = oc′,η,
+ei2π/3ησzoc′f,ηe−i2πη/3σz = oc′f,η
+C2x :
+σxoc′,ησx = oc′,η,
+σxoc′f,ησx = oc′f,η
+C2zT :
+σx(oc′,η)∗σx = oc′,η,
+σx(oc′f,η)∗σx = oc′f,η
+(S90)
+Then we can introduce a single real number χc′
+0 , χc′f
+0
+to characterize the density matrices
+Oc′
+αηs,α′η′s′ = χc′
+0 δα,α′δη,η′,
+Oc′f
+αηs,α′η′s′ = χc′f
+0 δα,α′δη,η′δs,s′ .
+(S91)
+Since the filling of Γ3 c-electrons is νc′ = Tr[Oc′] = 8χc′
+0 , we let χc′
+0 = νc′/8. Therefore, instead of calculating the original
+density matrices in Eq. S84, we can calculate the following quantities
+νf =
+1
+NM
+�
+R,αηs
+⟨Ψ| : f †
+R,αηsfR,αηs : |Ψ⟩
+νc′ =
+1
+NM
+�
+|k|<Λc,a=1,2,ηs
+⟨Ψ| : c†
+k,aηsck,aηs : |Ψ⟩
+νc′′ =
+1
+NM
+�
+|k|<Λc,a=3,4,ηs
+⟨Ψ| : c†
+k,aηsck,aηs : |Ψ⟩
+χc′f =
+1
+8NM
+√NM
+�
+|k|<Λc,R,αηs
+e−ik·R⟨Ψ|c†
+k,αηsfR,αηs|Ψ⟩
+(S92)
+and construct density matrices via Eq. S60 and Eq. S91. In addition, the filling constraints in Eq. S85 becomes
+ν = νf + νc′ + νc′′
+(S93)
+Combining Eq. S92 and Eq. S93, we have a complete set of the mean-field self-consistent equations of symmetric state.
+Here we discuss the differences and similarities between the symmetric solution in the KL model (Sec. S4 E) and the sym-
+metric solution in the THF model as introduced in this section.
+• Both Kondo symmetric (KS) state in KL model and the symmetric state in the THF model preserve all the symmetries.
+• The KS state is adiabatically connected to the symmetric state in the THF model.
+• The mean-field solutions are exact at N = ∞.
+
+29
+FIG. S6. Wilson loop spectrum of symmetric state in the THF model at ν = 0, −1, −2
+• To obtain a more precise description of the Kondo state, we need to introduce a Gutzwiller projector to our symmetric-
+state wavefunction in the THF model. The Gutzwiller projector will suppress the charge fluctuations of f-electrons and is
+expected to further lower the energy of the symmetric state.
+• We also comment that, in the THF model, the flat bands are mainly formed by f-electrons with f-electron orbital weights
+80%, 85% and 87% at ν = 0, −1, −2 respectively. However, in the KL model, the flat bands are mainly formed by
+Γ1 ⊕Γ2 electrons as discussed in Sec. S4 E. This is because, in the KL model, we observed an enhanced f-c hybridization
+driven by Kondo interactions which is absent in the symmetric state solution of THF model. We expect the enhanced
+hybridization will be recovered after introducing the Gutzwiller projector.
+• We find the spectrum of the Wilson loop in the symmetric state of the THF model has winding number one and three
+crossing points(Fig. S6). However, in the SK state of the KL model, the spectrum of the Wilson loop spectrum has
+winding number one and one crossing point (Fig. S5). However, at ν = 0, for both the THF model and the KL model, the
+symmetric state has a stable topology with an odd Winding number of Wilson loop spectrum [85]. We also mention the
+difference in the Wilson loop spectrum comes from the absence of enhanced f-c hybridization in the THF model.
+B.
+Mean-field equations of the symmetric state in the presence of strain
+We now discuss the mean-field solution of the symmetric state in the presence of strain. We add the following term to the
+Hamiltonian (Eq. S12)
+ˆHstrain = α
+�
+R,ηs
+(f †
+R,1ηsfR,2ηs + h.c.)
+(S94)
+We note that ˆHstrain only breaks C3z symmetry. To show this, we rewrite the ˆHstrain as
+ˆHstrain =
+�
+R,αηs,α′η′s′
+αf †
+R,αηs[hstrain]αηs,α′η′s′fR,α′η′s′
+hstrain = σxτ0ς0
+(S95)
+where
+the
+matrix
+structure
+of
+ˆHstrain
+is
+denoted
+by
+hstrainσxτ0ς0.
+We
+now
+show
+it
+commutes
+with
+Df(T), Df(C2x), Df(C2zT), Df(gSU(2)η(θµ
+η )), Df(gU(1)v((θv)), Df(gU(1)c((θc)) (Eq. S31, Eq. S32) and hence ˆHstrain
+preserves all symmetries except for C3z
+[hstrain, Df(T)] = [σxτ0ς0, σ0τxε0] = 0
+[hstrain, Df(C2x)] = [σxτ0ς0, σxτ0ς0] = 0
+[hstrain, Df(C2zT)] = [σxτ0ς0, σxτ0ς0] = 0
+[hstrain, Df(gU(1)c((θc))] = [σxτ0ς0, e−iθcσ0τ0ς0] = 0
+[hstrain, Df(gU(1)v((θv))] = [σxτ0ς0, σ0e−iθvτzς0] = 0
+[hstrain, Df(gSU(2)η(θµ
+η ))] = [σxτ0ς0, σ0e−i �
+µ θη
+µ
+τ0+ητz
+4
+ςµ] = 0
+
+30
+In the presence of strain, the symmetric state is defined as the state that preserves all the symmetries except for C3z which
+is broken by the strain. In the presence of C3z-breaking strain, Eq. S57 and Eq. S89 still hold, because the system still has
+U(1)c × U(1)v × SU(2)+ × SU(2)− symmetry. As for Eq. S58 and Eq. S90, we only need to consider T, C2x, C2zT and we
+find
+Of
+αηs,α′η′s′ =
+�
+χf
+0σ0 + χf
+1σx
+�
+α,α′δη,η′δs,s′
+Oc′
+αηs,α′η′s′ =
+�
+χc′
+0 σ0 + χc′
+1 σx
+�
+α,α′
+δη,η′δs,s′,
+Oc′′
+αηs,α′η′s′ =
+�
+χc′′
+0 σ0 + χc′′
+1 σx
+�
+α,α′
+δη,η′δs,s′,
+Oc′f
+αηs,α′η′s′ =
+�
+χc′f
+0 σ0 + χc′f
+1 σx
+�
+α,α′δη,η′δs,s′
+(S96)
+where χf
+0, χf
+1, χc′
+0 , χc′
+1 , χc′′
+0 , χc′′
+1 , χc′f
+0 , χc′f
+1
+are real numbers tha characterize the density matrices. Combining Eq. S38, Eq. S48
+and Eq. S96, we find
+χf
+0 =
+1
+8NM
+�
+R,αetas
+⟨Ψ| : f †
+R,αηsfR,αηs : |Ψ⟩,
+χf
+1 =
+1
+8NM
+�
+R,ηs
+⟨Ψ|f †
+R,1ηsfR,2ηs + f †
+R,2ηsfR,1ηs|Ψ⟩
+χc′
+0 =
+1
+8NM
+�
+|k|<Λc,=1,2,ηs
+⟨Ψ| : c†
+k,aηsc†
+k,aηs : |Ψ⟩,
+χc′
+1 =
+1
+8NM
+�
+|k|<Λc,ηs
+⟨Ψ|c†
+k,1ηsck,2ηs + c†
+k,2ηsck,1ηs|Ψ⟩
+χc′′
+0 =
+1
+8NM
+�
+|k|<Λc,=3,4,ηs
+⟨Ψ| : c†
+k,aηsc†
+k,aηs : |Ψ⟩,
+χc′′
+1 =
+1
+8NM
+�
+|k|<Λc,ηs
+⟨Ψ|c†
+k,3ηsck,4ηs + c†
+k,4ηsck,3ηs|Ψ⟩
+χc′f
+0
+=
+1
+8NM
+√NM
+�
+|k|<Λc,R,αηs
+e−ik·R⟨Ψ|c†
+k,αηsfR,αηs|Ψ⟩,
+χc′f
+1
+=
+1
+8NM
+√NM
+�
+|k|<Λc,R,ηs
+e−ik·R⟨Ψ|c†
+k,1ηsfR,2ηs + c†
+k,2ηsfR,1ηs|Ψ⟩
+χc′′f
+0
+=
+1
+8NM
+√NM
+�
+|k|<Λc,R,αηs
+e−ik·R⟨Ψ|c†
+k,α+2ηsfR,αηs|Ψ⟩,
+χc′′f
+1
+=
+1
+8NM
+√NM
+�
+|k|<Λc,R,ηs
+e−ik·R⟨Ψ|c†
+k,3ηsfR,2ηs + c†
+k,4ηsfR,3ηs|Ψ⟩
+(S97)
+where we also have χf
+0 = νf/8, χc′
+0 = νc′/8, χc′′
+0 = νc′′/8. As for V3, V4, we only consider the T, C2x, CzT of Eq. S61, which
+indicates
+V3 = V4 = V ∗
+3 = V ∗
+4
+(S98)
+Combining Eq. S44, Eq. S93, Eq. S97 and Eq. S98 with filling constraints ν = νf + νc′ + νc′′, we have a complete set of the
+mean-field self-consistent equations of symmetric state in the presence of strain. We perform calculations with non-zero strain
+at ν = 0, −1, −2, −3. We initialize the calculations with the fully symmetric solutions derived at zero strain, and the procedure
+converges within 500 iterations. The results are illustrated and discussed in Sec. S5 D.
+C.
+Effect of doping
+We now discuss the effect of doping at zero strain. For hole doping at ν = 0, −1, −2, −3 and electron doping at ν = 0, we
+mainly dope electrons to the light bands that are mostly formed by c-electrons (Fig. S7). Consequently, the energy difference
+between the symmetric state and the ordered state decreases since we have more conduction c-electrons near the Fermi energy,
+and the system favors the symmetric state.
+We now point out the complexity of electron dopings at ν = −1, −2. Doping electrons at ν = −1, −2 is equivalent to dope
+electrons to the heavy bands that are mostly formed by f-electrons (Fig. S7). The heavy (flat) bands become closer to the Fermi
+energy, and hence, the energy cost of putting f-electrons into flat bands will be small. Then we can fill the heavy (flat) bands
+with a small energy cost. By filling the heavy (flat) bands, the type of orders formed by f-electrons can change a lot. To observe
+
+31
+(a)
+(b)
+FIG. S7. Dipsersions of KIVC state at ν = 0, KIVC+VP state at ν = −1, KIVC state at ν = −2 and VP state at ν = −3. The color represents
+the weight of f-(yellow) and c-(blue) electrons.
+FIG. S8. Evolution of order parameter as a function of doping at ν = 0, −1, −2
+the change of the ordered states, we consider the following order parameters
+Ox =
+1
+NM
+�
+R,αηs,α′η′s′
+f †
+R,αηs[ox]αηs,α′η′s′fR,α′η′s′,
+x ∈ {KIV C, Sz, Vz, Vy}
+oKIV C = σyτyς0,
+oSz = σ0τ0ςz,
+oVz = σ0τzς0,
+oVy = σ0τyς0
+(S99)
+We measure the expectation values of Ox=KIV C,Sz,Vz,Vy with respect to the ordered states at each filling. In Fig. S8, we show
+the evolution of ⟨Ox⟩ as a function of doping. We find for hole doping at ν = 0, −1, −2 and electron doping at ν = 0 (where
+carriers go to light bands in both cases), the system stays in the same ordered states (compared to the integer filling). However,
+for electron doping at ν = −1, −2, we can observe the changes of the order parameters. This is because we are mainly dope
+f-electrons for electron doping at ν = −1, −2. We thus conclude that electron doping at ν = −1, −2 will introduce sizeable
+changes of the order parameters. Both the change of order parameters and the doping effect will affect the energy competition
+between the symmetric state and the ordered state.
+Finally, we comment on the ν = −3 case. At ν = −3, the actual ground state might be a CDW state which breaks the
+translational symmetry [145] and is beyond our current consideration. In addition, at ν = −3, even for the valley polarized state
+we currently considered, electron doping is equivalent to doping both heavy and light bands (Fig. S7), which is different from
+ν = −1, −2. We leave the detailed study of ν = −3 for future study.
+D.
+Effect of strain
+We next analyze the effect of the strain. We first note that as we increase α (Eq. S95), the strain will gradually suppress the
+KIVC order (OKIV C, Eq. S99). This can be observed from
+{oKIV C, hstrain} = {σyτyς0, σxτ0ς0} = 0
+(S100)
+Heuristically, the anti-commuting nature indicates the competition between oKIV C and hstrain. Thus, as we increase hstrain,
+oKIV C will be suppressed. We also find the spin-polarization OSz and valley polarization OVz commute with hstrain
+[oSz, hstrain] = [σ0τ0ςz, σxτ0ς0] = 0,
+[oVz, hstrain] = [σ0τzς0, σxτ0ς0] = 0
+(S101)
+
+32
+Heuristically, this indicates the valley and spin polarization do not directly compete with hstrain. However, as we will show in
+this section, a sufficiently large strain could still destroy the valley and spin polarization in the THF model.
+For future convenience, we also introduce the eigenstates of the strain Hamiltonian ˆHstrain (Eq. S95)
+d†
+R,1ηs =
+1
+√
+2(f †
+R,1ηs − f †
+R,2ηs),
+d†
+R,2ηs =
+1
+√
+2(f †
+R,1ηs + f †
+R,2ηs)
+(S102)
+We will call d†
+R,1ηs and d†
+R,2ηs as d1 and d2 electrons (orbitals), respectively, for short. We mention that d1, d2 are f-electrons.
+The strain Hamiltonian can then be written as
+ˆHstrain = α
+�
+R,αηs
+(−d†
+R,1ηsdR,1ηs + d†
+R,2ηsdR,2ηs)
+(S103)
+Thus, for a positive strain amplitude α > 0, the energy of d1 electrons will be lowered and the energy of d2 electrons will be
+raised. We introduce ⟨hstrain⟩ to characterize the population imbalance between d1 and d2 electrons
+⟨hstrain⟩ = ⟨ 1
+NM
+�
+R,αηs,α′η′s′
+f †
+R,αηs[hstrain]αηs,α′η′s′fR,α′η′s′⟩ =
+1
+NM
+�
+R,αηs
+⟨d†
+R,1ηsdR,1ηs − d†
+R,2ηsdR,2ηs⟩
+(S104)
+In Fig. S9, we plot the evolution of various order parameters and also |⟨hstrain⟩| where the expectation value is taken with
+respect to the ordered state solution. In all cases, |⟨hstrain⟩| increases as we increase α, since α linearly coupled to hstrain
+term. The KIVC order will be suppressed and fully destroyed at sufficient strong strain at ν = 0, −1, −2. At ν = 0, after the
+destruction of the KIVC order, self-consistent calculation produces a symmetric ground state that only breaks C3z symmetry,
+even though we initialize the mean-field calculation with an ordered state.
+However, at ν = −1, −2, after the destruction of KIVC order, the spin polarization and valley polarization still exist. By
+further increasing the strain, the ordered states will finally become unstable (Fig. S9), which means the mean-field calculations
+that are initialized with ordered solutions converge to a symmetric state.
+We next analyze the transition from an ordered state to a symmetric state at a large strain at ν = −1, −2.
+1.
+ν = −1
+We first consider the ν = −1 with 4meV ≲ α ≲ 18meV. In this parameter region, the KIVC order is destroyed but valley
+and spin polarization persist (Fig. S9). In Fig. S10 (a) (b), we plot the band structures in this parameter region. We note that
+flat bands that are mostly formed by f-electrons (marked by red circles, Fig. S10) move towards the Fermi level, as we increase
+strain. Near the transition point to the symmetric state, the flat bands are very close to the Fermi level. This signals an instability
+of the ordered states since we can fill the flat band without any energy cost. By diagonalizing the mean-field Hamiltonian, we
+find the flat bands (marked by red circles, Fig. S10) correspond to d1 electrons (Eq. S103). By filling the flat bands, we have
+more populations in d1 orbitals, which increase |⟨hstrain⟩| (Eq. S104) and drive the system to a symmetric state.
+We now estimate the critical value of strain αc at which a transition from an ordered state to a symmetric state happens. At
+αc, the flat bands (marked by red circles, Fig. S10) are very close to the Fermi energy and induce the transition. To estimate αc,
+we calculate the excitation gap of the flat bands (marked by red circles, Fig. S10): ∆Eflat. Then we have
+∆Eflat
+����
+α≈αc
+= 0
+(S105)
+We estimate ∆Eflat using the zero-hybridization limit [2] of the model, where γ = 0, v′
+⋆ = 0 (Eq. S15). In addition, we also
+set J = 0 to simplify the calculation (Eq. S17). The zero-hybridization model with non-zero strains are
+ˆHzero-hyb = ˆHU + ˆHW + ˆHV + ˆHstain + ˆHc + ˆHµ
+(S106)
+where ˆHU, ˆHW , ˆHV , ˆHstrain, ˆHc, ˆHµ are defined in Eq. S16, Eq. S18, Eq. S19, Eq. S95, Eq. S13 and Eq. S12 respectively,
+and ˆHV are treated with mean-field methods. In the zero-hybridization model, the filling of f-electrons νf and c-electrons-νc
+are good quantum numbers. We solve the zero-hybridization model at fixed total filling ν with the assumption that the ground
+state does not break translational symmetry (fillings of f-electrons are uniform) [2]. To estimate the excitation gap of the flat
+bands, we calculate the energy cost of adding one dR,1ηs electron. We mention that, in our mean-field calculations with finite
+f-c hybridization, the relevant flat bands (marked by red circles in Fig. S10) correspond to d1 electrons. We let |Ψzero-hyb⟩ denote
+the ground state of the zero-hybridization model. The state with one-more dR,1ηs is
+|Ψexct
+zero-hyb⟩ = d†
+R,1ηs|Ψzero-hyb⟩ .
+(S107)
+
+33
+We next calculate
+∆Eflat = ⟨Ψexct
+zero-hyb| ˆHzero-hyb|Ψexct
+zero-hyb⟩ − ⟨Ψzero-hyb| ˆHzero-hyb|Ψzero-hyb⟩
+(S108)
+The energy loss from Hubbard interaction term is
+∆EU = ⟨Ψexct
+zero-hyb| ˆHU|Ψexct
+U
+⟩ − ⟨Ψzero-hyb| ˆHU|Ψzero-hyb⟩ = U
+2 (νf + 1)2 − U
+2 ν2
+f = U(νf + 1
+2)
+The energy loss from ˆHW term is
+∆EW =⟨Ψexct
+zero-hyb| ˆHW |Ψexct
+zero-hyb⟩ − ⟨Ψzero-hyb| ˆHW |Ψzero-hyb⟩
+=
+�
+a=1,2,3,4
+Waνc,a(νf + 1) −
+�
+a=1,2,3,4
+Waνc,aνf =
+�
+a=1,2,3,4
+Waνc,a
+(S109)
+where νc,a denotes the filling of c-electrons in orbital a. The energy change from ˆHV is
+∆EV = ⟨Ψexct
+zero-hyb| ˆHV |Ψexct
+zero-hyb⟩ − ⟨Ψzero-hyb| ˆHV |Ψzero-hyb⟩ = 0
+(S110)
+The energy change from ˆHstrain is
+∆Estrain = ⟨Ψexct
+zero-hyb| ˆHstrain|Ψexct
+zero-hyb⟩ − ⟨Ψzero-hyb| ˆHstrain|Ψzero-hyb⟩ = −α
+(S111)
+The energy change from chemical potential ˆHµ is
+∆Eµ = ⟨Ψexct
+zero-hyb| ˆHµ|Ψexct
+zero-hyb⟩ − ⟨Ψzero-hyb| ˆHµ|Ψzero-hyb⟩ − µ
+(S112)
+Then the excitation energy of adding one dR,1ηs electron is
+∆Eflat = ∆EU + ∆EW + ∆strain + ∆Eµ = U
+2 (νf + 1/2) +
+�
+a
+Waνc,a − µ − α
+(S113)
+We further take the following approximation: W1,2,3,4 = W = 47meV (the difference between W1,2,3,4 is about 15%). Then
+∆Eflat ≈ U
+2 (νf + 1/2) + Wνc − µ − α
+(S114)
+At ν = −1 and 0meV ≤ α ≤ 43meV, the ground state of the zero-hybridization model has νf = −1, νc = ν − νf = 0
+(Fig. S11). Then
+∆Eflat = −U
+2 − µ − α
+(S115)
+We next determine chemical potential µ. Chemical potential µ is determined by requiring the c-electrons filling to be νc = 0.
+The single-particle Hamiltonian of c-electron in the zero-hybridization limit takes the form of
+ˆHc,zero-hyb = ˆHc +
+�
+k,aηs
+(Wνf + V (0)
+Ω0
+νc − µ)c†
+k,aηsck,aηs
+(S116)
+where we have set W1,2,3,4 = W. We note that when
+Wνf − V (0)
+Ω0
+νc − µ = 0
+(S117)
+ˆHc,zero-hyb = ˆHc and we have νc = 0. Therefore,
+µ = Wνf + V (0)
+Ω0
+νc = −W
+(S118)
+
+34
+where we take νc = 0, νf = −1 (Fig. S11). Using Eq. S115 and Eq. S118, we find
+∆Eflat = W − U
+2 − α
+(S119)
+Then the flat bands reach Fermi energy when ∆Eflat = 0, which leads to
+∆Eflat = 0 ⇒ αc = W − U
+2 = 18meV
+(S120)
+which is close to the value (also around α = 18meV as shown in Fig. S9 (b)) from self-consistent calculations of the finite-
+hybridization model. Here, the finite-hybridization model refers to the original THF model with finite γ, v′
+⋆. Therefore, we
+conclude the transition from an ordered state to a symmetric state happens at α = αc ≈ 18meV at ν = −1.
+We also discuss the solutions of the zero-hybridization model here. In Fig. S11 (a), we show the ground state properties of the
+zero-hybridization model at various strains and ν = −1, where
+νf
+1 =
+1
+NM
+�
+R,ηs
+: d†
+R,1ηsdR,1ηs :,
+νf
+2 =
+1
+NM
+�
+R,ηs
+: d†
+R,2ηsdR,2ηs :
+(S121)
+denotes the filling of d1 and d2 electrons respectively with νf = νf
+1 + νf
+2 . We find a transition happens at α ≈ 25meV.
+We note that this transition is described by filling one more dR,1ηs electrons at each site. After the transition, there will be
+4 f-electrons filling d1 orbitals, and zero f-electrons filling the d1 orbitals. Thus for d1 orbitals, all the valleys and spins are
+filled, but for d2 orbitals all the valleys and spins are empty. Therefore, there is no room to develop order and the ground state
+is a symmetric state. We note that the transition in the zero-hybridization limit and the transition in the finite-hybridization
+model (at ν = −1, α ≈ 16meV, Fig. S9) share the same origin. They are both driven by filling electrons in d1 orbitals (in the
+finite-hybridization model, we fill the flat bands) and, after the transition, both ground states are symmetric. Thus, the results
+between zero-hybridization and finite-hybridization models are consistent. However, the critical values αc for the two models
+are different, since we have finite f-c hybridization in the finite-hybridization model.
+2.
+ν = −2
+We next discuss the transition from an ordered state to a symmetric state at ν = −2. We focus on the parameter region
+10meV ≲ α ≲ 45meV, where the KIVC order is destroyed but valley and spin polarization exist (Fig. S9). In Fig. S10 (c) (d),
+we plot the band structures in this parameter region. As we increase strain, we note that flat bands (marked with red circles,
+Fig. S10), move towards the Fermi level. Similar to the ν = −1 case, when the flat bands reach the Fermi energy, a transition to
+the symmetric state happens. However, at ν = −2, we need a much larger strain to destroy the ordered state as shown in Fig. S9
+(d). To understand this, we start from the zero-hybridization limit of the model (Eq. S106). As shown in Eq. S114, the excitation
+energy of the relevant flat bands (marked by red circles in Fig. S10)
+∆Eflat = U
+2 (νf + 1/2) + Wνc − µ − α
+(S122)
+By solving the zero-hybridization model, we find the ground states have νf = −1 and νc = −1 in the parameter region we
+focused 4meV ≲ α ≲ 18meV, as shown in Fig. S11. Then
+∆Eflat = −U
+2 − W − µ − α
+(S123)
+We now calculate the chemical potential. µ is determined by requiring the filling of c-electrons to be νc = −1. The single-
+particle Hamiltonian of c-electron in the zero-hybridization limit (Eq. S116) takes the form of
+ˆHc,zero-hyb = ˆHc +
+�
+k,aηs
+(Wνf + V (0)
+Ω0
+νc − µ)c†
+k,aηsck,aηs
+(S124)
+where we have set W1,2,3,4 = W, and take the mean-field treatment of ˆHV (Eq. S20). At M = 0 limit (M = 3.697meV, which
+is relatively small), the dispersion of c-electrons are Ek = ±v⋆|k| − Ec, where we define
+Ec = Wνf + V (0)
+Ω0
+νc − µ
+(S125)
+
+35
+Then all the c-states with energy smaller than 0 will be filled. The corresponding Fermi momentum kF is
+|v⋆kF | = Ec ⇒ kF =
+1
+|v⋆|Ec
+(S126)
+Then the filling of c-electrons is
+νc = −
+8
+AMBZ
+�
+|k|<kF
+dkxdky = − 8πk2
+F
+AMBZ
+= −
+8πE2
+c
+AMBZ|v⋆|2
+(S127)
+where the prefactor 8 comes from the 8-fold degeneracy of the bands (2 spins, 2 valleys, and 2 orbitals) and AMBZ is the area
+of MBZ. We require νc = −1 which indicates
+νc = −
+8πE2
+c
+AMBZ|v⋆|2 = −1 ⇒ Ec =
+�
+AMBZ|v⋆|2
+8π
+(S128)
+Using Eq. S125 and Eq. S128, we find
+µ = −W − V (0)
+Ω0
+−
+�
+AMBZ|v⋆|2
+8π
+(S129)
+where we take νf = −1, νc = −1. Combining Eq. S123 and Eq. S128, we find
+∆Eflat = −U
+2 + V (0)
+Ω0
++
+�
+AMBZ|v⋆|2
+8π
+− α
+(S130)
+And the transition happens at
+∆Eflat = 0 ⇒ αc = −U
+2 + V (0)
+Ω0
++
+�
+AMBZ|v⋆|2
+8π
+= 62meV
+(S131)
+We note that our estimation of transition in Eq. S131 is based on the single-particle picture and has not included the effect
+of hybridization. Thus the estimated value is larger than the transition value α ≈ 45meV from mean-field self-consistent
+calculations with finite f-c hybridization.
+We also discuss the solution obtained from the zero-hybridization model at ν = −2. In Fig. S11 (b), we show the ground state
+properties of the zero-hybridization model at various strains and ν = −2. We observe two transitions, both characterize by the
+increments of νf
+1 . After the first transition α ∼ 15meV, we have three d1 electrons at each site. Then not all valleys and spins
+of d1 orbitals are filled. This gives the possibility of developing spin and valley polarized order after we turn on the ˆHJ and f-c
+hybridization. Indeed, from Fig. S9, we can observe the non-vanishing order at 15meV ≲ α ≲ 40meV. By further increasing
+strain, at α ≈ 50meV, we observe a second transition in the zero-hybridization model. After the second transition, we have
+νf
+1 = 2, νf
+2 = −2 and there is no room for f-electrons to develop order. Correspondingly, for the finite-hybridization model, we
+also observe a transition to the symmetric stat at α ≈ 45meV. Thus we conclude the consistency between zero-hybridization and
+finite-hybridization models. We also point out that, the transition value αc ≈ 62meV suggested from single-particle excitation
+in Eq. S131 is also larger than the transition value αc ≈ 50meV obtained by solving many-body wavefunction in the zero-
+hybridization model. This points out the limitation of the single-particle picture.
+Finally, we comment on the difference between ν = −1 and ν = −2. Eq. S131 and Eq. S120 indicate that a relatively large
+strain is required to destroy the ordered state at ν = −2 compared to ν = −1. This is because, near the transition point with
+α < αc, c-electrons have filling νc = 0 at ν = −1, but have filling νc = −1 at ν = −2 (Fig. S11). In other words, we dope
+more holes to the c-bands at ν = −2 compared to ν = −1. In order to dope more holes to c-bands, the chemical potential (or
+Fermi energy) needs to be increased. The increment of chemical potential also leads to a larger Fermi surface, which has been
+observed in the model with finite hybridization (Fig. S10), where the Fermi surface at ν = −2 is larger than the Fermi surface
+at ν = −1. In the zero-hybridization model, the change of chemical potential can be observed analytically from Eq. S118 and
+Eq. S129. We note that the excitation gap of the flat band (∆Eflat) is measured with respect to the Fermi energy (or chemical
+potential). Then at fixed α, we find the value of ∆Eflat at ν = −2 is much larger than its value at ν = −1 (Eq. S119 and
+Eq. S130) due to the larger chemical potential at ν = −2. Thus, at ν = −2, a much larger strain is needed to make ∆Eflat = 0
+and destroy the ordering.
+
+36
+(a)
+(b)
+(c)
+(d)
+FIG. S9. (a), (b), (c) Evolution of order parameters as a function of strain amplitude α at ν = 0, −1, −2. (d) Evolution of order parameters at
+ν = −2 with an extended parameter region 0meV ≤ α ≤ 55meV
+(a)
+(b)
+(c)
+(d)
+FIG. S10. Band structure at ν = −1 (a), (b) and ν = −2 (c), (d). Red circles mark the relevant flat bands.
+3.
+Discussions about ν = −3
+We now discuss ν = −3. We first comment that, at ν = −3, other low energy states that break translational symmetry exists
+even at zero strain [145]. Furthermore, even in the zero-hybridization limit at zero strain, ν = −3 is close to the transition point
+between νf = −3 and νf = −2 states [1]. These all point out the complexity of ν = −3 even in the absence of strain. Thus, we
+leave a more systematical analysis at ν = −3 for future study.
+E.
+Strain
+We now discuss the derivation of strain term in the Eq. S95. Here we will take a minimal model to capture the effect of strain,
+and leave a detailed analysis (based on Ref. [150]) for future study. We take the single-particle Hamiltonian of twisted bilayer
+(a)
+(b)
+FIG. S11. Evolution of filling as a function of the strain in the zero-hybridization model at ν = −1, −2.
+
+37
+graphene at valley η and with non-zero strain [148]
+Hη(k) =
+�Hη
+1 (k)
+U
+U †
+Hη
+2 (k)
+�
+(S132)
+where Hl(k) is a 2 × 2 matrix that characterizes the Hamiltonian of l-layer graphene. U is the interlayer coupling matrix. In
+real space, U takes the form of
+U =
+3
+�
+j=1
+Ujeiηqj·r,
+Uj =
+�
+w0
+w1e−iη2(j−1)π/3
+w1eiη2(j−1)π/3
+w0
+�
+with w0, w1 are two constant parameters that characterize the interlayer couplings. Hl(k) is given by
+Hη
+l=1,2(k) = −ℏvF
+��
+R(∓θ) + El
+�−1
+(k + e
+ℏAl,η)
+�
+· ση
+(S133)
+where ση = (ησx, σy) , R(θ) is a rotation matrix, θ is the twist angle and [148]
+El =
+�
+ϵ(l)
+xx
+−Ω(l) + ϵ(l)
+xy
+−Ω(l) + ϵ(l)
+xy
+ϵ(l)
+yy
+�
+Al,η = η 3βγ0
+2evF
+�
+ϵ(l)
+−
+ϵ(l)
+xy
+�
+,
+γ0 = 2.7eV,
+β ≈ 3.14
+ϵ(l)
+± = 1
+2(ϵ(l)
+xx ± ϵ(l)
+yy) .
+(S134)
+ϵ(l)
+xx, ϵ(l)
+yy are normal strains along x and y directions respectively, ϵ(l)
+xy is the sheer strain. Ω(l) denotes a rotation from the
+twist angles [148]. Since the flat band is sensitive to ϵ− = (ϵ(1)
+− − ϵ(2)
+− )/2 and ϵxy = (ϵ(1)
+xy − ϵ(2)
+xy )/2 [148], we consider the
+heterostrain with Ω(l) = 0, ϵ(l)
++ = 0, ϵ(l)
+− = (−1)l+1ϵ−, ϵ(l)
+xy = (−1)l+1ϵxy. We further focus on the sheer strain ϵxy and set the
+normal anisotropic strain to be zero ϵ− = 0. For small twist angle, we approximately have
+Hη
+l=1,2(k) ≈ − ℏvF
+��
+R(±θ) − El
+�
+(k + e
+ℏAl,η)
+�
+· ση
+≈ − ℏvF
+�
+R(±θ)k − Elk + e
+ℏAl,η
+�
+· ση
+(S135)
+The effect of strain is then captured by the following single particle Hamiltonian of valley η
+Hη
+strain(k) = Hη
+strain,1 + Hη
+strain,2
+Hη
+strain,1 = −3βγ0η
+2
+ϵxy
+�
+σy
+−σy
+�
+,
+Hη
+strain,2 = ℏvF ϵxy
+�
+ηkyσx + kxσy
+−ηkyσx − kxσy
+�
+(S136)
+We next project the Hη
+strain(k) to the Wannier basis of f-electrons. We take the following wavefunction of f electrons at R
+with orbital α, valley η and spin s [1]
+|WR,α=1,η,s⟩ =
+�
+2πλ2
+0
+Ωtot
+�
+l=±
+�
+k
+�
+Q∈Qlη
+ei π
+4 lη−ik·R− 1
+2 λ2
+0(k−Q)2|k, Q, 1, η, s⟩
+|WR,α=2,η,s⟩ =
+�
+2πλ2
+0
+Ωtot
+�
+l=±
+�
+k
+�
+Q∈Qlη
+e−i π
+4 lη−ik·R− 1
+2 λ2
+0(k−Q)2|k, Q, 2, η, s⟩
+(S137)
+where λ0 = 0.1aM, Ωtot is the total area of the sample and Qlη = {±q1 + n1bM,1 + n2bM,2|n1, n2 ∈ Z}, q1 = kθ(0, −1) [1].
+From Eq. S136 and Eq. S137, we can calculate
+hstrain
+αηs,α2η2s2 =
+1
+NM
+�
+R
+⟨WR,α,η,s|
+�
+k,α′l′,α′
+2l′
+2,η′,s′
+ψ†
+k,l,α′,η′,s′[Hη′
+strain(k)]l′α′,l′
+2α′
+2ψk,l′
+2α′,η′,s′|WR,α2,η2,s2⟩
+(S138)
+
+38
+where ψk,l,α,η,s is the electron operator of the original electron basis with momentum k, layer l, sublattice α, valley η and spin
+s. Here, we take the average over all the moir´e unit cells (sum over R), and thus we only keep the momentum-independent (in
+f-electron basis) contributions. We leave the momentum-dependent term for future studies. In the f-basis, the strain can be
+characterized by
+ˆHstrain =
+�
+R,αηs,α2η2s2
+hstrain
+αηs,α2η2s2f †
+R,αηsfR,α2η2s2
+(S139)
+We next evaluate hstrain from Eq. S138. We separate hstrain into two parts (Eq. S136)
+hstrain,1
+αηs,α2η2s2 =
+1
+NM
+�
+R
+⟨WR,α,η,s|
+�
+k,α′l′,α′
+2l′
+2,η′,s′
+ψ†
+k,l,α′,η′,s′[Hη′
+strain,1(k)]l′α′,l′
+2α′
+2ψk,l′
+2α′,η′,s′|WR,α2,η2,s2⟩
+hstrain,2
+αηs,α2η2s2 =
+1
+NM
+�
+R
+⟨WR,α,η,s|
+�
+k,α′l′,α′
+2l′
+2,η′,s′
+ψ†
+k,l,α′,η′,s′[Hη′
+strain,2(k)]l′α′,l′
+2α′
+2ψk,l′
+2α′,η′,s′|WR,α2,η2,s2⟩
+(S140)
+For hstrain,1, using Eq. S136, Eq. S137 and Eq. S140, we find
+hstrain,1
+αηs,α2η2s2
+=δη,η2δs,s2δα,3−α2
+2πλ2
+0
+Ωtot
+−3βγ0
+2
+ϵxy
+�
+l=±
+�
+Q∈Qlη
+�
+k
+i(−1)αlηe−λ2
+0(k−Q)2e−i πηl
+2 (−1)α+1
+=δη,η2δs,s2δα,3−α2
+3πλ2
+0βγ0
+Ωtot
+ϵxy
+�
+k
+� �
+Q∈Q+
+e−λ2
+0(k−Q)2 +
+�
+Q∈Q−
+e−λ2
+0(k−Q)2�
+(S141)
+In a more compact form, we find
+hstrain,1 = ασxτ0ς0
+α = 3πλ2
+0βγ0
+Ωtot
+ϵxy
+�
+k
+� �
+Q∈Q+
+e−λ2
+0(k−Q)2 +
+�
+Q∈Q−
+e−λ2
+0(k−Q)2�
+≈
+�
+1.3 × 104ϵxy
+�
+meV .
+(S142)
+We next calculate hstrain,2, using Eq. S136, Eq. S137 and Eq. S140, we find
+hstrain,2
+αηs,α2η2s2
+=δη,η2δs,s2δα,3−α2
+2πλ2
+0ℏϵxy
+Ωtot
+�
+l=±
+�
+Q∈Qlη
+�
+k
+l
+�
+η(ky − Qy) + i(−1)α(kx − Qx)
+�
+ei π
+2 lη(−1)αe−λ2
+0(k−Q)2
+=δη,η2δs,s2δα,3−α2
+2πλ2
+0ℏϵxy
+Ωtot
+�
+k
+� �
+Q∈Qη
+�
+η(ky − Qy) + i(−1)α(kx − Qx)
+�
+iη(−1)αe−λ2
+0(k−Q)2
++
+�
+Q∈Q−η
+�
+η(ky − Qy) + i(−1)α(kx − Qx)
+�
+iη(−1)αe−λ2
+0(k−Q)2�
+=δη,η2δs,s2δα,3−α2
+2πλ2
+0ℏϵxy
+Ωtot
+�
+k
+�
+Q∈Q0
+�
+η(ky − Qy − ηq1,y) + i(−1)α(kx − Qx − ηq1,x)
+�
+iη(−1)αe−λ2
+0(k−Q−ηq1)2
++
+�
+η(ky − Qy + ηq1,y) + i(−1)α(kx − Qx + ηq1,x)
+�
+iη(−1)αe−λ2
+0(k−Q+ηq1)2�
+(S143)
+where we let Q0 = {n1bM,1 + n2bM,2|n1, n2 ∈ Z}. Since q1 = kθ(0, 1), q1,x = 0, q1,y = kθ, we find
+hstrain,2
+αηs,α2η2s2
+=δη,η2δs,s2δα,3−α2
+2πλ2
+0ℏϵxy
+Ωtot
+�
+k
+�
+Q∈Q0
+�
+i(−1)α(ky − Qy)f+(k − Q) − iq1,y(−1)αf−(k − Q)
+− (kx − Qx)ηf+(k − Q)
+(S144)
+
+39
+where we define
+f±(k − Q) = ±e−λ2
+0(k−Q+q1)2 + e−λ2
+0(k−Q−q1)2
+(S145)
+Since q1 = kθ(0, 1), f+(k − Q) is an even function of both (k − Q)x and (k − Q)y and f−(k − Q) is an even function of
+(k − Q)x and an odd function of (k − Q)y. Therefore,
+�
+k
+�
+Q∈Q0
+(ky − Qy)f+(k − Q) = 0,
+�
+k
+�
+Q∈Q0
+f−(k − Q) = 0,
+�
+k
+�
+Q∈Q0
+(kx − Qx)f+(k − Q) = 0
+(S146)
+Thus,
+hstrain,2
+αηs,α2η2s2 = 0
+(S147)
+However, we mention that, even though hstrain,2 = 0, Hη
+strain,2 in Eq. S136 can contribute a k-denpendent term in the THF
+model which has been omitted here.
+Combining Eq. S142 and Eq. S147, we have
+ˆHstrain =
+�
+R,αηs,α2η2s2
+ααηs,α2η2s2f †
+R,αηsfR,α2η2s2 =
+α
+�
+R,αηs,α2η2s2
+[hstrain]αηs,α2η2s2f †
+R,αηsfR,α2η2s2
+(S148)
+which is equivalent to the Eq. S95. α is connected to ϵxy via
+α = 3πλ2
+0βγ0
+Ωtot
+ϵxy
+�
+k
+� �
+Q∈Q+
+e−λ2
+0(k−Q)2 +
+�
+Q∈Q−
+e−λ2
+0(k−Q)2�
+≈
+�
+1.3 × 104ϵxy
+�
+meV .
+(S149)
+For a typical strain at the order of ϵxy ∼ 10−3 [148], α ∼ 13meV. We mention that the strain could also induce addition terms
+to the conduction c-electron blocks (c†c) and f-c hybridization blocks (c†f, f †c), which are expected to be small. Here, we
+only consider a minimal model, that is zeroth order in k, to capture the essential effect of the strain. We leave a comprehensive
+construction of strain term (based on Ref. [150]) for future study.
+S6.
+DYNAMICAL MEAN FIELD THEORY: IMPLEMENTATION
+We study dynamical effects of the f-f density-density term using DMFT. The c-f and c-c density terms are included in this
+model on the Hartree level. We neglect the f-c exchange interaction ˆHJ. We treat a Hamiltonian of the form ˆHc + ˆHfc + ˆHU +
+ˆHMF
+W
++ ˆHMF
+V
+where the first two terms compose the non-interacting THF model and the remaining terms are given by
+ˆHU =U
+2
+�
+(α,η,σ)̸=(α′,η′,σ′)
+f †
+α,η,σfα,η,σf †
+α′,η′,σ′fα′,η′,σ′
+− 3.5U
+�
+(α,η,σ)
+f †
+α,η,σfα,η,σ
+(S150)
+ˆHMF
+W
+=Wνf
+�
+(a,η,σ)
+c†
+a,η,σca,η,σ + Wνc
+�
+(α,η,σ)
+f †
+α,η,σfα,η,σ,
+(S151)
+ˆHMF
+V
+=V νc
+�
+(a,η,σ)
+c†
+a,η,σca,η,σ.
+(S152)
+Here, the last two terms are mean-field decoupled and therefore composed of single particle terms, and we have neglected all
+constant terms as an overall shift does not affect the observables we are interested in. The two particle term ˆHU is restricted to the
+f- subspace. We proceed with an LDA+DMFT-style calculation with a correlated subspace made up of f- electrons embedded
+in a larger space of f- and c- electrons. By allowing an arbitrary shift of the chemical potential µ and fixing the total filling
+ν = νf + νc, we can consider an effective interacting Hamiltonian restricted to the the f- subspace:
+ˆHeff
+I
+= ˆHff + ˆHDC
+(S153)
+ˆHff =U
+2
+�
+(α,η,σ)̸=(α′,η′,σ′)
+f †
+α,η,σfα,η,σf †
+α′,η′,σ′fα′,η′,σ′
+(S154)
+ˆHDC = (−3.5U + (ν − 2νf)W − V (ν − νf))
+�
+��
+�
+µDC(νf )
+�
+(α,η,σ)
+f †
+α,η,σfα,η,σ.
+(S155)
+
+40
+The f-electron occupation νf is self-consistently calculated along with the the f- subspace self-energy Σf in at each step of
+the DMFT loop for a set of different total fillings (ν) and temperatures (T). Written in this way, the single particle terms can
+be treated as an iteration-dependent double counting potential, µ(i)
+DC = µDC
+�
+ν(i)
+f
+�
+, where the superscript labels the iteration
+number.
+In the ith iteration of the self-consistency loop, we perform the following sequence of steps:
+1. Embed the old impurity self-energy Σ(i−1)
+f
+into the lattice Green’s function associated to the Hamiltonian ˆHc + ˆHfc +
+ˆH(i−1)
+DC
+.
+2. Fix the total filling to ν.
+3. Calculate the new orbital-resolved fillings ν(i)
+f
+and ν(i)
+c .
+4. Determine the new double counting term µ(i)
+DC.
+5. Use a CT-QMC solver to obtain the new self energy Σ(i)
+f .
+We performed our calculations in parallel using the w2dynamics suite [153, 154]1 as well as the TRIQS family of packages
+[155–157].
+In Fig. 3(a) of the main text, the zero frequency spectral weight and the scattering rate are computed by extrapolating the green
+function and the self-energy respectively on the Matsubara axis. We approximate these by using the value at the first Matsubara
+frequency so that
+Γ = −ImΣ(ω = 0) ≈ −ImΣ(ω0),
+(S156)
+A(ω = 0) = − 1
+π
+��
+k
+TrG(k, ω = 0)
+�
+≈ − 1
+π
+��
+k
+TrG(k, ω0)
+�
+.
+(S157)
+1 https://github.com/w2dynamics/w2dynamics
+
diff --git a/FdE3T4oBgHgl3EQftAvj/content/tmp_files/load_file.txt b/FdE3T4oBgHgl3EQftAvj/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..a913e0e45743bc946817eec2382237b6131fb82e
--- /dev/null
+++ b/FdE3T4oBgHgl3EQftAvj/content/tmp_files/load_file.txt
@@ -0,0 +1,2225 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf,len=2224
+page_content='Symmetric Kondo Lattice States in Doped Strained Twisted Bilayer Graphene  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Hu,1 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Rai,2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Crippa,3 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
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+page_content=' C˘alug˘aru,4 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Wehling,2, 5  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Sangiovanni,3 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Tsvelik,7 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Bernevig4, 1, 8, ∗  1Donostia International Physics Center, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain  2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Institute of Theoretical Physics, University of Hamburg, Notkestrasse 9, 22607 Hamburg, Germany  3Institut f¨ur Theoretische Physik und Astrophysik and W¨urzburg-Dresden Cluster  of Excellence ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='qmat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Universit¨at W¨urzburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 97074 W¨urzburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Germany  4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Princeton University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Princeton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' New Jersey 08544,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' USA  5The Hamburg Centre for Ultrafast Imaging,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 22761 Hamburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Germany  6Institut f¨ur Theoretische Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Goethe Universit¨at Frankfurt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Max-von-Laue-Strasse 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 60438 Frankfurt am Main,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Germany  7Division of Condensed Matter Physics and Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Brookhaven National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Upton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' NY 11973-5000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' USA  8IKERBASQUE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Basque Foundation for Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Bilbao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Spain  We use the topological heavy fermion (THF) model [1] and its Kondo Lattice (KL) formulation [2] to study  the possibility of a symmetric Kondo state in twisted bilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Via a large-N approximation, we find  a symmetric Kondo state in the KL model at fillings ν = 0, ±1, ±2 where a KL model can be constructed [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  In the symmetric Kondo state, all symmetries are preserved and the local moments are Kondo screened by the  conduction electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' At the mean-field level of the THF model at ν = 0, ±1, ±2, ±3 we also find a similar  symmetric state that is adiabatically connected to the symmetric Kondo state [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We study the stability of the  symmetric state by comparing its energy with the ordered (symmetry-breaking) states found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [1] and  find the ordered states to have lower energy at ν = 0, ±1, ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, moving away from integer fillings  by doping holes to the light bands, our mean-field calculations find the energy difference between the ordered  state and the symmetric state to be reduced, which suggests the loss of ordering and a tendency towards Kondo  screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We expect that including the Gutzwiller projection in our mean-field state will further reduce the  energy of the symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In order to include many-body effects beyond the mean-field approximation, we  also performed dynamical mean-field theory (DMFT) calculations on the THF model in the non-ordered phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  The spin susceptibility follows a Curie behavior at ν = 0, ±1, ±2 down to ∼ 2K where the onset of screening  of the local moment becomes visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This hints to very low Kondo temperatures at these fillings, in agreement  with the outcome of our mean-field calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' At non-integer filling ν = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5, ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='8, ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2 DMFT shows  deviations from a 1/T-susceptibility at much higher temperatures, suggesting a more effective screening of local  moments with doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Finally, we study the effect of a C3z-rotational-symmetry-breaking strain via mean-field  approaches and find that a symmetric phase (that only breaks C3z symmetry) can be stabilized at sufficiently  large strain at ν = 0, ±1, ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Our results suggest that a symmetric Kondo phase is strongly suppressed at  integer fillings, but could be stabilized either at non-integer fillings or by applying strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Introduction— The experiments on magic-angle (θ  =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='05◦) twisted bilayer graphene (MATBLG) [4–6] have es-  tablished the existence of a variety of interesting phases [7–  28], including correlated insulating phases [29–39] and super-  conductivity [40–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Their discovery has been followed by  considerable theoretical efforts [45–69] aimed at understand-  ing their origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' An extended Hubbard model has been con-  structed to analyze the interacting physics [60, 70–82], how-  ever, due to the non-trivial topology of the flat bands [83–  91], certain symmetries become non-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Alternatively, an  approach based on a momentum space model has been con-  sidered [92–100], in which correlated insulators [101–108],  superconductivity [109–114], and other correlated quantum  phases [115–119] have been identified and studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Besides,  various numerical calculations [120–127] have also been per-  formed to investigate the correlated nature of the phenom-  ena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, the active phase diagram including the states  at non-integer fillings is not well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The exact map-  ping between the MATBLG and topological heavy-fermion  ∗ bernevig@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='edu  model constructed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [1] could be used for develop-  ments in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This mapping establishes a bridge  between heavy-fermions [3, 128–131] and moir´e systems [1,  2, 132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The presence of localized moments in MATBG is  supported by recent entropy measurements which have found  a Pomeranchuk-type transition [19, 133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' A large entropy ob-  served at high-temperatures, originates from weakly interact-  ing local moments whose fluctuations are quenched at low  temperatures [19, 133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Since a similar behavior is observed  in heavy fermion systems [3, 128], where the fluctuating lo-  cal moments are screened by conduction electrons (Kondo  effect), this observation is suggestive of a Kondo state with  screened local moments in MATBLG [128, 134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  In this paper we use the KL model [2], to describe and study  the symmetric Kondo (SK) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We focus on integer fillings  ν = 0, ±1, ±2, where a KL model can be constructed [2] (a  KL description fails at ν = ±3 as demonstrated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  The SK phase preserves all symmetries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' the local moments  are screened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We discuss the topology and the band struc-  ture of the SK state and extend the study to the THF model  where we identify the symmetric state that is adiabatically  connected to the SK state [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In order to address integer  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='04673v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='str-el]  11 Jan 2023  2  and fractional fillings on equal footing, we perform both a  mean-field and a dynamical mean-field theory (DMFT) calcu-  lations of the THF defining a “periodic Anderson model” with  a momentum-dependent hybridization between the correlated  f- and the dispersive c-electrons in the non-ordered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Our mean-field calculations indicate that the energy of the  symmetric state is higher than that of the ordered (symmetry-  breaking) states found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [1] at integer filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We thus  conclude that ordered states are more energetically favored  at integer fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' DMFT supports this picture as we obtain  a Curie behavior of the local spin susceptibility at integer  fillings, down to very low temperatures ∼ 2K, hinting to a  very small Kondo scale (lower than ∼ 2K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Together with the  mean-field results we would then expect an ordered state to be  favored at low temperatures for these fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Turning to the effect of doping, instead, from our mean-  field analysis, we find that the energy difference between the  symmetric phase and the ordered phase can be sizeably re-  duced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Doping hence suppresses the ordering and enhances  the Kondo screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This conclusion is further supported  by the DMFT results at non-integer fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Here, we find  clear deviations from the Curie behavior in the entire range  from 10K down to ∼1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Even though it is computationally  too demanding to go further down in temperature, we point  out that our evidence of a clear-cut difference in the screening  properties between integer and fractional fillings is reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  DMFT treats indeed local quantum fluctuations exactly [135]  and hence takes into account the many-body processes that  can potentially lead to the screening of local moments at any  filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Since realistic samples have intrinsic strains, we finally  study the effect of a C3z-breaking strain on the symmetric  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Our mean-field calculations show that the order is sup-  pressed by the strain effect and a symmetric state can be sta-  bilized at a sufficiently large strain at ν = 0, ±1, ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  In summary, we conclude that a symmetric Kondo phase  is absent at integer fillings of MATBLG, but could in princi-  ple be stabilized either at non-integer fillings or by applying  strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Topological Heavy Fermion model and the Kondo lattice  model— The THF model [1] contains two types of electrons:  topological conduction c-electrons (ck,aηs) and localized f-  electrons (fR,αηs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The operator ck,aηs annihilates conduc-  tion c-electron with momentum k, orbital a ∈ {1, 2, 3, 4},  valley η ∈ {+, −} and spin s ∈ {↑, ↓}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' At the ΓM-point for  each valley and each spin projection, c-electrons in the orbital  1 and 2 transform according to the Γ3 irreducible represen-  tation (of magnetic space group P6′2′2) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The remaining  c-electrons (a = 3, 4) at the same valley with the same spin  projection transform in the Γ1 ⊕ Γ2 reducible representation  (of magnetic space group P6′2′2) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We will call them Γ3 c-  electrons (a = 1, 2) and Γ1⊕Γ2 c-electrons (a = 3, 4) respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' fR,αηs is the annihilation operator of the f-electron at  the moir´e unit cell R with orbital α ∈ {1, 2}, valley η and  spin s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The Hamiltonian of the THF model [1, 136] is  ˆHT HF = ˆHc + ˆHfc + ˆHU + ˆHW + ˆHV + ˆHJ  (1)  where ˆHc describes the kinetic term of conduction elec-  trons,  ˆHfc describes the hybridization between f-c elec-  trons [1, 136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The interactions include an on-site Hubbard  interaction of f-electrons ( ˆHU with U = 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='95meV), a re-  pulsion between f- and c-electrons ( ˆHW with W = 48meV),  a Coulomb interaction between c-electrons ( ˆHV with V (q =  0)/Ω0 = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='33meV and Ω0 the area of moir´e unit cell), and a  ferromagnetic exchange coupling between f-and c-electrons  ( ˆHJ with J = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='38meV) [1, 136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Based on the THF model [1], a KL model of MATBLG  has been constructed via a generalized Schrieffer–Wolff (SW)  transformation as shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The KL model is de-  scribed by the following Hamiltonian  ˆHKondo = ˆHc + ˆHcc + ˆHK + ˆHJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (2)  where  ˆHc, ˆHJ come from the original THF model and  ˆHcc, ˆHK emerge from the SW transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' ˆHcc is the one-  body scattering term of Γ3 c-electrons with the form of  ˆHcc =  �  |k|<Λc  �  a,a′∈{1,2}  η,s  e−|k|2λ2 : c†  k,aηsck,a′ηs :  �  −1  Dνc,νf  +  −1  Dνc,νf  � �  γ2/2  γv′  ⋆(ηkx − iky)  γv′  ⋆(ηkx + iky)  γ2/2  �  a,a′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (3)  λ is the damping factor of the f-c hybridization in the THF  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' γ, v′  ⋆ characterize the zeroth order and linear order  f-c hybridization of the THF model with v′  ⋆ characterizing  a k-dependent hybridization matrix [1, 136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' D1,νc,νf and  D2,νc,νf are defined as  D1,νc,νf = (U − W)νf − U  2 + (−V (0)  Ω0  + W)νc  D2,νc,νf = (U − W)νf + U  2 + (−V (0)  Ω0  + W)νc ,  (4)  where νf, νc are the filling of f- and c-electrons deter-  mined from the calculations of the THF model at the zero-  hybridization limit [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We point out that in the single-orbital  Kondo model, the one-body scattering term merely introduces  a chemical potential shift [3, 137] of the c-electrons and is  usually omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, in our model, ˆHcc cannot be ig-  nored for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' First, ˆHcc is k-dependent and thus  introduces additional kinetic energy to the conduction elec-  trons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S24, we observe the k-dependency mainly  comes from the linear k term that is proportional to v′  ⋆ and  can be traced back to the k-dependency of the hybridization  matrix in the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Secondly, even if we drop the v′  ⋆  term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S24 (v′  ⋆ = 0 corresponding to the chiral limit [1]),  ˆHcc still produces an energy shift for the Γ3 c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Thus  ˆHcc leads to the energy splitting between Γ3 and Γ1 ⊕ Γ2 c-  electrons and cannot be simply treated as a shift of the chem-  ical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  ˆHK is the Kondo interaction between f- and Γ3 c-electrons  whose explicit form is given in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [2, 136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The Kondo  interaction consists of two parts: the zeroth order Kondo in-  teraction proportional to γ2/Dνc,νf and the first order Kondo  3  interaction proportional to γv′  ⋆/Dνc,νf , where D−1  νc,νf  =  −D−1  1,νc,νf + D−1  2,νc,νf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The zeroth order Kondo interaction  term describes the antiferromagnetic interaction between the  U(8) moments of the f- and the Γ3 c-electrons and has a U(8)  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The linear-order Kondo interaction only has a flat  U(4) symmetry and is k-dependent [1, 136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  ˆHJ is the fer-  romagnetic exchange interaction between Γ1 ⊕ Γ2 c- and f-  electrons that already exists in the TFH model [1, 136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We  also note that, for both the THF model and the KL model,  ground states at filling ν and −ν are connected by a charge-  conjugation transformation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This can be broken by other  one-body terms which we did not consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Therefore, in  what follows, we only focus on ν ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Mean-field Hamiltonian of the Kondo model— We next per-  form a mean-field study of the KL model [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This MF sup-  presses the RKKY interaction and essentially restores the hy-  bridization term ˆHfc of the original periodic Anderson model,  but in a renormalized form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' It becomes exact in the N → ∞  limit (we have N = 4 here which corresponds to the approxi-  mate flat U(4) symmetry of the KL Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' At  the mean-field level, the Kondo interaction ˆHK can be written  as (see Supplementary Materials (SM))  ˆHMF  K  =  �  R,|k|<Λc  �  αηs  eik·R−|k|2λ2/2  √NMDνc,νf  �  − f †  R,αηsck,aηs  �  γ2V ∗  1 + γv′  ⋆V ∗  2  V ∗  1 (ηkx − iky)  V ∗  1 (ηkx + iky) γ2V ∗  1 + γv′  ⋆V ∗  2  �  α,a  + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  �  + NM  �  γ2|V ∗  1 |2 + γv′  ⋆(V ∗  1 V ∗  2 + V ∗  2 V ∗  1 )  �  + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (5)  where we have introduced the following mean fields  V ∗  1 =  �  R,|k|<Λc  �  αηs  eik·R−|k|2λ2/2  √NMNM  ⟨Ψ|f †  R,αηsck,αηs|Ψ⟩  V ∗  2 =  �  R,|k|<Λc  �  αaηs  eik·R−|k|2λ2/2  √NMNM  (ηkxσx + kyσy)αa  ⟨Ψ|f †  r,αηsck,αηs|Ψ⟩  (6)  with |Ψ⟩ being the mean-field ground state, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' denotes  the Hartree term (⟨f †f⟩, ⟨c†c⟩) whose explicit formula is in  the Supplementay Materials (SM) [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Several points are  in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' First, as we have mentioned above, the mean field  restores the hybridization of the original Anderson model, but  in a renormalized form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' V ∗  1 , V ∗  2 describe the renormalized  hybridization between the f- and Γ3 c-electrons driven by the  Kondo interactions between two types of electrons(f and Γ3  c) [3, 138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Second, it is necessary to keep the Hartree con-  tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In the canonical Kondo model, the Hartree term  merely produces a chemical potential shift (in the case without  symmetry breaking) and hence can be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Here, Hartree  contributions (see SM [136]) are k-dependent because of the  k-dependency of the Kondo interactions, and thus contribute  to the dispersion of the conduction c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Furthermore,  since only Γ3 c-electrons contribute to the Kondo interaction,  the Hartree term also produces an energy splitting between the  Γ3 and the Γ1 ⊕ Γ2 c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  As for ˆHJ, we perform a similar mean-field decoupling  ˆHMF  J  =J  �  R,|k|<Λc,αηs  eik·R  √NM  �  V3δ1,η(−1)α+1f †  R,αηsck,α+2ηs  + V4δ−1,η(−1)α+1ηf †  R,αηsck,α+2ηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  �  − JNM  �  |V3|2 + |V4|2  �  + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (7)  where we have introduced the following two mean-field aver-  ages that describe the f-c hybridization:  V3 =  �  R,|k|<Λc  �  αη,s  eik·Rδ1,η(−1)α+1  √NMNM  ⟨Ψ|f †  R,αηsck,α+2ηs|Ψ⟩  V4 =  �  R,|k|<Λc  �  αη,s  eik·Rδ−1,η(−1)α+1  √NMNM  ⟨Ψ|ηf †  R,αηsck,α+2ηs|Ψ⟩ ,  (8)  To impose the filling of the f-electrons to be νf, we intro-  duce the Lagrange multiplier [134, 136, 138]:  ˆHλf =  �  R,αηs  λf  �  : f †  R,αηsfR,αηs : −νf  �  (9)  with λf to be determined self-consistently [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Finally, we  introduce a chemical potential µc to the c-electrons  ˆHµc = −µc  �  |k|<Λc,aηs  : c†  k,aηsck,aηs : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (10)  In the calculation, we tune µc and λc together to fix both the  total filling ν = νf + νc and the filling of f-electrons [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  The final mean-field Hamiltonian of the KL model now is  ˆHMF  Kondo = ˆHc + ˆHcc + ˆHMF  K  + ˆHMF  J  + ˆHλf + ˆHµc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (11)  We then self-consistently solve the mean-field equations  (see SM [136]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' At ν = νf = 0, −1, −2 (where a KL model  can be constructed), we identify a SK state that preserves  all the symmetries and is characterized by V ∗  1  ̸= 0, V ∗  2  ̸=  0, V ∗  3 = 0, V ∗  4 = 0 [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We comment that the exchange  interaction ˆHJ [1] between f- and Γ1 ⊕ Γ2 c-electrons is fer-  romagnetic, and hence disfavors the singlet formation or hy-  bridization (V3, V4) between f- and Γ1 ⊕ Γ2 c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We  find that V3, V4 vanish (their numerical amplitudes are smaller  than 10−5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In fact, ˆHJ favors the triplet formation or pair-  ing formation (f †c†), where both lead to a symmetry-breaking  state at the mean-field level and are beyond our current con-  sideration of SK state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Properties of the symmetric Kondo phase— In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 1, we  plot the band structure of the SK phase and compare it with  the non-interacting THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We find the hybridization in  4  (a)  (b)  (c)  (d)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (a) Band structure of the non-interaction THF model at ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' (b), (c), (d) Band structure of the SK phase at ν = 0, −1, −2  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  the SK state defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 5 to be enhanced compared to the  non-interacting limit of THF model, which is clear from the  increase of the gap of the Γ3 states at the Γ point [1] from  its non-interacting value 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='75meV at ν = 0, to 168meV,  190meV, 213meV at ν = 0, −1, −2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We also  find that in the SK phase the bandwidths of the flat bands at  ν = −1, −2 become 16meV, 53meV, which are (much) larger  than the non-interacting flat-band bandwidth (= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='4meV) of  the THF model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, at ν = 0, the flat-band  bandwidth is the same as the non-interacting flat-band band-  width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This is because, at ν = 0, the one-body scattering  term and the Hartree contributions from ˆHK, ˆHJ all van-  ish [136], and the enhanced hybridization pushes the remote  bands away from the Fermi energy and does not change much  the band structures of the flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In addition, unlike the  non-interacting case, here we found the flat bands are mostly  formed by Γ1⊕Γ2 c-electrons with orbital weights larger than  70% at ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This is because the large f-c hy-  bridization induced by V1, V2 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 5) pushes the energy of Γ3  c- and f-electrons away from the Fermi energy and reduces  their orbital weights [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  The flat bands in the SK phase form Γ1 ⊕ Γ2, M1 ⊕ M2  and K2K3 representations at ΓM, MM, KM respectively, and  have the same topology as the flat bands in the non-interacting  THF model [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' More explicitly, the flat bands for each val-  ley and each spin projection belong to a fragile topology [1]  at ν = −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' At ν = 0, due to the additional particle-hole  symmetry, flat bands have a stable topology [1, 85, 91, 136],  which is characterized by the odd winding number of the Wil-  son loop as shown in supplementary material [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We men-  tion that the interplay between Kondo effect and the topologi-  cal bands has also been studied in various other systems [139–  144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Symmetric phase in the topological heavy-fermion model—  We next investigate the similar symmetric phase in the THF  model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We first focus on integer fillings ν  =  0, −1, −2, −3 and perform the mean-field calculations of  THF as introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [1, 136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' By enforcing the mean-  field Hamiltonian to preserve all the symmetries, we are able  to identify a symmetric state that preserves all the symmetries  at ν = 0, −1, −2, −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' To observe the stability of the sym-  metric phase, we compare its energy (Esym) with the energy  (Eorder) of the ordered (symmetry-breaking) ground states  derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The ordered ground states in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [1]  are a Kramers inter-valley-coherent (KIVC) state at ν = 0,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5  0  10  20  30  40  50  E/meV  = -3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5  0  10  20  30  40  50  = -2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5  0  10  20  30  40  50  = -1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5  0  10  20  30  40  50  = 0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Doping dependence of the ground state energy difference  ∆E = Esym − Eorder near integer fillings νt = 0, −1, −2, −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  a KIVC+valley polarized (VP) state at ν = −1, a KIVC state  at ν = −2 and a VP state at ν = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We point out that at  ν = −3 other states with lower energy exist [145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In our  numerical calculations, we find ∆E = Esym − Eorder =  47meV, 40meV, 33meV, 23meV at ν = 0, −1, −2, −3 re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In all integer filling cases, the symmetric states  have higher energy, and the ground states cannot be the sym-  metric state, which is consistent with the previous calcula-  tions of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [1, 2, 103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Note that our mean-field calcula-  tion does not include a Gutzwiller projection to fix the fill-  ing of f-electrons at each site, and hence we expect the en-  ergy of projected symmetric states will be lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, as  we show later, after including the effect of local correlations  via the DMFT approach, we confirm that the Kondo phase,  which is adiabatically connected to the symmetric phase in  the mean-field calculations, is strongly suppressed at integer  fillings (down to temperatures ∼ 1-2K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This further supports  the development of ordering at integer fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Effects of doping— We next investigate the effects of dop-  ing, first at the level of mean-field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We stick to a nar-  row region ν ∈ [νint−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5, νint+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5] near each integer filling  νint = 0, −1, −2, −3 and compare the energies of the ordered  states Eord and the symmetric states Esym in the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  To find the ordered state solutions, we first initialize the cal-  culations with the mean-field solutions at integer filling νint  and fill the mean-field bands up to current filling ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We then  self-consistently solve the mean-field equations and calculate  the energy of the resulting states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We obtain the symmetric-  state solution in a similar manner but take the symmetric so-  lution at νint as initialization and enforce the symmetry of  the mean-field Hamiltonian during the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 2  displays a plot of the difference of the ground state energies  5  ∆E = Esym −Eorder as a function of doping ∆ν = ν −νint  near νint = 0, −1, −2, −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We observe that hole doping at  ν = 0, −1, −2, −3 and electron doping at ν = 0 decreases  the ∆E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Doping holes at ν = 0, −1, −2, −3 and doping elec-  trons at ν = 0 to the ordered states is equivalent to doping the  light (dispersive) bands which are mostly formed by conduc-  tion c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' After doping, the conduction electrons will  stay close to the Fermi energy, and then enhance the tendency  towards the Kondo effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  However, doping electrons at ν = −1, −2 to the ordered  states is equivalent to doping heavy (flat) bands which mostly  come from the f-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Because of the flatness of the  band, we find the nature of the ordered states will change with  doping (see SM [136]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' For example at ν = 2, the KIVC  order is suppressed by the electron doping (see SM [136]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Thus, ∆E will be affected by both, changes of ordering and  doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, a sizeable change of the order parameters  is not observed for hole doping at ν = 0, −1, −2, −3 and also  electron doping at ν = 0 (see SM [136]), because we are dop-  ing conduction c-electrons in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We also point out  the complexity of ν = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' First, other states that break trans-  lational invariant [145] could have lower energy than the VP  state we currently considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Second, even for the VP state,  doping electrons is equivalent to doping both heavy and light  bands [1], since both light and heavy bands appear in the elec-  tron doping case [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In practice, as we increase ∆ν, we find  that, at ν = −1, −2, ∆E will first decrease and then increase  and, at ν = −3, ∆E will always increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  In summary, we conclude that hole doping can suppress the  long-range order and enhance the tendency towards the Kondo  effect near ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Electron doping, depending on the  fillings, could also enhance the tendency toward the Kondo ef-  fect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, on the electron doping side, the change of or-  der moments indicates the importance of the correlation effect  which could be underestimated in the mean-field approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In  the next section, we provide a more comprehensive study of  the doping effect using the DMFT calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Dynamical mean-field theory results of the THF model—  In the following, we present the dynamical mean-field the-  ory resultss of the THF model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 1), where we describe the  local quantum many-body effects of the density-density Hub-  bard term ˆHU within the f-subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The ˆHW and ˆHV in-  teractions involving density fluctuations on the c orbitals are  accounted for at the static mean-field level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We neglect or-  dered phases and perform calculations in the non-ordered one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  There, we focus in particular on lifetime effects, quasiparticle  weights and exploit the ability of DMFT to take local vertex  corrections to the spin-spin correlation function into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  First, DMFT finds a qualitative difference between the  strong quasiparticle renormalization when the f+c manifold  is occupied with an integer number of electrons and a lighter  Fermi liquid at fractional fillings: this can be seen from the  scattering rate Γf = −ImΣf(ω = 0) which is shown as a  function of the total filling ν at T = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='6K (light blue empty  circles) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The largest scattering rates are found close  to ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0, -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0 and -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0, progressively decreasing as one  moves away from the charge neutrality point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Correspond-  ingly, the spectral weight at the Fermi level (black and grey  solid circles) is suppressed at these fillings, with a residual  nonzero value due to the finite temperature on the one hand  and the resilient f/c hybridization on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3(b) illustrates the temperature-dependent screening of  the local magnetic moment on the f orbitals at different fill-  ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' A flat T · χloc  spin(ω = 0) indicates Curie behavior and a  well-defined effective local moment, while deviations signal  the onset of screening and a crossover towards a renormalized  Pauli-like behavior, in agreement with the general expecta-  tion of zero-temperature Fermi-liquid in the periodic Ander-  son model [146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' While at ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0, -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0 and -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0 the 1/T  local spin susceptibility persists down to 1-2 K, the fractional  fillings deviate from Curie at much higher temperatures, in  line with the better Fermi-liquid nature signaled by the single-  particle quantities in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  As in the Hartree-Fock treatment of the THF model, DMFT  confirms the difference between electron doping and hole dop-  ing (particle-hole asymmetry) near integer filling ν = −1 and  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Here, DMFT reveals particle-hole asymmetric scattering  rates (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 3(a)) and also in the difference of effective local  moments at ν = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='8 and −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  In summary, our DMFT calculations confirm that the  Kondo phase is strongly suppressed at integer fillings ν =  0, −1, −2, increasing the propensity towards long-range order  at these fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, by doping the system, the develop-  ment of Kondo screening (starting from ∼ 10K) is observed,  which suggests that doping could enhance the Kondo effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  This picture is consistent with our mean-field calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Effects of strain— Since twisted bilayer graphene samples  exhibit intrinsic strain [147] and the ordered states are disfa-  vored by strain, we investigate the effect of strain on the sym-  metric state of THF model via mean-field approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We focus  on ν = 0, −1, −2, −3 and introduce the following Hamilto-  nian [136] that qualitatively characterizes the effect of strain  ˆHstrain = α  �  R,ηs  (f †  R,1ηsfR,2ηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=')  where α is proportional to the strain amplitude (we leave the  construction of a realistic strain Hamiltonian [148–150] for  a future study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' A non-zero α breaks the C3z symmetry but  preserves all other symmetries [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We compare the ground  state energies of the symmetric states (Estrain  sym  ) and the or-  dered states (Estrain  ord  ) at non-zero strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' To obtain the sym-  metric state solution, we solve the mean-field equations by  requiring the mean-field Hamiltonian to satisfy all symme-  tries except for the C3z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We obtain the solution of the or-  dered states by initializing the mean-field calculations with  the ordered ground states at zero strain and then perform self-  consistent calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 4, we plot the difference be-  tween the ground state energies of the symmetric and the or-  dered states ∆Estrain = Estrain  sym  − Estrain  order as a function of  the effective strain amplitude α with 0meV < α < 20meV at  ν = 0, −1, −2, −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We observe that ∆E at ν = 0, −1 van-  ishes at sufficiently large strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' A detailed analysis [136] of  the wavefunction shows that the ordered state cannot be sta-  bilized and converged to a C3z broken symmetric solution at  large strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' By further increasing strain, we find a symmet-  ric state at ν = −2 can also be stabilized at α ∼ 45meV  6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5  A(  = 0)  a  Atot  Af  Ac  f  0  2  4  6  8  10  T [K]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0  T  loc  spin(  = 0)  b  total  eff  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='39  total  eff  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='34  total  eff  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='19  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='00  =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='50  =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='80  =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='00  =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='20  =  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='00  2  1  0  2  0  f/c  f  c  0  20  40  f [meV]  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' DMFT solution of the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' (a) Doping ν dependent low-energy spectral function at the Fermi level (A(ω = 0)) for the  full system Atot, the c- (Ac) and the f-electrons (Af) at 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Also shown is the scattering rate Γf as extracted from the local f-electron  self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' (b) Effective local moment T · χloc  spin(ω = 0) as a function of temperature T for different doping levels ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='0  /meV  0  2  4  6  8  10  12  14  16  Estrain/meV  =  0  =  1  =  2  =  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Energy difference ∆Estrain = Estrain  sym  − Estrain  ord  between  the symmetric state that only breaks C3z symmetry (Estrain  sym  ) and  the ordered state (Estrain  ord  ) as a function of α - a parameter charac-  terizing the strain amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We note that even at zero strain α = 0,  a symmetric state that only breaks C3z symmetry has lower energy  than the fully symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Thus ∆Estrain at α = 0 is smaller  than the corresponding ∆E in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (see SM [136]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We conclude that a symmetric phase can be  stabilized by sufficiently large strain at ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' As  for ν = −3, we mention that other ordered states, that break  translational symmetry and have lower energy than the VP  state, exist even at zero strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We leave a systematical analy-  sis of ν = −3 for future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Finally, we comment that even  at zero strain, a symmetric state that breaks C3z symmetry has  lower energy than the fully symmetric state that preserves all  the symmetries (including C3z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Therefore, ∆Estrain (energy  difference between a symmetric state that only breaks C3z and  an ordered state) at zero strain α = 0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 4 is smaller  than the corresponding ∆E (energy difference between a fully  symmetric state and an ordered state) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Discussion and summary— Our main result is that an or-  dered state, instead of a SK state, will be the ground state of  the system at integer filling ν = 0, −1, −2, −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Our mean-  field calculations of THF model indicate ground state energy  of the symmetric state is higher than the one of the ordered  states at these fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Via DMFT calculations, we find the  Kondo temperature to be substantially smaller than 2K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Thus,  we conclude the Kondo effect is suppressed at integer filling  ν = 0, −1, −2, −3, and the ground state is likely to be an  ordered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, our mean-field calculations suggest  doping can reduce the energy difference between the symmet-  ric state and the ordered state enhancing the tendency towards  the SK state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This has also been confirmed by the DMFT  calculations which show a strong deviation from the Curie  Weiss law at non-integer fillings ν = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='8, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2 al-  ready around 10K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Furthermore, we show that a sufficiently  large C3z breaking strain could also stabilize a symmetric  state that only breaks the C3z symmetry at ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Therefore, we conclude both doping and strain enhance the  Kondo effect and could, in principle, stabilize a SK state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Our  results may explain the recent entropy experiments [18, 19]  which reveal a high-temperature phase with fluctuating mo-  ments and a low-temperature Fermi liquid phase with unpolar-  ized isospins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This could be understood as a sign of screening  of the local moments and the development of the SK phase at  low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  As far as the SK state is concerned, we have performed a  systematic study of its band structure and topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Via the  mean-field approach, we successfully identified the SK state  in the KL model, and a symmetric state, that is adiabatically  connected to the SK state, in the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' For the SK  state in the KL model, we find that the Γ3 states near the ΓM  point have been pushed away, and the bandwidth of the flat  bands is enlarged at ν = −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The hybridization between  f-electrons and Γ3 c-electons is enhanced by the Kondo in-  teractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Consequently, the flat bands are mostly formed by  Γ1 ⊕ Γ2 c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The topology of the flat bands remains  the same as in the non-interacting case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, for the sym-  metric state in the THF model, the enhanced f-c hybridization  does not appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We mention that the mean-field solution of  7  the symmetric state in the THF model underestimates the cor-  relation effect, which could be the origin of the weak f-c hy-  bridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We expect introducing a Gutzwiller projector will  give a more precise description of the symmetric state in the  THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Note added— After finishing this work, we learned that re-  lated, but not identical, results have also recently been ob-  tained by the S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Das Sarma’s [151], P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Coleman’s [134], and  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Song’s groups [152].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We also mention that results from Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Song’s group are compatible with our DMFT results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Acknowledgements— B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' B.’s work was primarily sup-  ported by the DOE Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  DE-SC0016239, the Si-  mons Investigator Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 404513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' was supported  by the European Research Council (ERC) under the Euro-  pean Union’s Horizon 2020 research and innovation program  (Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 101020833).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' was supported  by a Hertz Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' was supported by the DOE  Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' DE-SC0016239 and the Simons Investigator Grant  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 404513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' was supported by the Office of Ba-  sic Energy Sciences, Material Sciences and Engineering Divi-  sion, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Department of Energy (DOE) under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  DE-SC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' R, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
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+page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' thank  the Deutsche Forschungsgemeinschaft (DFG, German Re-  search Foundation) for funding through QUAST FOR 5249-  449872909 (Projects P4 and P5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' acknowledges funding  from the European Commission via the Graphene Flagship  Core Project 3 (grant agreement ID: 881603).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
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+page_content='00175 [cond-mat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  [157] Markus Aichhorn, Leonid Pourovskii, Priyanka Seth, Veron-  ica Vildosola, Manuel Zingl, Oleg E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Peil, Xiaoyu Deng,  Jernej Mravlje, Gernot J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Kraberger, Cyril Martins, Michel  Ferrero, and Olivier Parcollet, “TRIQS/DFTTools: A TRIQS  application for ab initio calculations of correlated materials,”  Computer Physics Communications 204, 200–208 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  13  Supplementary Materials  CONTENTS  References  7  S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Toplogical heavy-fermion model  14  S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Kondo lattice model  15  S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Symmetry  16  S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Mean-field solutions of the Kondo lattice model  16  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Mean-field decoupling of ˆHK  17  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Fock term  17  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Hartree term  18  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Fock and Hartree terms  19  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Mean-field decoupling of ˆHJ  20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Fock term  20  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Hartree term  21  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Fock and Hartree terms  21  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Filling constraints and mean-field equations  21  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Mean-field equations of the symmetric Kondo state  22  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Properties of the symmetric Kondo state  24  S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Mean-field solutions of the topological heavy-fermion model  27  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Mean-field equations of fully symmetric state  27  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Mean-field equations of the symmetric state in the presence of strain  29  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Effect of doping  30  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Effect of strain  31  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' ν = −1  32  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' ν = −2  34  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Discussions about ν = −3  36  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Strain  36  S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Dynamical mean field theory: implementation  39  14  S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  TOPLOGICAL HEAVY-FERMION MODEL  The topological heavy-fermion (THF) model introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [1] takes the following Hamiltonian  ˆH = ˆHc + ˆHfc + ˆHU + ˆHJ + ˆHW + ˆHV + ˆHµ  (S12)  The single-particle Hamiltonian of conduction c-electrons has the form of  ˆHc =  �  η,s,a,a′,|k|<Λc  H(c,η)  a,a′ (k)c†  kaηscka′ηs  ,  H(c,η)(k) =  �  02×2  v⋆(ηkxσ0 + ikyσz)  v⋆(ηkxσ0 − ikyσz)  Mσx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  �  (S13)  where σ0,x,y,z are identity and Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' ckaηs represents the annihilation operator of the a(= 1, 2, 3, 4)-th conduction  band basis of the valley η(= ±) and spin s(=↑, ↓) at the moir´e momentum k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' At ΓM point (k = 0) of the moir´e Brillouin zone  (MBZ), ck1ηs, ck2ηs form a Γ3 irreducible representation (of P6′2′2 group), ck3ηs, ck4ηs form a Γ1 ⊕ Γ2 reducible (into Γ1 and  Γ2 - as they are written, the ck3ηs, ck4ηs are just the σx linear combinations of Γ1 ± Γ2 ) representation (of P6′2′2 group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Λc is  the momentum cutoff for the c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' N is the total number of moir´e unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The parameter values are v⋆ = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='303eV·˚A,  M = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='697meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  The hybridization between f and c electrons has the form of  ˆHfc =  1  √NM  �  |k|<Λc  R  �  αaηs  �  eik·R− |k|2λ2  2  H(fc,η)  αa  (k)f †  Rαηsckaηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  �  ,  (S14)  where fRαηs represents the annihilation operators of the f electrons with orbital index α(= 1, 2), valley index η(= ±) and spin  s(=↑, ↓) at the moir´e unit cell R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' NM is the number of moir´e unit cells and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3376aM is the damping factor, where aM is  the moir´e lattice constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The hybridization matrix H(fc,η) has the form of  H(fc,η)(k) =  �γσ0 + v′  ⋆(ηkxσx + kyσy), 02×2  �  (S15)  which describe the hybridization between f electrons and Γ3 c electrons (a = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The parameter values are γ = −24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='75meV,  v′  ⋆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='622eV · ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  ˆHU (U = 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='89meV) describes the on-site interactions of f-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  ˆHU = U  2  �  R  : nf  R :: nf  R :,  (S16)  where nf  R = �  αηs f †  RαηsfRαηs is the f-electrons density and the colon symbols represent the normal ordered operator with  respect to the normal state: : f †  Rα1η1s1fRα2η2s2 := f †  Rα1η1s1fRα2η2s2 − 1  2δα1η1s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α2η2s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  The ferromagnetic exchange interaction between f and c electrons ˆHJ is defined as  HJ = −  J  2NM  �  Rs1s2  �  αα′ηη′  �  |k1|,|k2|<Λc  ei(k1−k2)·R(ηη′ + (−1)α+α′) : f †  Rαηs1fRα′η′s2 :: c†  k2,α′+2,η′s2ck1,α+2,ηs1 :  (S17)  where J = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='38meV and : c†  k2,α′+2,η′s2ck1,α+2,ηs1 := c†  k2,α′+2,η′s2ck1,α+2,ηs1 − 1  2δk1,k2δα,α′δη,η′δs1,s2  The repulsion between f and c electrons ˆHW has the form of  ˆHW =  �  η,s,η′,s′,a,α  �  |k|<Λc,|k+q|<Λc  Wae−iq·R : f †  R,aηsfR,aηs :: c†  k+q,aη′s′ck,aη′s′ :  (S18)  where we take W1 = W2 = 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='03meV and W3 = W4 = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='20meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  The Coulomb interaction between c electrons has the form of  ˆHV =  1  2Ω0N  �  η1s1a1  �  η2s2a2  �  |k1|,|k2|<Λc  �  q  |k1+q|,|k2+q|<Λc  V (q) : c†  k1a1η1s1ck1+qa1η1s1 :: c†  k2+qa2η2s2ck2a2η2s2 :  (S19)  15  where Ω0 is the area of the moir´e unit cell and V (q = 0)/Ω0 = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='33meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We will always treat ˆHV at mean-field level (int  both the THF model and the Kondo lattice (KL) model) [1]  ˆHV ≈ V (0)  Ω0  νc  �  |k|<Λc,a,η,s  c†  k,aηsck,aηs − V (0)  2Ω0  NMν2  c + V (0)  Ω0  �  |k|<Λc  8νc  (S20)  where νc is the filling of c electrons νc =  1  NM  �  |k|<Λc,a,η,s⟨Ψ| : c†  k,aηsck,aηs : |Ψ⟩ with |ψ⟩ the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Finally, we introduce a chemical potential term  ˆHµ = −µ  �  |k|<Λc,aηs  c†  k,aηsck,aηs − µ  �  R,αηs  f †  R,αηsfR,αηs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (S21)  S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  KONDO LATTICE MODEL  The Kondo lattice model is derived by performing a generalized Schrieffer-Wolff (SW) transformation on the topological  heavy fermion model (detailed derivation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The Hamiltonian has the form of  ˆHKondo = ˆHc + ˆHV + ˆHW + ˆHJ + ˆHK + ˆHcc − ˆHµc  (S22)  where ˆHc (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S13), ˆHV (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S19) and ˆHW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S18) and ˆHJ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S17) come from the original TFH model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The Kondo  interactions and the one-body scattering term are  ˆHK =  �  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='|k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='|k′|<Λc  �  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s′  ei(k−k′)R−|k|2λ2/2−|k′|2λ2/2  NMDνc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='νf  : f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′ :: c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′η′s′ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηs :  �  γ2δα′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′δα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a + γv′  ⋆δα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a[η′k′  xσx − k′  yσy]α′a′ + γv′  ⋆δα′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′[ηkxσx + kyσy]αa  �  (S23)  and  ˆHcc = −  �  |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s  �  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′∈{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2}  e−|k|2λ2�  1  D1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='νc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='νf  +  1  D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='νc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='νf  � �  γ2/2  γv′  ⋆(ηkx − iky)  γv′  ⋆(ηkx + iky)  γ2/2  �  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′ : c†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′ηs : (S24)  where  D1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='νc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='νf = (U − W)νf − U  2 + (−V0  Ω0  + W)νc  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='νc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='νf = (U − W)νf + U  2 + (−V0  Ω0  + W)νc  Dνc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='νf =  �  −  1  D1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='νc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='νf  +  1  D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='νc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='νf  �−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (S25)  We point out that, at ν = νf = νc = 0, D1,νc,νf = −D2,νc,νf and the on-body term ˆHcc(= 0) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We note that in the Kondo model the filling of f electron at each site is fixed to be νf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Then we can replace �  αηs :  f †  R,αηsfR,αηs : with νf and ˆHW becomes  ˆHW =  �  |k|<Λc,|k′|<Λc,aηs  �  Q  Wνf : c†  k,aη′s′ck′,aηsδk,k′+Q  (S26)  where Q ∈ {mbM1 + mbM2|m, n ∈ Z} and bM1, bM2 are the reciprocal lattice vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' If we focus on the conduction  electrons within the first MBZ, we can replace δk,k′+Q by δk,k′ and  ˆHW =  �  |k|<Λc,aηs  �  Q  Wνf : c†  k,aη′s′ck,aηs  (S27)  which is a chemical shift of conduction electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We also set W1 = W2 = W3 = W4 = W = 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='12meV in ˆHW to simplify  the SW transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The realistic values of W1,2,3,4 are not identical but the difference is about 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Finally, we introduce a chemical potential µc to tune the filling of the system  ˆHµc = −µc  �  |k|<Λc,aηs  : c†  k,aηsck,aηs :  (S28)  16  S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  SYMMETRY  We now provide the symmetry transformation of electron operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  For a given symmetry operation g, we let  Df(g), Dc′(g), Dc′′(g) denote the representation matrix of f-, Γ3 c- and Γ1 ⊕ Γ2 c-electrons:  gf †  R,αηsg−1 =  �  α′η′s′  f †  gR,α′η′s′Df(g)α′η′s′,αηs  gc†  k,aηsg−1 =  �  a′∈{1,2},η′s′  c†  gk,a′η′s′Dc′(g)a′η′s′,aηs,  a ∈ {1, 2}  gc†  k,aηsg−1 =  �  a′∈{3,4},η′s′  c†  gk,a′η′s′Dc′′(g)a′+2η′s′,a+2ηs,  a ∈ {3, 4}  (S29)  We consider the following symmetry operations as given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  T, C3z, C2x, C2zT  (S30)  with the following representation matrices  T :  Df(T) = σ0τxς0,  Dc′(T) = σ0τxς0,  Dc′′(T) = σ0τxς0  C3z :  Df(C3z) = ei 2π  3 σzτzς0,  Dc′(C3z) = ei 2π  3 σzτzς0,  Dc′′(C3z) = σ0τ0ς0  C2x :  Df(C2x) = σxτ0ς0,  Dc′(C2x) = σxτ0ς0,  Dc′′(C2x) = σxτ0ς0  C2zT :  Df(C2xT) = σxτ0ς0,  Dc′(C2xT) = σxτ0ς0,  Dc′′(C2zT) = σxτ0ς0  (S31)  where σx,y,z,0, τx,y,z,0, ςx,y,z,0 are Pauli or identity matrices of orbital, valley and spin degrees of freedom respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  At M ̸= 0, v′  ⋆ ̸= 0, we also have U(1)c charge symmetry, U(1)v valley symmetry and SU(2)η spin symmetry for each  valley η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We also mention that at M = 0, we have an enlarged flat U(4) symmetry and at v′  ⋆ = 0 we have an enlarged  chiral U(4) symmetry [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' At M = 0, v′  ⋆ = 0, we have a U(4) × U(4) symmetry [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Here, we consider the case of  M ̸= 0, v′  ⋆ ̸= 0, where we only have a U(1)c ×U(1)v ×SU(2)η=+ ×SU(2)η=− symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We comment that M = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='698meV  is relatively small and we have an approximate flat U(4) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Under U(1)c transformation gU(1)c(θc) (characterized  by a real number θc),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' U(1)v transformation gU(1)v(θv) (characterized by a real number θv) and SU(2)η spin transformation  gSU(2)η(θµ  η ) (characterized by three real numbers θµ  η ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' µ ∈ {x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' z} ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' we have  U(1)c :  Df(gU(1)c((θc)) = e−iθcσ0τ0ς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Dc′(gU(1)c((θc)) = e−iθcσ0τ0ς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Dc′′(gU(1)c((θc)) = e−iθcσ0τ0ς0  U(1)v :  Df(gU(1)v((θv)) = σ0e−iθvτzς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Dc′(gU(1)v((θv)) = σ0e−iθvτzς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Dc′′(gU(1)v((θv)) = σ0e−iθvτzς0  SU(2)η :  Df(gSU(2)η(θµ  η )) = σ0e−i �  µ θη  µ  τ0+ητz  4  ςµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Dc′(gSU(2)η(θµ  η )) = σ0e−i �  µ θη  µ  τ0+ητz  4  ςµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Dc′′(gSU(2)η(θµ  η )) = σ0e−i �  µ θη  µ  τ0+ητz  4  ςµ  (S32)  S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  MEAN-FIELD SOLUTIONS OF THE KONDO LATTICE MODEL  The Kondo Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S22 contains two single-particle term ˆHc and ˆHcc and two interaction terms ˆHK + ˆHJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We  now discuss the mean-field decoupling of ˆHK, ˆHJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  17  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Mean-field decoupling of ˆHK  We treat the interaction terms via mean-field decoupling  ˆHK ≈ ˆHMF  K  =  �  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='|k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='|k′|<Λc  �  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s′  ei(k−k′)R−|k|2λ2/2−|k′|2λ2/2  NMDνc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='νf  �  γ2δα′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′δα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a + γv′  ⋆δα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a[η′k′  xσx − k′  yσy]α′a′ + γv′  ⋆δα′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′[ηkxσx + kyσy]αa  �  �  ⟨f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηs⟩⟨c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′⟩ − ⟨f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηs⟩c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′ − f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηs⟨c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′⟩  − ⟨: f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′ :⟩⟨: c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′η′s′ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηs :⟩ + ⟨: f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′ :⟩ : c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′η′s′ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηs : + : f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′ : ⟨: c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′η′s′ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηs :⟩  �  (S33)  where for an operator O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' ⟨O⟩ = ⟨Ψ|O|Ψ⟩ with |Ψ⟩ the mean-field ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Fock term  We first consider the Fock term (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' ), which takes the form of  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' =  �  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='|k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='|k′|<Λc  �  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s′  ei(k−k′)R−|k|2λ2/2−|k′|2λ2/2  NMDνc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='νf  �  γ2δα′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′δα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a + γv′  ⋆δα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a[η′k′  xσx − k′  yσy]α′a′ + γv′  ⋆δα′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′[ηkxσx + kyσy]αa  �  �  ⟨f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηs⟩⟨c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′⟩ − ⟨f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηs⟩c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′ − f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηs⟨c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′⟩  �  =  �  R  1  Dνc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='νf  �  γ2⟨  �  |k|<Λc  �  αηs  eik·R−|k|2λ2/2  √NM  f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηs⟩⟨  �  |k′|<Λc  �  α′η′s′  e−ik′·R−|k′|2λ2/2  √NM  c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′⟩  −  �  γ2 �  |k|<Λc  �  αηs  eik·R−|k|2λ2/2  √NM  f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηs⟨  �  |k′|<Λc  �  α′η′s′  e−ik′·R−|k′|2λ2/2  √NM  c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′⟩ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  �  + γv′  ⋆⟨  �  |k|<Λc  �  αηs  eik·R−|k|2λ2/2  √NM  f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs⟩⟨  �  |k′|<Λc  �  a′α′η′s′  e−ik′·R−|k′|2λ2/2[η′k′  xσx − k′  yσy]α′a′  √NM  c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′⟩  + γv′  ⋆⟨  �  |k|<Λc  �  αaηs  eik·R−|k|2λ2/2[ηkxσx + kyσy]αa  √NM  f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs⟩⟨  �  |k′|<Λc  �  α′η′s′  e−ik′·R−|k′|2λ2/2  √NM  c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′⟩  − γv′  ⋆  � �  |k|<Λc  �  αaηs  eik·R−|k|2λ2/2[ηkxσx + kyσy]αa  √NM  f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs⟨  �  |k′|<Λc  �  α′η′s′  e−ik′·R−|k′|2λ2/2  √NM  c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′⟩  +  �  |k|<Λc  ⟨  �  αaηs  eik·R−|k|2λ2/2[ηkxσx + kyσy]αa  √NM  f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs⟩  �  |k′|<Λc  �  α′η′s′  e−ik′·R−|k′|2λ2/2  √NM  c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  ��  (S34)  18  We introduce the following mean-field expectation values  V1 =  �  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='|k|<Λc  �  αηs  eik·R−|k|2λ2/2  NM  √NM  ⟨Ψ|f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs|Ψ⟩  V2 =  �  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='|k|<Λc  �  αaηs  eik·R−|k|2λ2/2  NM  √NM  (ηkxσx + kyσy)αa⟨Ψ|f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηs|Ψ⟩  (S35)  and assume the ground state is translational invariant such that  �  |k|<Λc  �  αηs  eik·R−|k|2λ2/2  √NM  ⟨Ψ|f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs|Ψ⟩ =  1  NM  �  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='|k|<Λc  �  αηs  eik·R−|k|2λ2/2  √NM  ⟨Ψ|f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs|Ψ⟩ = V1  �  |k|<Λc  �  αaηs  eik·R−|k|2λ2/2  √NM  (ηkxσx + kyσy)αa⟨Ψ|f †  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs|Ψ⟩  = 1  NM  �  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='|k|<Λc  �  αaηs  eik·R−|k|2λ2/2  √NM  (ηkxσx + kyσy)αa⟨Ψ|f †  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs|Ψ⟩ = V2  (S36)  Then the Fock term (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S34) becomes  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' = −  γ2  Dνc,νf  �  R,|k|<Λc  �  αη,s  eik·R−|k|2λ2/2  √NM  �  V ∗  1 f †  R,αηsck,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  �  + NMγ2|V1|2  Dνc,νf  −  γv′  ⋆  Dνc,νf  �  R,|k|<Λc  �  α,a,η,s  eik·R−|k|2λ2/2  √NM  �  V ∗  1 (ηkxσx + kyσy)αaf †  R,αηsck,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  �  + NMγv′  ⋆V ∗  1 V2  Dνc,νf  −  γv′  ⋆  Dνc,νf  �  R,|k|<Λc  �  α,η,s  eik·R−|k|2λ2/2  √NM  �  V ∗  2 f †  R,αηsck,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  �  + NMγv′  ⋆V ∗  2 V1  Dνc,νf  (S37)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Hartree term  For the Hartree term (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' ), we introduce the following density matrices Of, Oc′,1, Oc′,2, where Of have also been used in  the mean-field calculations of the THF model as shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [1] (however, Oc′,1, Oc′,2, V1, V2 are absent in the THF model)  Of  αηs,α′η′s′ =  1  NM  �  R  ⟨Ψ| : f †  R,αηsfR,α′η′s′ : |Ψ⟩  Oc′,1  aηs,a′η′s′ =  1  NM  �  |k|<Λc  e−|k|2λ2⟨Ψ| : c†  k,aηsck,a′η′s′ : |Ψ⟩,  a, a′ ∈ {1, 2}  Oc′,2  a′η′s′,αηs =  1  NM  �  |k|<Λc  �  a=1,2  e−|k|2λ2(ηkxσx + kyσy)αa⟨Ψ| : c†  k,a′η′s′ck,aηs : |Ψ⟩,  a′, α ∈ {1, 2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (S38)  We then assume the ground state is translational invariance such that  ⟨Ψ| : f †  R,αηsfR,α′η′s′ : |Ψ⟩ =  1  NM  �  R  ⟨Ψ| : f †  R,αηsfR,α′η′s′ : |Ψ⟩ = Of  αηs,α′η′s′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (S39)  19  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S38 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S39, the Hartree term can be written as  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  =  �  R,  |k|<Λc,|k′|<Λc  �  α,α′,a,a′,  η,η′,s,s′  ei(k−k′)R−|k|2λ2/2−|k′|2λ2/2  NMDνc,νf  �  γ2δα′,a′δα,a + γv′  ⋆δα,a[η′k′  xσx − k′  yσy]α′a′ + γv′  ⋆δα′,a′[ηkxσx + kyσy]αa  �  �  − ⟨: f †  R,αηsfR,α′η′s′ :⟩⟨: c†  k′,a′η′s′ck,aηs :⟩ + ⟨: f †  R,αηsfR,α′η′s′ :⟩ : c†  k′,a′η′s′ck,aηs : + : f †  R,αηsfR,α′η′s′ : ⟨: c†  k′,a′η′s′ck,aηs :⟩  �  =  �  α,α′,  η,η′,s,s′  NM  Dνc,νf  �  − γ2Of  αηs,αη′s′Oc′1  α′η′s′,αηs −  �  γv′  ⋆Of  αηs,α′η′s′Oc′,2  α′η′s′,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  ��  +  �  |k|<Λc  �  α,α′,  η,η′,s,s′  �  Of  αηs,α′η′s′e−|k|2λ2 : c†  k,a′η′s′ck,aηs : δα,aδα′,a′ +  �  γv′  ⋆Of  αηs,α′η′s′δα,a[η′kxσx − kyσy]α′a′e−|k|2λ2  : c†  k,a′η′sck,aηs : +h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  ��  +  �  R  �  α,α′,  η,η′,s,s′  �  : f †  R,αηsfR,α′η′s′ : Oc′,1  α′η′s′,αηs +  �  γv′  ⋆ : f †  R,αηsfR′,α′η′s′ : Oc′,2  α′η′s′,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  ��  (S40)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Fock and Hartree terms  Combining Fock and Hartree (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S37 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S37) terms, we have  ˆHK ≈ ˆHMF  K  =F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  = −  γ2  Dνc,νf  �  R,|k|<Λc  �  αη,s  eik·R−|k|2λ2/2  √NM  ��  V ∗  1 f †  R,αηsck,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  ��  + NMγ2|V1|2  Dνc,νf  −  γv′  ⋆  Dνc,νf  �  R,|k|<Λc  �  α,a,η,s  eik·R−|k|2λ2/2  √NM  ��  V ∗  1 (ηkxσx + kyσy)αaf †  R,αηsck,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  ��  + NMγv′  ⋆V ∗  1 V2  Dνc,νf  −  γv′  ⋆  Dνc,νf  �  R,|k|<Λc  �  α,η,s  eik·R−|k|2λ2/2  √NM  ��  V ∗  2 f †  R,αηsck,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  �  − V ∗  2 V1  �  + NMγv′  ⋆V ∗  2 V1  Dνc,νf  +  �  α,α′,  η,η′,s,s′  NM  Dνc,νf  �  − γ2Of  αηs,αη′s′Oc′1  α′η′s′,αηs −  �  γv′  ⋆Of  αηs,α′η′s′Oc′,2  α′η′s′,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  ��  +  �  |k|<Λc  �  α,α′,  η,η′,s,s′  �  Of  αηs,α′η′s′e−|k|2λ2 : c†  k,a′η′s′ck,aηs : δα,aδα′,a′ +  �  γv′  ⋆Of  αηs,α′η′s′δα,a[η′kxσx − kyσy]α′a′e−|k|2λ2  : c†  k,a′η′sck,aηs : +h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  ��  +  �  R  �  α,α′,  η,η′,s,s′  �  : f †  R,αηsfR,α′η′s′ : Oc′,1  α′η′s′,αηs +  �  γv′  ⋆ : f †  R,αηsfR′,α′η′s′ : Oc′,2  α′η′s′,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  ��  (S41)  V1, V2 describes the Fock contribution that characterize the hybridization between f- and Γ3 c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Of, Oc′,1, Oc′,2 are  the mean fields taking the form of ⟨f †f⟩, ⟨c†c⟩ which represent the Fock contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  20  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Mean-field decoupling of ˆHJ  We now perform a mean-field decoupling of the ferromagnetic exchange coupling term [1]  ˆHJ ≈ ˆHMF  J  = −  J  2NM  �  R  �  αα′ηη′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='ss′  �  |k|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='|k′|<Λc  ei(k−k′)·R(ηη′ + (−1)α+α′)  �  ⟨f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='ηs⟩⟨c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′⟩  − ⟨f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='ηs1⟩c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′ − f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs1ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
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+page_content='ηs :⟩+ : f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
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+page_content='ηs :⟩  + ⟨: f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
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+page_content='α′η′s′ :⟩ : c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
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+page_content='  Fock term  The Fock term takes the form of  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' = −  J  2NM  �  R  �  αα′ηη′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
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+page_content='α′η′s′⟩  − ⟨f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
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+page_content='ηs1⟩c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
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+page_content='α′η′s′ − f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
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+page_content='ηs⟨c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′⟩  �  = − J  �  R  �  ξ=±  �  ⟨  �  |k′|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs  ei(−k′)·R  √NM  δξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η(−1)α+1f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='ηs⟩⟨  �  |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′  δξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η′(−1)α′+1 eik·R  √NM  c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′⟩  −  �  |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs  ei(−k′)·R  √NM  δξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η(−1)α+1f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='ηs⟨  �  |k′|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs  δξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η′(−1)α′+1 eik·R  √NM  �  α′η′s′  c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′⟩  − ⟨  �  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs  ei(−k′)·R  √NM  δξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η(−1)α+1f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='ηs⟩  �  |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′  eik·R  √NM  δξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η′(−1)α′+1c†  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′  �  (S43)  We then introduce the following mean-fields  V3 =  �  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='|k|<Λc  �  αη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s  eik·Rδ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η(−1)α+1  NM  √NM  ⟨Ψ|f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α+2ηs|Ψ⟩  V4 =  �  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='|k|<Λc  �  αη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s  eik·Rδ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η(−1)α+1  NM  √NM  ⟨Ψ|ηf †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α+2ηs|Ψ⟩  (S44)  and assume the ground state is translational invariant such that  �  |k|<Λc  �  αη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s  eik·Rδ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η(−1)α+1  √NM  ⟨Ψ|f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α+2ηs|Ψ⟩ =  1  NM  �  R  �  |k|<Λc  �  αη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s  eik·Rδ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η(−1)α+1  √NM  ⟨Ψ|f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α+2ηs|Ψ⟩ = V3  �  |k|<Λc  �  αη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s  eik·Rδ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η(−1)α+1  √NM  ⟨Ψ|f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α+2ηs|Ψ⟩ =  1  NM  �  R  �  |k|<Λc  �  αη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s  eik·Rδ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η(−1)α+1  √NM  ⟨Ψ|f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α+2ηs|Ψ⟩ = V4  (S45)  Then the Fock term can be written as  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' = − JNM[|V3|2 + |V4|2]  + J  �  R,|k|<Λc,αηs  �ei(−k′)·R  √NM  �  δ1,η(−1)α+1f †  R,αηsck′,α+2,ηsV ∗  3 + δ−1,η(−1)α+1f †  R,αηsck′,α+2,ηsV ∗  4 +  �  + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  �  (S46)  21  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Hartree term  The Hartree term takes the form of  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' = −  J  2NM  �  R  �  αα′ηη′,ss′  �  |k|,|k′|<Λc  ei(k−k′)·R(ηη′ + (−1)α+α′)  �  − ⟨: f †  R,αηsfR,α′η′s′ :⟩⟨: c†  k′,α′+2,η′s′ck,α+2,ηs :⟩  + : f †  R,αηsfR,α′η′s′ : ⟨: c†  k′,α′+2,η′s′ck,α+2,ηs :⟩ + ⟨: f †  R,αηsfR,α′η′s′ :⟩ : c†  k′,α′+2,η′s′ck,α+2,ηs :  �  (S47)  We introduce the following density matric which has also been used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [1]  Oc′′  aηs,a′η′s′ =  1  NM  �  |k|<Λc  ⟨Ψ| : c†  k,a+2ηsck,a′+2η′s′ : |Ψ⟩,  a, a′ ∈ {1, 2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (S48)  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S38 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S48, the Hartree term becomes  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' = − JNM  2  �  αα′ηη′ss′  (ηη′ + (−1)α+α′)Of  αηs,α′η′s′Oc′′  α′η′s′,αηs  + J  2  �  Rc,αα′ηη′ss′  : f †  R,αηsfR,α′η′s′ : Oc′′  α′η′s′,αηs + J  2  �  |k|<Λc,αα′ηη′ss′  Of  αηs,α′η′s′ : c†  k′,α′+2,η′s′ck,α+2,ηs :  (S49)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Fock and Hartree terms  Combing Hartree and Fock terms (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S46 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S49), we have  ˆHJ ≈ ˆHMF  J  = − JNM  �  ξ=±  |V3|2 + |V4|2  + J  �  R,|k|<Λc,αηs  �ei(−k′)·R  √NM  �  δ1,η(−1)α+1f †  R,αηsck′,α+2,ηsV ∗  3 + δ−1,η(−1)α+1f †  R,αηsck′,α+2,ηsV ∗  4  �  + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  �  − JNM  2  �  αα′ηη′ss′  (ηη′ + (−1)α+α′)Of  αηs,α′η′s′Oc′′  α′η′s′,αηs  + J  2  �  Rc,αα′ηη′ss′  : f †  R,αηsfR,α′η′s′ : Oc′′  α′η′s′,αηs + J  2  �  |k|<Λc,αα′ηη′ss′  Of  αηs,α′η′s′ : c†  k′,α′+2,η′s′ck,α+2,ηs :  (S50)  V3, V4 describes the Fock contribution that characterize the hybridization between f- and Γ1 ⊕ Γ2 c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Of, Oc′′ are the  mean fields taking the form of ⟨f †f⟩, ⟨c†c⟩ which represent the Fock contribution and have also been used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Filling constraints and mean-field equations  We note that in the Kondo model the filling of f electrons is fixed to be νf at each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' To simplify the calculation, we take  a common approximation that only requires the average filling of f-electron to be νf [3, 138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In other words, we only require  1  NM  �  R,αηs⟨Ψ| : f †  R,αηsfR,αηs : |Ψ⟩ = νf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We then add the following term to the Hamiltonian  ˆHλf =  �  R,αηs  λf  �  : f †  R,αηsfR,αηs : −νf  �  (S51)  and determine the Langrangian multiplier λf from the following equation  1  NM  �  R,αηs  ⟨Ψ| : f †  R,αηsfR,αηs : |Ψ⟩ = νf  (S52)  22  In practice, we perform calculations at fixed total filling ν = νf + νc, where νf and νc are the average fillings of f and c  electrons respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Since νf is also fixed in the Kondo model, we will self-consistently determine the chemical potential µc  (in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S28) by requiring  1  NM  �  |k|<Λc,aηs  ⟨Ψ| : c†  k,aηsck,aηs : |Ψ⟩ = νc = ν − νf  (S53)  Finally, our mean-field Hamiltonian takes the form of  ˆHMF = ˆHc + ˆHcc + ˆHMF  K  + ˆHMF  J  + ˆHλf + ˆHµc  (S54)  and we determine V1, V2, V3, V4, Of, Oc′,1, Oc′,2, Oc′′, λf, µc from the self-consistent equations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S35, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S38, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S44,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S48, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S52, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' During the self-consistent solution, at each step, we will adjust λf, µc according to the current  filling of f- and c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We use νi  f and νi  c denote the filling of f and c at i-th step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' For the i + 1-th step, we will update  λf, µc as λf → λf + r(νi  f − νf), µc → µc − r(νi  c − νc), where r(> 0) will be manually adjusted to improve the convergence  (in practice, we take r ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Mean-field equations of the symmetric Kondo state  We focus on the symmetric Kondo phase without any symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Therefore, we require our density matrix of f-  and c- electrons (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S38, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S48) to be symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We can then utilize symmetry to simplify the self-consistent equations  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S38, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We first consider the U(1)v symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S32, a U(1)v symmetric solution satisfies  Of  αηs,α′η′s′ = Of  αηs,α′η′s′e−iθν(η−η′) ⇒ Of  αηs,α−ηs′ = 0  Oc′,1  aηs,a′η′s′ = Oc′,1  aηs,a′η′s′e−iθν(η−η′) ⇒ Oc′,1  aηs,a′−η′s′ = 0  Oc′,2  aηs,α′η′s′ = Oc′,2  aηs,α′η′s′e−iθν(η−η′) ⇒ Oc′,2  aηs,α′−η′s′ = 0  Oc′′  aηs,a′η′s′ = Oc′′  aηs,a′η′s′e−iθν(η−η′) ⇒ Oc′′  aηs,a′−ηs′ = 0  (S55)  and V1, V2, V3, V4 are invariant under U(1)v transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S55, Of, Oc′,1, Oc′,2, Oc′′ are block diagonalized in  valley index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We next consider a SU(2)η transformation acting on the valley η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We find  �  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s′  [ei �  µ θη  µσµ]s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='sOf  αηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′ηs′[ei �  µ θη  µσµ]s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s′  2 = Of  αηs2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′ηs′  2 ⇒ Of  aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′ηs′ ∝ Is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s′  �  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s′  [ei �  µ θη  µσµ]s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='sOc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='1  aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′ηs′[ei �  µ θη  µσµ]s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s′  2 = Oc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='1  aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′ηs′ ⇒ Oc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='1  aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′ηs′ ∝ Is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s′  �  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s′  [ei �  µ θη  µσµ]s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='sOc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2  aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′ηs′[ei �  µ θη  µσµ]s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s′  2 = Oc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2  aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′ηs′ ⇒ Oc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2  aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′ηs′ ∝ Is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s′  �  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s′  [ei �  µ θη  µσµ]s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='sOc′′  aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′ηs′[ei �  µ θη  µσµ]s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s′  2 = Oc′′  aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′ηs′ ⇒ Oc′′  aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′ηs′ ∝ Is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s′  (S56)  where I is an 2 × 2 identical matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In addition, V1, V2, V3, V4 are invariant under SU(2)η transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S55 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S56, the density matrices Of, Oc′,1, Oc′′ are diagonalized in valley and spin incdies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We then introduce  2 × 2 matrices, of,η, oc′,1,η, oc′,2,η, oc′′,η, such that  Of  αηs,α′η′s′ = of,η  α,α′δη,η′δs,s′,  Oc′,1  aηs,a′η′s′ = oc′,1,η  a,a′ δη,η′δs,s′,  Oc′,2  aηs,a′η′s′ = oc′,2,η  a,a′ δη,η′δs,s′,  Oc′′  aηs,a′η′s′ = oc′′,η  a,a′ δη,η′δs,s′  (S57)  We now consider the effect of discrete symmetries in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S31 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S57, we find  T :  (of,η)∗ = of,−η,  (oc′,1,η)∗ = oc′,1,−η,  (oc′,2,η)∗ = oc′,2,−η,  (oc′′,η)∗ = oc′′,−η  C3z :  ei 2πη  3 σzof,ηe−i 2πη  3 σz = of,η,  ei 2πη  3 σzoc′,1,ηe−i 2πη  3 σz = oc′,1,η,  ei η2π  3 oc′,2,ηe−i η2π  2 σz = oc′,2,η,  oc′′,η = oc′′,η  C2x :  σxof,ησx = of,η,  σxoc′,1,ησx = oc′,1,η,  σxoc′,2,ησx = oc′,2,η,  σxoc′′,ησx = oc′′,η  C2zT :  (σxof,ησx)∗ = of,η,  (σxoc′,1,ησx)∗ = oc′,1,η,  (σxoc′,2,ησx)∗ = −oc′,2,−η,  (σxoc′′,ησx)∗ = oc′′,η  (S58)  23  From the definition (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S38), Of, Oc′,1, Oc′′ are Hermitian matrices and then of, oc′,1, oc′′ are also Hermitian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Com-  bining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S58 and the Hermitian properties, we can introduce real numbers χf  0, χc′,1  0  , χc′′  0 , χc′′  1  and then of, oc′, oc′′ take the  following structure  of,η = of,−η = χf  0σ0,  oc′,1,η = oc′,−η = χc′,1  0  σ0,  oc′,2,η = 0,  oc′′,η = χc′′  0 σ0 + χc′′  1 σx  (S59)  where σ0, σx,y,z are identity and Pauli matrices respectively with row and column indices α = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Since the filling of f- and  Γ1 ⊕ Γ2 c- electrons are νf = Tr[Of], νc′′ = Tr[Oc′′] respectively, we find χf  0 = νf/8, χc′′  0  = νc′′/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S59 and  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S57, for the symmetric solution, we finally have  Of  αηs,α′η′s′ = δα,α′δη,η′δs,s′νf/8  Oc′,1  aηs,a′η′s′ = δa,a′δη,η′δs,s′χc′,1  0  ,  Oc′,2  aηs,a′η′s′ = 0,  a, a′ ∈ {1, 2}  Oc′′  aηs,a′η′s′ = δη,η′δs,s′(δa,a′νc′′/8 + δa,3−a′χc′′  x ),  a, a′ ∈ {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2}  (S60)  As for the hybridization fields, we find discrete symmetries will not impose constraints on V1, V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' As for V3, V4, we have  T : V3 = V ∗  4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  C3z : V3 = ei2π/3V3,  V4 = ei2π/3V4  C2x : V3 = V4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  C2zT : V3 = V ∗  4  (S61)  therefore  V3 = V4 = 0  (S62)  In summary, instead of solving self-consistent equations of Of, Oc′,1, Oc′′, V3, V4 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S38, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S44, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S48), we can use  νf =  1  NM  �  R,αηs  ⟨Ψ| : f †  R,αηsfR,αηs : |Ψ⟩  χc′,1  0  =  1  8NM  �  |k|<Λc,a=1,2,ηs  ⟨Ψ|e−|k|2λ2 : c†  k,aηsck,aηs : |Ψ⟩  νc′′ =  1  NM  �  |k|<Λc,a=3,4,ηs  ⟨Ψ| : c†  k,aηsck,aηs : |Ψ⟩  χc′′  1 =  1  8NM  �  |k|<Λc,ηs  ⟨Ψ|c†  k,3ηsck,4ηs + c†  k,4ηsck,3ηs|Ψ⟩  V3 = V4 = 0  (S63)  and obtain Of, Oc, Oc′′ via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We note that the first equation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S63 is equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In summary, combining  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S35, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S53 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S63, we have a complete set of mean-field self-consistent equations for the symmetric Kondo state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We comment that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S63 are the same mean-field equations as we derived in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S4 A, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S4 B and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S4 C, but with  additional symmetry requirement, that is the ground states satisfy all symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We mention that, at ν = νf = νc = 0, we have Of  αηs,α′η′s′ = 0, Oc′  αηs,α′η′s′ = 0 and the Hartree term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S41 vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We now prove the Hartree term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S50 also vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We note the only non-zero components of Oc′′ are Oc′′  1ηs,2ηs, Oc′′  2ηs,1ηs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S49, the Hartree term takes the form of (with Of = 0)  −  J  2NM  �  Rs1s2  �  αα′ηη′  �  |k1|,|k2|<Λc  ei(k1−k2)·R(ηη′ + (−1)α+α′)  �  : f †  Rαηs1fRα′η′s2 : Oc′′  α′η′s2,αηs1  �  = −  J  2NM  �  Rs  �  αη  �  |k1|,|k2|<Λc  ei(k1−k2)·R(ηη + (−1)α+3−α)  �  : f †  RαηsfRα′ηs : Oc′′  3−αηs,αηs  �  = −  J  2NM  �  Rs  �  αη  �  |k1|,|k2|<Λc  ei(k1−k2)·R(0)  �  : f †  RαηsfRα′ηs : Oc′′  3−αηs,αηs  �  =0  (S64)  and hence vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In summary, at ν = 0, we only need to consider V1, V2, and other mean fields vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  24  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Properties of the symmetric Kondo state  We solve the self-consistent equations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S35, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S44, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S52, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S53 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S63 at integer filling ν = 0, −1, −2  with νf  = ν, νc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We identify the symmetric Kondo (SK) states at ν = 0, −1, −2 which are characterized by  V1 ̸= 0 (|γ2V1/Dνf ,νc| = 95meV, 111meV, 209meV at ν = 0, −1, −2 respectively), V2 ̸= 0 (|v′  ⋆γV2/Dνf ,νc| =  80meV, 97meV, 197meV at ν = 0, −1, −2 respectively) and V3 = 0, V4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Even if we allow non-zero V3, V4 and ini-  tialize the mean-field calculations with non-zero V3, V4, V3, V4, we still find V3 = V4 = 0 after self-consistent calculations  (amplitudes smaller than 10−5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This is because ˆHJ describes ferromagnetic interactions and disfavors the development of non-  zero V3, V4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We also comment that the non-zero V1, V2, introduce an effective f-c hybridization (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S37) and characterize the  Kondo physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We next discuss the topological feature of the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Since the SK states preserve all the symmetries, it is sufficient to only  consider the bands in valley + and spin ↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We find at ν = 0, −1, −2, the representations formed by flat bands at Γ, K, M are  Γ1 ⊕ Γ2, K2K3, M1 ⊕ M2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We note that the representations formed by flat bands here are equivalent to that of the  non-interacting THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Thus, the flat bands form a fragile topology at ν = −1, −2 and a stable topology at ν = 0 due to  the additional particle-hole symmetry at ν = 0 [1, 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  In addition, we also calculate the Wilson loop of the flat bands (valley + spin ↑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In the calculation of Wilson loop, we  let k1 ∈ { i  N }i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=',N−1, k2 = { j  N }j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=',N−1 and k1 ∈ { i  N }i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=',N−1, k2 = { j  N }j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=',N−1 and k = k1bM,1 + k2bM,2,  where bM,1 =  4π  3aM (  √  3, 0), bM,2 =  4π  3aM (  √  3  2 , 3  2) are two moir´e reciprocal lattice vectors and aM is the moir´e lattice constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We then let |un,k⟩ denote the n-th eigenvectors of the single-particle Hamiltonian H(k) (of valley + spin ↑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We focus on  the subset of the bands, which we denote with band indices n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='., nband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Here, we take the flat bands as the subset of  the bands that we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We then define the matrix Uk as a matrix formed by the eigenvectors of the flat bands  Uk = [|u1,k⟩, |u2,k⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=', |unband,k⟩].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The Wilson loop [85] along the k2 direction is defined as  W(k1) = U †  k1,k2=0  N−1  �  j=1  �  Uk1,k2= 2πj  N U †  k1,k2= 2π(j+1)  N  �  V (k1=0,k2=2π)Uk1,k2=2π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (S65)  where V G is defined as H(k+G) = V GH(k)V G,†, G = nbM,1 +mbM,2, n, m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We mention that c-electrons are defined  in the momentum space that can be larger than the first MBZ (depending on the momentum cutoff Λc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Thus we introduce  V G that maps ck to ck+G to restore the periodic condition H(k + G) = V GH(k)V G,†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The corresponding Wilson loop  Hamiltonian [85] is  H(k1) = −i ln(W(k1))  (S66)  We plot the Wilson loop spectrum (eigenvalues of H(k1)) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S5, where we observe the Wilson loop has winding number 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [85], in the presence of additional particle-hole symmetry at ν = 0 [1, 85], (−1)n with n the winding number  of Wilson loop is a stable topological index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We conclude that at ν = 0, the symmetric Kondo state has a stable topology that is  characterized by the odd winding number of the Wilson loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S5, we observe the behaviors of the Wilson loop are similar at different fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We check the overlapping of the  flat-band wavefunctions between different bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We let {|uν  i,k⟩}i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=',nband denote the wavefunction of flat bands at filling ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We define the overlapping between wavefunctions at fillings ν and ν′ as  Overlap(ν, ν′) = 1  N  �  k  �  i,j∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=',nband}  ⟨uν  i,k|uν′  j,k⟩⟨uν′  j,k|uν  i,k⟩  (S67)  We find Overlap(0, −1) = 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='1%, Overlap(−1, −2) = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The large overlapping of wavefunctions between different  fillings indicates similar behaviors of the Wilson loop at different fillings as we showed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Finally, we analyze the mean-field Hamiltonian of the symmetric Kondo state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The mean-field single-particle Hamiltonian of  valley η and spin ↑ (spin ↑ and spin ↓ are equivalent) of the Kondo symmetric state can be approximately written as  ˜H(η)(k) =  � ˜H(f,η)(k)  ˜H(fc,η)(k)  ˜H(fc,η),†(k)  ˜H(c,η)(k)  �  (S68)  ˜H(f,η)(k) = EfI2×2  ˜H(c,η)(k) =  �  EcI2×2  v⋆(ηkxσ0 + ikyσz)  v⋆(ηkxσ0 − ikyσz)  Ec′′I2×2  �  ˜H(fc,η)(k) =  �  ˜γσ0 + ˜v′⋆(ηkxσx + kyσy) 02×2  �  (S69)  25  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Wilson loop spectrum of flat bands of SK states at ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  where ˜H(f,η), ˜H(c,η), ˜H(fc,η) denote the single-particle Hamiltonian of the f-block, c-block and fc-block respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Ef, Ec, Ec′′ denote the energy shifting induced by the Hartree term, one-body scattering term ˆHcc and chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Ef, Ec can be k dependent and we only keep its k-independent part which makes dominant contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Ec′′ comes from  the Hartree contribution of ˆHJ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S49, which is relatively small and we set Ec′′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We also set M = 0, since it is small  compared to the other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' ˜γ, ˜v⋆  ′ denote the renormalized f-c hybridization emerged from Kondo interactions (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We also drop the damping factor e−|k|2λ2/2 to simplify the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In practice, we find |˜γ| =  1  Dνc,νf |γ2V ∗  1 + γv′  ⋆V ∗  2 | ≈  175meV, 209meV, 406meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In the chiral limit v′  ⋆ = 0, we also have ˜v′⋆ = 0 ( ˜v′⋆ = v′  ⋆ = γv′  ⋆V ∗  1 /Dνc,νf ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We note that  |˜v′  ⋆||k| can reach a similar amplitude as the k-independent hybridization |˜γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, we expect in most regions of MBZ, |˜γ|  makes the dominant contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We, therefore, drop the set ˜v′⋆ = 0 or equivalently v′  ⋆ = 0 as an approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' By setting  v′  ⋆ = 0, we can further separate ˜H(η)(k) into two blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The first block corresponds to the row and column indices 1, 3, 5  with electron operators,fk,1ηs, ck,1ηs, ck,3ηs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The second block corresponds to the row and column indices 2, 4, 6 with electron  operators,fk,2ηs, ck,2ηs, ck,4ηs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We focus on the first block whose single-particle Hamiltonian is  ˜h(η)(k) =  �  �  Ef  ˜γ  0  ˜γ  Ec  v⋆(ηkx + iky)  0  v⋆(ηkx − iky)  0  �  �  (S70)  We next analyze the eigensystems of ˜h(η)(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We note that ˜γ provides the largest energy scales near ΓM point and will gap  out f- and Γ3 c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' To observe this, we first consider the first 2 × 2 block of ˜h(η)(k) which describes the single-particle  Hamiltonian of f- and Γ3 c-electrons  �  Ef  ˜γ  ˜γ  Ec  �  (S71)  The eigenvalues and eigenvectors are  E1 = Ec + Ef  2  −  �  ˜γ2 + (Ec − Ef)2  4  ,  E2 = Ec + Ef  2  +  �  ˜γ2 + (Ec − Ef)2  4  v1 =  1  �  2Efc(Efc − E3)  �E3 − Efc ˜γ�T ,  v2 =  1  �  2Efc(Efc + E3)  �E3 + Efc ˜γ�T  (S72)  where E3 = Ef −Ec  2  , Efc =  �  ˜γ2 + E2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Since ˜γ is larger than Ec, Ef, Γ3 c-electrons and f-electrons are gapped out by the  hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Consequently, the flat bands are mostly formed by Γ1 ⊕ Γ2 c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Numerically, we indeed find the orbital  weights of Γ1 ⊕ Γ2 c-electrons are large (71%, 77%, 89% at ν = 0, −1, −2 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We next treat v⋆ perturbatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We find the dispersion of the flat band becomes  Eflat  k  ≈ −Ef|k|2(v⋆)2  E1E2  = Ef|k|2(v⋆)2  ˜γ2 − EcEf  (S73)  At ν = 0 with particle-hole symmetry, Ef = Ec = 0 and Eflat  k  ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, at ν = −1, −2, where Ef ̸= 0, Ec ̸= 0, flat  bands become dispersive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We observe that |Ec| is much smaller than |˜γ| at ν = −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Ef increases as we change from ν = 0  to ν = −2, because we are doping more holes to the f-orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' At ν = −2, Ef can reach ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5|˜γ|, but at ν = −1, Ef ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='1|˜γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Approximately, the dispersion of the flat band is Eflat  k  ≈ (v⋆)2Ef/˜γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' At ν = −1, we have Ek ≈ 13meV · ˚A2|k|2 and, at  26  ν = −2, we have Ek ≈ 45meV · ˚A2|k|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This indicates a larger dispersion at ν = −2, which is consistent with our numerical  result shown in the main text Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We next analyze the wavefunctions of the flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The corresponding electron operator of the flat band d†  flat,k is  d†  flat,k ≈ 1  Ak  �  c†  k,3ηs + v⋆(ηkx − iky)  EcEf − ˜γ2  �  − Efc†  k,1ηs + ˜γf †  k,1ηs  ��  (S74)  where the normalization factor  Ak =  �  1 +  |v⋆|2|k|2(E2  f + ˜γ2)  (EcEf − ˜γ2)2  (S75)  We observe that in the large |˜γ| limit, the flat bands are mostly formed by Γ1 ⊕ Γ2 c-electrons (c†  k,3ηs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We also provide the  Berry curvature derived from the wavefunction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S74  Ω(k) =  −2(EcEf − ˜γ2)2(E2  f + ˜γ2)v2  ⋆  �  (EcEf − ˜γ2)2 + (E2  f + ˜γ2)v2⋆|k|2  �2  (S76)  We next calculate the Wilson loop from the wavefunction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The wavefunctions of d†  flat,k is  u(k) = 1  Ak  �  1  v⋆ηkx−iky  EcEf −˜γ2 (−Ef)  v⋆ηkx−iky  EcEf −˜γ2 ˜γ  �T  (S77)  where the first, second and third rows denote c†  k,3ηs, f †  k,1ηs, c†  k,1ηs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We then parametrize the momentum as  k = x1aMbM,1 + x2aMbM,2,  bM,1 =  4π  3aM  (  √  3, 0),  bM,2 =  4π  3aM  (  √  3  2 , 3  2)  (S78)  x1, x2 ∈ [−1  2, 1  2] 1  aM  (S79)  and define |u(x1, x2)⟩ as |u(k)⟩ with k = x1aMbM,1 + x2aMbM,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The Wilson loop can be written as  W(x1) =  N−1  �  j=0  ⟨u(x1, x2 = x2,i)|u(x1, x2 = x2,j+1⟩⟨u(x1, x2,N)|u(x1, x2,0)⟩,  x2,i = −  1  2aM  +  1  aM  i  N  (S80)  The spectrum of the Wilson loop is  N(x1) = −i ln(W(x1)) = −i  �  1  2aM  −  1  2aM  ⟨u(x1, x2)|∂x2|u(x1, x2)⟩dx2 − i ln(⟨u(x1, 1/(2aM))|u(x1, −1/(2aM)⟩)  (S81)  In the continuous limit with aM → 0, we find  N(x1) = −i  � ∞  −∞  ⟨u(x1, x2)|∂x2|u(x1, x2)⟩dx2 − i ln(⟨u(x1, ∞)|u(x1, −∞⟩)  (S82)  Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S77 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S82, we find  N(x1) =π  �  1 +  v2  ⋆x1  �  x2  1v2⋆ + (˜γ2−EcEf )2  E2  f +˜γ2  �  (S83)  Even though Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S82 is calculated from the perturbative wavefunction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S77, it qualitatively captures the behaviors of the  Wilson loop shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We observe that N(−∞) = 0 and N(∞) = 2π, which indicates a 2π winding at ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We also mention that current calculations correspond to one of the two flat bands for each valley and each spin, because we only  pick one block of the single-particle Hamiltonian as we discussed near Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The other flat band can be derived in the same  manner and has similar behaviors, since it has a similar single-particle Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  27  S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  MEAN-FIELD SOLUTIONS OF THE TOPOLOGICAL HEAVY-FERMION MODEL  We now discuss the mean-field equations of topological heavy-fermion mode in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We use a similar Hartree-Fock ap-  proximation as introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, we decouple ˆHJ via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The mean-field expectation values we considered  are  Of  αηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′ =  1  NM  �  R  ⟨Ψ| : f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′ : |Ψ⟩  Oc′  aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′η′s′ =  1  NM  �  |k|<Λc  ⟨Ψ| : c†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′η′s′ : |Ψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  a ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 2}  Oc′′  aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′η′s′ =  1  NM  �  |k|<Λc  ⟨Ψ| : c†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a+2ηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a′+2η′s′ : |Ψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' a′ ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 2}  Oc′f  aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′ =  1  √  NN  �  |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='R  e−ik·R⟨Ψ|c†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α′η′s′|Ψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  a ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 2}  V3 =  �  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='|k|<Λc  �  αη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s  eik·Rδ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η(−1)α+1  NM  √NM  ⟨Ψ|f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α+2ηs|Ψ⟩  V4 =  �  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='|k|<Λc  �  αη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s  eik·Rδ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η(−1)α+1  NM  √NM  ⟨Ψ|ηf †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α+2ηs|Ψ⟩  (S84)  where Of,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Oc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' V3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' V4 have also been used in the Kondo lattice mean-field calculations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S38, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S48 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  In addition, THF model also has a chemical potential term ˆHµ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S21) and we determine µ by requiring the total filling of  f- and c-electrons to be ν:  ν = Tr[Of] + Tr[Oc′] + Tr[Oc′′]  (S85)  where we note that the filling of f-, Γ3 c- and Γ1 ⊕ Γ2 c-electrons are  νf = Tr[Of],  νc′ = Tr[Oc′],  νc′′ = Tr[Oc′′] ,  (S86)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We discuss the difference and similarities between the mean-field equations of the KL model and that of the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  For the THF model, we introduce mean fields Of, Oc′, Oc′′, Oc′f, V3, V4 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S84) (for a generic state without enforcing any  symmetries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' As for KL model, we introduce mean fields V1, V2, Of, Oc′,1, Oc′,2, Oc′′, V3, V4 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S38, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S35, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S48,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S44) (for a generic state without enforcing any symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  For both models, Oc′′, Of, V3, V4 are part of mean fields and contribute the mean-field decoupling of ˆHJ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  In the THF model, we introduce a chemical potential term µ that couples to both the f-electron density operators and c-  electron density operator (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We enforce the total filling of f- and c-electrons to be ν by tuning µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In the KL model,  we introduce a Lagrangian multiplier λf (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S51) that couples to f-electron density operators, and a chemical potential  µc (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S28) that couples to the c-electron density operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We enforce the fillings of f-electrons and c-electrons to be  νf and νc respectively by tuning λf and µc in the KL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We also mention that Oc′ in THF model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S84) and Oc′,1 in the KL model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S38) are different, where the latter one  has included an additional damping factor e−|k|2λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  In the THF model, we do not need hybridization fields V1, V2 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S35), since both come from the decoupling of Kondo  interactions that only appear in the KL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Mean-field equations of fully symmetric state  We next discuss the solution of the symmetric state The fully symmetric state is characterized by density matrices  Of, Oc′, Oc′′, Oc′f and hybridization fields V3, V4 that satisfy all symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The structures of Of, Oc′′ in the fully sym-  metric state are given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We also prove that V3 = V4 = 0 in a fully symmetric state (near Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We now discuss  28  the symmetry properties of Oc′, Oc′f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S32, a U(1)v symmetric solution satisfies  Oc′  aηs,α′η′s′ = Oc′  aηs,α′η′s′e−iθν(η−η′) ⇒ Oc′  aηs,α−ηs′ = 0  Oc′f  aηs,α′η′s′ = Oc′f  aηs,α′η′s′e−iθν(η−η′) ⇒ Oc′f  aηs,α−ηs′ = 0  (S87)  Then, Oc′, Oc′f is block diagonalized in valley indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We next consider a SU(2)η transformation acting on the valley η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' It  indicates  �  s,s′  [ei �  µ θη  µσµ]s2,sOc′  αηs,α′ηs′[ei �  µ θη  µσµ]s′,s′  2 = Oc′  αηs2,α′ηs′  2 ⇒ Oc′  αηs,α′ηs′  �  s,s′  [ei �  µ θη  µσµ]s2,sOc′f  αηs,α′ηs′[ei �  µ θη  µσµ]s′,s′  2 = Oc′f  αηs2,α′ηs′  2 ⇒ Oc′f  αηs,α′ηs′  (S88)  Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S87 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S88, we can introduce 2 × 2 matrices oc′,η, oc′f,η, such that  Oc′  αηs,α′η′s′ = oc′  α,α′δs,s′δη,η′,  Oc′f  αηs,α′η′s′ = oc′f  α,α′δs,s′δη,η′  (S89)  We now consider the effect of discrete symmetries in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S31 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S89, we find  T :  (oc′,η)∗ = oc′,−η,  (oc′f,η)∗ = oc′f,−η  C3z :  ei2π/3ησzoc′,ηe−i2πη/3σz = oc′,η,  ei2π/3ησzoc′f,ηe−i2πη/3σz = oc′f,η  C2x :  σxoc′,ησx = oc′,η,  σxoc′f,ησx = oc′f,η  C2zT :  σx(oc′,η)∗σx = oc′,η,  σx(oc′f,η)∗σx = oc′f,η  (S90)  Then we can introduce a single real number χc′  0 , χc′f  0  to characterize the density matrices  Oc′  αηs,α′η′s′ = χc′  0 δα,α′δη,η′,  Oc′f  αηs,α′η′s′ = χc′f  0 δα,α′δη,η′δs,s′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (S91)  Since the filling of Γ3 c-electrons is νc′ = Tr[Oc′] = 8χc′  0 , we let χc′  0 = νc′/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Therefore, instead of calculating the original  density matrices in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S84, we can calculate the following quantities  νf =  1  NM  �  R,αηs  ⟨Ψ| : f †  R,αηsfR,αηs : |Ψ⟩  νc′ =  1  NM  �  |k|<Λc,a=1,2,ηs  ⟨Ψ| : c†  k,aηsck,aηs : |Ψ⟩  νc′′ =  1  NM  �  |k|<Λc,a=3,4,ηs  ⟨Ψ| : c†  k,aηsck,aηs : |Ψ⟩  χc′f =  1  8NM  √NM  �  |k|<Λc,R,αηs  e−ik·R⟨Ψ|c†  k,αηsfR,αηs|Ψ⟩  (S92)  and construct density matrices via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S60 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In addition, the filling constraints in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S85 becomes  ν = νf + νc′ + νc′′  (S93)  Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S92 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S93, we have a complete set of the mean-field self-consistent equations of symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Here we discuss the differences and similarities between the symmetric solution in the KL model (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S4 E) and the sym-  metric solution in the THF model as introduced in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Both Kondo symmetric (KS) state in KL model and the symmetric state in the THF model preserve all the symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  The KS state is adiabatically connected to the symmetric state in the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  The mean-field solutions are exact at N = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  29  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Wilson loop spectrum of symmetric state in the THF model at ν = 0, −1, −2  To obtain a more precise description of the Kondo state, we need to introduce a Gutzwiller projector to our symmetric-  state wavefunction in the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The Gutzwiller projector will suppress the charge fluctuations of f-electrons and is  expected to further lower the energy of the symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We also comment that, in the THF model, the flat bands are mainly formed by f-electrons with f-electron orbital weights  80%, 85% and 87% at ν = 0, −1, −2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, in the KL model, the flat bands are mainly formed by  Γ1 ⊕Γ2 electrons as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This is because, in the KL model, we observed an enhanced f-c hybridization  driven by Kondo interactions which is absent in the symmetric state solution of THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We expect the enhanced  hybridization will be recovered after introducing the Gutzwiller projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We find the spectrum of the Wilson loop in the symmetric state of the THF model has winding number one and three  crossing points(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, in the SK state of the KL model, the spectrum of the Wilson loop spectrum has  winding number one and one crossing point (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, at ν = 0, for both the THF model and the KL model, the  symmetric state has a stable topology with an odd Winding number of Wilson loop spectrum [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We also mention the  difference in the Wilson loop spectrum comes from the absence of enhanced f-c hybridization in the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Mean-field equations of the symmetric state in the presence of strain  We now discuss the mean-field solution of the symmetric state in the presence of strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We add the following term to the  Hamiltonian (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S12)  ˆHstrain = α  �  R,ηs  (f †  R,1ηsfR,2ηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=')  (S94)  We note that ˆHstrain only breaks C3z symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' To show this, we rewrite the ˆHstrain as  ˆHstrain =  �  R,αηs,α′η′s′  αf †  R,αηs[hstrain]αηs,α′η′s′fR,α′η′s′  hstrain = σxτ0ς0  (S95)  where  the  matrix  structure  of  ˆHstrain  is  denoted  by  hstrainσxτ0ς0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We  now  show  it  commutes  with  Df(T), Df(C2x), Df(C2zT), Df(gSU(2)η(θµ  η )), Df(gU(1)v((θv)), Df(gU(1)c((θc)) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S31, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S32) and hence ˆHstrain  preserves all symmetries except for C3z  [hstrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Df(T)] = [σxτ0ς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' σ0τxε0] = 0  [hstrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Df(C2x)] = [σxτ0ς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' σxτ0ς0] = 0  [hstrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Df(C2zT)] = [σxτ0ς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' σxτ0ς0] = 0  [hstrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Df(gU(1)c((θc))] = [σxτ0ς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' e−iθcσ0τ0ς0] = 0  [hstrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Df(gU(1)v((θv))] = [σxτ0ς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' σ0e−iθvτzς0] = 0  [hstrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Df(gSU(2)η(θµ  η ))] = [σxτ0ς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' σ0e−i �  µ θη  µ  τ0+ητz  4  ςµ] = 0  30  In the presence of strain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' the symmetric state is defined as the state that preserves all the symmetries except for C3z which  is broken by the strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In the presence of C3z-breaking strain, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S57 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S89 still hold, because the system still has  U(1)c × U(1)v × SU(2)+ × SU(2)− symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' As for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S58 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S90, we only need to consider T, C2x, C2zT and we  find  Of  αηs,α′η′s′ =  �  χf  0σ0 + χf  1σx  �  α,α′δη,η′δs,s′  Oc′  αηs,α′η′s′ =  �  χc′  0 σ0 + χc′  1 σx  �  α,α′  δη,η′δs,s′,  Oc′′  αηs,α′η′s′ =  �  χc′′  0 σ0 + χc′′  1 σx  �  α,α′  δη,η′δs,s′,  Oc′f  αηs,α′η′s′ =  �  χc′f  0 σ0 + χc′f  1 σx  �  α,α′δη,η′δs,s′  (S96)  where χf  0, χf  1, χc′  0 , χc′  1 , χc′′  0 , χc′′  1 , χc′f  0 , χc′f  1  are real numbers tha characterize the density matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S38, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S48  and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S96,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' we find  χf  0 =  1  8NM  �  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αetas  ⟨Ψ| : f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs : |Ψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  χf  1 =  1  8NM  �  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='ηs  ⟨Ψ|f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='1ηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2ηs + f †  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2ηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='1ηs|Ψ⟩  χc′  0 =  1  8NM  �  |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='ηs  ⟨Ψ| : c†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηsc†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηs : |Ψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  χc′  1 =  1  8NM  �  |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='ηs  ⟨Ψ|c†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='1ηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2ηs + c†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2ηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='1ηs|Ψ⟩  χc′′  0 =  1  8NM  �  |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='=3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='ηs  ⟨Ψ| : c†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηsc†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aηs : |Ψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  χc′′  1 =  1  8NM  �  |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='ηs  ⟨Ψ|c†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3ηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='4ηs + c†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='4ηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3ηs|Ψ⟩  χc′f  0  =  1  8NM  √NM  �  |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs  e−ik·R⟨Ψ|c†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs|Ψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  χc′f  1  =  1  8NM  √NM  �  |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='ηs  e−ik·R⟨Ψ|c†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='1ηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2ηs + c†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2ηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='1ηs|Ψ⟩  χc′′f  0  =  1  8NM  √NM  �  |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs  e−ik·R⟨Ψ|c†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α+2ηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='αηs|Ψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  χc′′f  1  =  1  8NM  √NM  �  |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='ηs  e−ik·R⟨Ψ|c†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3ηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2ηs + c†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='4ηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3ηs|Ψ⟩  (S97)  where we also have χf  0 = νf/8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' χc′  0 = νc′/8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' χc′′  0 = νc′′/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' As for V3, V4, we only consider the T, C2x, CzT of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S61, which  indicates  V3 = V4 = V ∗  3 = V ∗  4  (S98)  Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S44, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S93, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S97 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S98 with filling constraints ν = νf + νc′ + νc′′, we have a complete set of the  mean-field self-consistent equations of symmetric state in the presence of strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We perform calculations with non-zero strain  at ν = 0, −1, −2, −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We initialize the calculations with the fully symmetric solutions derived at zero strain, and the procedure  converges within 500 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The results are illustrated and discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S5 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Effect of doping  We now discuss the effect of doping at zero strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' For hole doping at ν = 0, −1, −2, −3 and electron doping at ν = 0, we  mainly dope electrons to the light bands that are mostly formed by c-electrons (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Consequently, the energy difference  between the symmetric state and the ordered state decreases since we have more conduction c-electrons near the Fermi energy,  and the system favors the symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We now point out the complexity of electron dopings at ν = −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Doping electrons at ν = −1, −2 is equivalent to dope  electrons to the heavy bands that are mostly formed by f-electrons (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The heavy (flat) bands become closer to the Fermi  energy, and hence, the energy cost of putting f-electrons into flat bands will be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Then we can fill the heavy (flat) bands  with a small energy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' By filling the heavy (flat) bands, the type of orders formed by f-electrons can change a lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' To observe  31  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Dipsersions of KIVC state at ν = 0, KIVC+VP state at ν = −1, KIVC state at ν = −2 and VP state at ν = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The color represents  the weight of f-(yellow) and c-(blue) electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Evolution of order parameter as a function of doping at ν = 0, −1, −2  the change of the ordered states, we consider the following order parameters  Ox =  1  NM  �  R,αηs,α′η′s′  f †  R,αηs[ox]αηs,α′η′s′fR,α′η′s′,  x ∈ {KIV C, Sz, Vz, Vy}  oKIV C = σyτyς0,  oSz = σ0τ0ςz,  oVz = σ0τzς0,  oVy = σ0τyς0  (S99)  We measure the expectation values of Ox=KIV C,Sz,Vz,Vy with respect to the ordered states at each filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S8, we show  the evolution of ⟨Ox⟩ as a function of doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We find for hole doping at ν = 0, −1, −2 and electron doping at ν = 0 (where  carriers go to light bands in both cases), the system stays in the same ordered states (compared to the integer filling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However,  for electron doping at ν = −1, −2, we can observe the changes of the order parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This is because we are mainly dope  f-electrons for electron doping at ν = −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We thus conclude that electron doping at ν = −1, −2 will introduce sizeable  changes of the order parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Both the change of order parameters and the doping effect will affect the energy competition  between the symmetric state and the ordered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Finally, we comment on the ν = −3 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' At ν = −3, the actual ground state might be a CDW state which breaks the  translational symmetry [145] and is beyond our current consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In addition, at ν = −3, even for the valley polarized state  we currently considered, electron doping is equivalent to doping both heavy and light bands (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S7), which is different from  ν = −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We leave the detailed study of ν = −3 for future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Effect of strain  We next analyze the effect of the strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We first note that as we increase α (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S95), the strain will gradually suppress the  KIVC order (OKIV C, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This can be observed from  {oKIV C, hstrain} = {σyτyς0, σxτ0ς0} = 0  (S100)  Heuristically, the anti-commuting nature indicates the competition between oKIV C and hstrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Thus, as we increase hstrain,  oKIV C will be suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We also find the spin-polarization OSz and valley polarization OVz commute with hstrain  [oSz, hstrain] = [σ0τ0ςz, σxτ0ς0] = 0,  [oVz, hstrain] = [σ0τzς0, σxτ0ς0] = 0  (S101)  32  Heuristically, this indicates the valley and spin polarization do not directly compete with hstrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, as we will show in  this section, a sufficiently large strain could still destroy the valley and spin polarization in the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  For future convenience, we also introduce the eigenstates of the strain Hamiltonian ˆHstrain (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S95)  d†  R,1ηs =  1  √  2(f †  R,1ηs − f †  R,2ηs),  d†  R,2ηs =  1  √  2(f †  R,1ηs + f †  R,2ηs)  (S102)  We will call d†  R,1ηs and d†  R,2ηs as d1 and d2 electrons (orbitals), respectively, for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We mention that d1, d2 are f-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  The strain Hamiltonian can then be written as  ˆHstrain = α  �  R,αηs  (−d†  R,1ηsdR,1ηs + d†  R,2ηsdR,2ηs)  (S103)  Thus, for a positive strain amplitude α > 0, the energy of d1 electrons will be lowered and the energy of d2 electrons will be  raised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We introduce ⟨hstrain⟩ to characterize the population imbalance between d1 and d2 electrons  ⟨hstrain⟩ = ⟨ 1  NM  �  R,αηs,α′η′s′  f †  R,αηs[hstrain]αηs,α′η′s′fR,α′η′s′⟩ =  1  NM  �  R,αηs  ⟨d†  R,1ηsdR,1ηs − d†  R,2ηsdR,2ηs⟩  (S104)  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S9, we plot the evolution of various order parameters and also |⟨hstrain⟩| where the expectation value is taken with  respect to the ordered state solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In all cases, |⟨hstrain⟩| increases as we increase α, since α linearly coupled to hstrain  term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The KIVC order will be suppressed and fully destroyed at sufficient strong strain at ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' At ν = 0, after the  destruction of the KIVC order, self-consistent calculation produces a symmetric ground state that only breaks C3z symmetry,  even though we initialize the mean-field calculation with an ordered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  However, at ν = −1, −2, after the destruction of KIVC order, the spin polarization and valley polarization still exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' By  further increasing the strain, the ordered states will finally become unstable (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S9), which means the mean-field calculations  that are initialized with ordered solutions converge to a symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We next analyze the transition from an ordered state to a symmetric state at a large strain at ν = −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  ν = −1  We first consider the ν = −1 with 4meV ≲ α ≲ 18meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In this parameter region, the KIVC order is destroyed but valley  and spin polarization persist (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S10 (a) (b), we plot the band structures in this parameter region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We note that  flat bands that are mostly formed by f-electrons (marked by red circles, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S10) move towards the Fermi level, as we increase  strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Near the transition point to the symmetric state, the flat bands are very close to the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This signals an instability  of the ordered states since we can fill the flat band without any energy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' By diagonalizing the mean-field Hamiltonian, we  find the flat bands (marked by red circles, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S10) correspond to d1 electrons (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S103).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' By filling the flat bands, we have  more populations in d1 orbitals, which increase |⟨hstrain⟩| (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S104) and drive the system to a symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We now estimate the critical value of strain αc at which a transition from an ordered state to a symmetric state happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' At  αc, the flat bands (marked by red circles, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S10) are very close to the Fermi energy and induce the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' To estimate αc,  we calculate the excitation gap of the flat bands (marked by red circles, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S10): ∆Eflat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Then we have  ∆Eflat  ����  α≈αc  = 0  (S105)  We estimate ∆Eflat using the zero-hybridization limit [2] of the model, where γ = 0, v′  ⋆ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In addition, we also  set J = 0 to simplify the calculation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The zero-hybridization model with non-zero strains are  ˆHzero-hyb = ˆHU + ˆHW + ˆHV + ˆHstain + ˆHc + ˆHµ  (S106)  where ˆHU, ˆHW , ˆHV , ˆHstrain, ˆHc, ˆHµ are defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S16, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S18, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S95, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S13 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S12 respectively,  and ˆHV are treated with mean-field methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In the zero-hybridization model, the filling of f-electrons νf and c-electrons-νc  are good quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We solve the zero-hybridization model at fixed total filling ν with the assumption that the ground  state does not break translational symmetry (fillings of f-electrons are uniform) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' To estimate the excitation gap of the flat  bands, we calculate the energy cost of adding one dR,1ηs electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We mention that, in our mean-field calculations with finite  f-c hybridization, the relevant flat bands (marked by red circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S10) correspond to d1 electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We let |Ψzero-hyb⟩ denote  the ground state of the zero-hybridization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The state with one-more dR,1ηs is  |Ψexct  zero-hyb⟩ = d†  R,1ηs|Ψzero-hyb⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (S107)  33  We next calculate  ∆Eflat = ⟨Ψexct  zero-hyb| ˆHzero-hyb|Ψexct  zero-hyb⟩ − ⟨Ψzero-hyb| ˆHzero-hyb|Ψzero-hyb⟩  (S108)  The energy loss from Hubbard interaction term is  ∆EU = ⟨Ψexct  zero-hyb| ˆHU|Ψexct  U  ⟩ − ⟨Ψzero-hyb| ˆHU|Ψzero-hyb⟩ = U  2 (νf + 1)2 − U  2 ν2  f = U(νf + 1  2)  The energy loss from ˆHW term is  ∆EW =⟨Ψexct  zero-hyb| ˆHW |Ψexct  zero-hyb⟩ − ⟨Ψzero-hyb| ˆHW |Ψzero-hyb⟩  =  �  a=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='4  Waνc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a(νf + 1) −  �  a=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='4  Waνc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='aνf =  �  a=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='4  Waνc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a  (S109)  where νc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a denotes the filling of c-electrons in orbital a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The energy change from ˆHV is  ∆EV = ⟨Ψexct  zero-hyb| ˆHV |Ψexct  zero-hyb⟩ − ⟨Ψzero-hyb| ˆHV |Ψzero-hyb⟩ = 0  (S110)  The energy change from ˆHstrain is  ∆Estrain = ⟨Ψexct  zero-hyb| ˆHstrain|Ψexct  zero-hyb⟩ − ⟨Ψzero-hyb| ˆHstrain|Ψzero-hyb⟩ = −α  (S111)  The energy change from chemical potential ˆHµ is  ∆Eµ = ⟨Ψexct  zero-hyb| ˆHµ|Ψexct  zero-hyb⟩ − ⟨Ψzero-hyb| ˆHµ|Ψzero-hyb⟩ − µ  (S112)  Then the excitation energy of adding one dR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='1ηs electron is  ∆Eflat = ∆EU + ∆EW + ∆strain + ∆Eµ = U  2 (νf + 1/2) +  �  a  Waνc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='a − µ − α  (S113)  We further take the following approximation: W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='4 = W = 47meV (the difference between W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='4 is about 15%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Then  ∆Eflat ≈ U  2 (νf + 1/2) + Wνc − µ − α  (S114)  At ν = −1 and 0meV ≤ α ≤ 43meV, the ground state of the zero-hybridization model has νf = −1, νc = ν − νf = 0  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Then  ∆Eflat = −U  2 − µ − α  (S115)  We next determine chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Chemical potential µ is determined by requiring the c-electrons filling to be νc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  The single-particle Hamiltonian of c-electron in the zero-hybridization limit takes the form of  ˆHc,zero-hyb = ˆHc +  �  k,aηs  (Wνf + V (0)  Ω0  νc − µ)c†  k,aηsck,aηs  (S116)  where we have set W1,2,3,4 = W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We note that when  Wνf − V (0)  Ω0  νc − µ = 0  (S117)  ˆHc,zero-hyb = ˆHc and we have νc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Therefore,  µ = Wνf + V (0)  Ω0  νc = −W  (S118)  34  where we take νc = 0, νf = −1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S115 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S118, we find  ∆Eflat = W − U  2 − α  (S119)  Then the flat bands reach Fermi energy when ∆Eflat = 0, which leads to  ∆Eflat = 0 ⇒ αc = W − U  2 = 18meV  (S120)  which is close to the value (also around α = 18meV as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S9 (b)) from self-consistent calculations of the finite-  hybridization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Here, the finite-hybridization model refers to the original THF model with finite γ, v′  ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Therefore, we  conclude the transition from an ordered state to a symmetric state happens at α = αc ≈ 18meV at ν = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We also discuss the solutions of the zero-hybridization model here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S11 (a), we show the ground state properties of the  zero-hybridization model at various strains and ν = −1, where  νf  1 =  1  NM  �  R,ηs  : d†  R,1ηsdR,1ηs :,  νf  2 =  1  NM  �  R,ηs  : d†  R,2ηsdR,2ηs :  (S121)  denotes the filling of d1 and d2 electrons respectively with νf = νf  1 + νf  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We find a transition happens at α ≈ 25meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We note that this transition is described by filling one more dR,1ηs electrons at each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' After the transition, there will be  4 f-electrons filling d1 orbitals, and zero f-electrons filling the d1 orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Thus for d1 orbitals, all the valleys and spins are  filled, but for d2 orbitals all the valleys and spins are empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Therefore, there is no room to develop order and the ground state  is a symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We note that the transition in the zero-hybridization limit and the transition in the finite-hybridization  model (at ν = −1, α ≈ 16meV, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S9) share the same origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' They are both driven by filling electrons in d1 orbitals (in the  finite-hybridization model, we fill the flat bands) and, after the transition, both ground states are symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Thus, the results  between zero-hybridization and finite-hybridization models are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, the critical values αc for the two models  are different, since we have finite f-c hybridization in the finite-hybridization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  ν = −2  We next discuss the transition from an ordered state to a symmetric state at ν = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We focus on the parameter region  10meV ≲ α ≲ 45meV, where the KIVC order is destroyed but valley and spin polarization exist (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S10 (c) (d),  we plot the band structures in this parameter region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' As we increase strain, we note that flat bands (marked with red circles,  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S10), move towards the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Similar to the ν = −1 case, when the flat bands reach the Fermi energy, a transition to  the symmetric state happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' However, at ν = −2, we need a much larger strain to destroy the ordered state as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S9  (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' To understand this, we start from the zero-hybridization limit of the model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S106).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' As shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S114, the excitation  energy of the relevant flat bands (marked by red circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S10)  ∆Eflat = U  2 (νf + 1/2) + Wνc − µ − α  (S122)  By solving the zero-hybridization model, we find the ground states have νf = −1 and νc = −1 in the parameter region we  focused 4meV ≲ α ≲ 18meV, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Then  ∆Eflat = −U  2 − W − µ − α  (S123)  We now calculate the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' µ is determined by requiring the filling of c-electrons to be νc = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The single-  particle Hamiltonian of c-electron in the zero-hybridization limit (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S116) takes the form of  ˆHc,zero-hyb = ˆHc +  �  k,aηs  (Wνf + V (0)  Ω0  νc − µ)c†  k,aηsck,aηs  (S124)  where we have set W1,2,3,4 = W, and take the mean-field treatment of ˆHV (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' At M = 0 limit (M = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='697meV, which  is relatively small), the dispersion of c-electrons are Ek = ±v⋆|k| − Ec, where we define  Ec = Wνf + V (0)  Ω0  νc − µ  (S125)  35  Then all the c-states with energy smaller than 0 will be filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The corresponding Fermi momentum kF is  |v⋆kF | = Ec ⇒ kF =  1  |v⋆|Ec  (S126)  Then the filling of c-electrons is  νc = −  8  AMBZ  �  |k|<kF  dkxdky = − 8πk2  F  AMBZ  = −  8πE2  c  AMBZ|v⋆|2  (S127)  where the prefactor 8 comes from the 8-fold degeneracy of the bands (2 spins, 2 valleys, and 2 orbitals) and AMBZ is the area  of MBZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We require νc = −1 which indicates  νc = −  8πE2  c  AMBZ|v⋆|2 = −1 ⇒ Ec =  �  AMBZ|v⋆|2  8π  (S128)  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S125 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S128, we find  µ = −W − V (0)  Ω0  −  �  AMBZ|v⋆|2  8π  (S129)  where we take νf = −1, νc = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S123 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S128, we find  ∆Eflat = −U  2 + V (0)  Ω0  +  �  AMBZ|v⋆|2  8π  − α  (S130)  And the transition happens at  ∆Eflat = 0 ⇒ αc = −U  2 + V (0)  Ω0  +  �  AMBZ|v⋆|2  8π  = 62meV  (S131)  We note that our estimation of transition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S131 is based on the single-particle picture and has not included the effect  of hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Thus the estimated value is larger than the transition value α ≈ 45meV from mean-field self-consistent  calculations with finite f-c hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We also discuss the solution obtained from the zero-hybridization model at ν = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S11 (b), we show the ground state  properties of the zero-hybridization model at various strains and ν = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We observe two transitions, both characterize by the  increments of νf  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' After the first transition α ∼ 15meV, we have three d1 electrons at each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Then not all valleys and spins  of d1 orbitals are filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This gives the possibility of developing spin and valley polarized order after we turn on the ˆHJ and f-c  hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Indeed, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S9, we can observe the non-vanishing order at 15meV ≲ α ≲ 40meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' By further increasing  strain, at α ≈ 50meV, we observe a second transition in the zero-hybridization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' After the second transition, we have  νf  1 = 2, νf  2 = −2 and there is no room for f-electrons to develop order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Correspondingly, for the finite-hybridization model, we  also observe a transition to the symmetric stat at α ≈ 45meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Thus we conclude the consistency between zero-hybridization and  finite-hybridization models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We also point out that, the transition value αc ≈ 62meV suggested from single-particle excitation  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S131 is also larger than the transition value αc ≈ 50meV obtained by solving many-body wavefunction in the zero-  hybridization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This points out the limitation of the single-particle picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Finally, we comment on the difference between ν = −1 and ν = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S131 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S120 indicate that a relatively large  strain is required to destroy the ordered state at ν = −2 compared to ν = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' This is because, near the transition point with  α < αc, c-electrons have filling νc = 0 at ν = −1, but have filling νc = −1 at ν = −2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In other words, we dope  more holes to the c-bands at ν = −2 compared to ν = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In order to dope more holes to c-bands, the chemical potential (or  Fermi energy) needs to be increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The increment of chemical potential also leads to a larger Fermi surface, which has been  observed in the model with finite hybridization (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S10), where the Fermi surface at ν = −2 is larger than the Fermi surface  at ν = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In the zero-hybridization model, the change of chemical potential can be observed analytically from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S118 and  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We note that the excitation gap of the flat band (∆Eflat) is measured with respect to the Fermi energy (or chemical  potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Then at fixed α, we find the value of ∆Eflat at ν = −2 is much larger than its value at ν = −1 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S119 and  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S130) due to the larger chemical potential at ν = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Thus, at ν = −2, a much larger strain is needed to make ∆Eflat = 0  and destroy the ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  36  (a)  (b)  (c)  (d)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' (a), (b), (c) Evolution of order parameters as a function of strain amplitude α at ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' (d) Evolution of order parameters at  ν = −2 with an extended parameter region 0meV ≤ α ≤ 55meV  (a)  (b)  (c)  (d)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Band structure at ν = −1 (a), (b) and ν = −2 (c), (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Red circles mark the relevant flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Discussions about ν = −3  We now discuss ν = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We first comment that, at ν = −3, other low energy states that break translational symmetry exists  even at zero strain [145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Furthermore, even in the zero-hybridization limit at zero strain, ν = −3 is close to the transition point  between νf = −3 and νf = −2 states [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' These all point out the complexity of ν = −3 even in the absence of strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Thus, we  leave a more systematical analysis at ν = −3 for future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Strain  We now discuss the derivation of strain term in the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Here we will take a minimal model to capture the effect of strain,  and leave a detailed analysis (based on Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [150]) for future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We take the single-particle Hamiltonian of twisted bilayer  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Evolution of filling as a function of the strain in the zero-hybridization model at ν = −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  37  graphene at valley η and with non-zero strain [148]  Hη(k) =  �Hη  1 (k)  U  U †  Hη  2 (k)  �  (S132)  where Hl(k) is a 2 × 2 matrix that characterizes the Hamiltonian of l-layer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' U is the interlayer coupling matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In  real space, U takes the form of  U =  3  �  j=1  Ujeiηqj·r,  Uj =  �  w0  w1e−iη2(j−1)π/3  w1eiη2(j−1)π/3  w0  �  with w0, w1 are two constant parameters that characterize the interlayer couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Hl(k) is given by  Hη  l=1,2(k) = −ℏvF  ��  R(∓θ) + El  �−1  (k + e  ℏAl,η)  �  ση  (S133)  where ση = (ησx, σy) , R(θ) is a rotation matrix, θ is the twist angle and [148]  El =  �  ϵ(l)  xx  −Ω(l) + ϵ(l)  xy  −Ω(l) + ϵ(l)  xy  ϵ(l)  yy  �  Al,η = η 3βγ0  2evF  �  ϵ(l)  −  ϵ(l)  xy  �  ,  γ0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='7eV,  β ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='14  ϵ(l)  ± = 1  2(ϵ(l)  xx ± ϵ(l)  yy) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (S134)  ϵ(l)  xx, ϵ(l)  yy are normal strains along x and y directions respectively, ϵ(l)  xy is the sheer strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Ω(l) denotes a rotation from the  twist angles [148].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Since the flat band is sensitive to ϵ− = (ϵ(1)  − − ϵ(2)  − )/2 and ϵxy = (ϵ(1)  xy − ϵ(2)  xy )/2 [148], we consider the  heterostrain with Ω(l) = 0, ϵ(l)  + = 0, ϵ(l)  − = (−1)l+1ϵ−, ϵ(l)  xy = (−1)l+1ϵxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We further focus on the sheer strain ϵxy and set the  normal anisotropic strain to be zero ϵ− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' For small twist angle, we approximately have  Hη  l=1,2(k) ≈ − ℏvF  ��  R(±θ) − El  �  (k + e  ℏAl,η)  �  ση  ≈ − ℏvF  �  R(±θ)k − Elk + e  ℏAl,η  �  ση  (S135)  The effect of strain is then captured by the following single particle Hamiltonian of valley η  Hη  strain(k) = Hη  strain,1 + Hη  strain,2  Hη  strain,1 = −3βγ0η  2  ϵxy  �  σy  −σy  �  ,  Hη  strain,2 = ℏvF ϵxy  �  ηkyσx + kxσy  −ηkyσx − kxσy  �  (S136)  We next project the Hη  strain(k) to the Wannier basis of f-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We take the following wavefunction of f electrons at R  with orbital α, valley η and spin s [1]  |WR,α=1,η,s⟩ =  �  2πλ2  0  Ωtot  �  l=±  �  k  �  Q∈Qlη  ei π  4 lη−ik·R− 1  2 λ2  0(k−Q)2|k, Q, 1, η, s⟩  |WR,α=2,η,s⟩ =  �  2πλ2  0  Ωtot  �  l=±  �  k  �  Q∈Qlη  e−i π  4 lη−ik·R− 1  2 λ2  0(k−Q)2|k, Q, 2, η, s⟩  (S137)  where λ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='1aM, Ωtot is the total area of the sample and Qlη = {±q1 + n1bM,1 + n2bM,2|n1, n2 ∈ Z}, q1 = kθ(0, −1) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S136 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S137, we can calculate  hstrain  αηs,α2η2s2 =  1  NM  �  R  ⟨WR,α,η,s|  �  k,α′l′,α′  2l′  2,η′,s′  ψ†  k,l,α′,η′,s′[Hη′  strain(k)]l′α′,l′  2α′  2ψk,l′  2α′,η′,s′|WR,α2,η2,s2⟩  (S138)  38  where ψk,l,α,η,s is the electron operator of the original electron basis with momentum k, layer l, sublattice α, valley η and spin  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Here, we take the average over all the moir´e unit cells (sum over R), and thus we only keep the momentum-independent (in  f-electron basis) contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We leave the momentum-dependent term for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' In the f-basis, the strain can be  characterized by  ˆHstrain =  �  R,αηs,α2η2s2  hstrain  αηs,α2η2s2f †  R,αηsfR,α2η2s2  (S139)  We next evaluate hstrain from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We separate hstrain into two parts (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S136)  hstrain,1  αηs,α2η2s2 =  1  NM  �  R  ⟨WR,α,η,s|  �  k,α′l′,α′  2l′  2,η′,s′  ψ†  k,l,α′,η′,s′[Hη′  strain,1(k)]l′α′,l′  2α′  2ψk,l′  2α′,η′,s′|WR,α2,η2,s2⟩  hstrain,2  αηs,α2η2s2 =  1  NM  �  R  ⟨WR,α,η,s|  �  k,α′l′,α′  2l′  2,η′,s′  ψ†  k,l,α′,η′,s′[Hη′  strain,2(k)]l′α′,l′  2α′  2ψk,l′  2α′,η′,s′|WR,α2,η2,s2⟩  (S140)  For hstrain,1, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S136, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S137 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S140, we find  hstrain,1  αηs,α2η2s2  =δη,η2δs,s2δα,3−α2  2πλ2  0  Ωtot  −3βγ0  2  ϵxy  �  l=±  �  Q∈Qlη  �  k  i(−1)αlηe−λ2  0(k−Q)2e−i πηl  2 (−1)α+1  =δη,η2δs,s2δα,3−α2  3πλ2  0βγ0  Ωtot  ϵxy  �  k  � �  Q∈Q+  e−λ2  0(k−Q)2 +  �  Q∈Q−  e−λ2  0(k−Q)2�  (S141)  In a more compact form, we find  hstrain,1 = ασxτ0ς0  α = 3πλ2  0βγ0  Ωtot  ϵxy  �  k  � �  Q∈Q+  e−λ2  0(k−Q)2 +  �  Q∈Q−  e−λ2  0(k−Q)2�  ≈  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3 × 104ϵxy  �  meV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (S142)  We next calculate hstrain,2, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S136, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S137 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S140,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' we find  hstrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2  αηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='α2η2s2  =δη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η2δs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s2δα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3−α2  2πλ2  0ℏϵxy  Ωtot  �  l=±  �  Q∈Qlη  �  k  l  �  η(ky − Qy) + i(−1)α(kx − Qx)  �  ei π  2 lη(−1)αe−λ2  0(k−Q)2  =δη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η2δs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s2δα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3−α2  2πλ2  0ℏϵxy  Ωtot  �  k  � �  Q∈Qη  �  η(ky − Qy) + i(−1)α(kx − Qx)  �  iη(−1)αe−λ2  0(k−Q)2  +  �  Q∈Q−η  �  η(ky − Qy) + i(−1)α(kx − Qx)  �  iη(−1)αe−λ2  0(k−Q)2�  =δη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='η2δs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='s2δα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3−α2  2πλ2  0ℏϵxy  Ωtot  �  k  �  Q∈Q0  �  η(ky − Qy − ηq1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='y) + i(−1)α(kx − Qx − ηq1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='x)  �  iη(−1)αe−λ2  0(k−Q−ηq1)2  +  �  η(ky − Qy + ηq1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='y) + i(−1)α(kx − Qx + ηq1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='x)  �  iη(−1)αe−λ2  0(k−Q+ηq1)2�  (S143)  where we let Q0 = {n1bM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='1 + n2bM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='2|n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' n2 ∈ Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Since q1 = kθ(0, 1), q1,x = 0, q1,y = kθ, we find  hstrain,2  αηs,α2η2s2  =δη,η2δs,s2δα,3−α2  2πλ2  0ℏϵxy  Ωtot  �  k  �  Q∈Q0  �  i(−1)α(ky − Qy)f+(k − Q) − iq1,y(−1)αf−(k − Q)  − (kx − Qx)ηf+(k − Q)  (S144)  39  where we define  f±(k − Q) = ±e−λ2  0(k−Q+q1)2 + e−λ2  0(k−Q−q1)2  (S145)  Since q1 = kθ(0, 1), f+(k − Q) is an even function of both (k − Q)x and (k − Q)y and f−(k − Q) is an even function of  (k − Q)x and an odd function of (k − Q)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Therefore,  �  k  �  Q∈Q0  (ky − Qy)f+(k − Q) = 0,  �  k  �  Q∈Q0  f−(k − Q) = 0,  �  k  �  Q∈Q0  (kx − Qx)f+(k − Q) = 0  (S146)  Thus,  hstrain,2  αηs,α2η2s2 = 0  (S147)  However, we mention that, even though hstrain,2 = 0, Hη  strain,2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S136 can contribute a k-denpendent term in the THF  model which has been omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S142 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S147, we have  ˆHstrain =  �  R,αηs,α2η2s2  ααηs,α2η2s2f †  R,αηsfR,α2η2s2 =  α  �  R,αηs,α2η2s2  [hstrain]αηs,α2η2s2f †  R,αηsfR,α2η2s2  (S148)  which is equivalent to the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' S95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' α is connected to ϵxy via  α = 3πλ2  0βγ0  Ωtot  ϵxy  �  k  � �  Q∈Q+  e−λ2  0(k−Q)2 +  �  Q∈Q−  e−λ2  0(k−Q)2�  ≈  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='3 × 104ϵxy  �  meV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (S149)  For a typical strain at the order of ϵxy ∼ 10−3 [148], α ∼ 13meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We mention that the strain could also induce addition terms  to the conduction c-electron blocks (c†c) and f-c hybridization blocks (c†f, f †c), which are expected to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Here, we  only consider a minimal model, that is zeroth order in k, to capture the essential effect of the strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We leave a comprehensive  construction of strain term (based on Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' [150]) for future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  DYNAMICAL MEAN FIELD THEORY: IMPLEMENTATION  We study dynamical effects of the f-f density-density term using DMFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The c-f and c-c density terms are included in this  model on the Hartree level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We neglect the f-c exchange interaction ˆHJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We treat a Hamiltonian of the form ˆHc + ˆHfc + ˆHU +  ˆHMF  W  + ˆHMF  V  where the first two terms compose the non-interacting THF model and the remaining terms are given by  ˆHU =U  2  �  (α,η,σ)̸=(α′,η′,σ′)  f †  α,η,σfα,η,σf †  α′,η′,σ′fα′,η′,σ′  − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5U  �  (α,η,σ)  f †  α,η,σfα,η,σ  (S150)  ˆHMF  W  =Wνf  �  (a,η,σ)  c†  a,η,σca,η,σ + Wνc  �  (α,η,σ)  f †  α,η,σfα,η,σ,  (S151)  ˆHMF  V  =V νc  �  (a,η,σ)  c†  a,η,σca,η,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (S152)  Here, the last two terms are mean-field decoupled and therefore composed of single particle terms, and we have neglected all  constant terms as an overall shift does not affect the observables we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' The two particle term ˆHU is restricted to the  f- subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We proceed with an LDA+DMFT-style calculation with a correlated subspace made up of f- electrons embedded  in a larger space of f- and c- electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' By allowing an arbitrary shift of the chemical potential µ and fixing the total filling  ν = νf + νc, we can consider an effective interacting Hamiltonian restricted to the the f- subspace:  ˆHeff  I  = ˆHff + ˆHDC  (S153)  ˆHff =U  2  �  (α,η,σ)̸=(α′,η′,σ′)  f †  α,η,σfα,η,σf †  α′,η′,σ′fα′,η′,σ′  (S154)  ˆHDC = (−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='5U + (ν − 2νf)W − V (ν − νf))  �  ��  �  µDC(νf )  �  (α,η,σ)  f †  α,η,σfα,η,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (S155)  40  The f-electron occupation νf is self-consistently calculated along with the the f- subspace self-energy Σf in at each step of  the DMFT loop for a set of different total fillings (ν) and temperatures (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Written in this way, the single particle terms can  be treated as an iteration-dependent double counting potential, µ(i)  DC = µDC  �  ν(i)  f  �  , where the superscript labels the iteration  number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  In the ith iteration of the self-consistency loop, we perform the following sequence of steps:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Embed the old impurity self-energy Σ(i−1)  f  into the lattice Green’s function associated to the Hamiltonian ˆHc + ˆHfc +  ˆH(i−1)  DC  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Fix the total filling to ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Calculate the new orbital-resolved fillings ν(i)  f  and ν(i)  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Determine the new double counting term µ(i)  DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' Use a CT-QMC solver to obtain the new self energy Σ(i)  f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  We performed our calculations in parallel using the w2dynamics suite [153, 154]1 as well as the TRIQS family of packages  [155–157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' 3(a) of the main text, the zero frequency spectral weight and the scattering rate are computed by extrapolating the green  function and the self-energy respectively on the Matsubara axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content=' We approximate these by using the value at the first Matsubara  frequency so that  Γ = −ImΣ(ω = 0) ≈ −ImΣ(ω0),  (S156)  A(ω = 0) = − 1  π  ��  k  TrG(k, ω = 0)  �  ≈ − 1  π  ��  k  TrG(k, ω0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='  (S157)  1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
+page_content='com/w2dynamics/w2dynamics' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'}
diff --git a/FtAzT4oBgHgl3EQfUfxu/content/tmp_files/2301.01267v1.pdf.txt b/FtAzT4oBgHgl3EQfUfxu/content/tmp_files/2301.01267v1.pdf.txt
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+arXiv:2301.01267v1  [math.PR]  3 Jan 2023
+Optimal convergence rates in stochastic
+homogenization in a balanced random environment
+Xiaoqin Guo ∗1 and Hung V. Tran †2
+1 Department of Mathematical Sciences, University of Cincinnati , 2815
+Commons Way, Cincinnati, OH 45221, USA
+2 Department of Mathematics, University of Wisconsin Madison, 480
+Lincoln Drive, Madison, WI 53706, USA
+January 4, 2023
+Abstract
+We consider random walks in a uniformly elliptic, balanced, i.i.d. ran-
+dom environment in ℤ푑 for 푑 ≥ 2. We first derive a quantitative law of large
+numbers for the invariant measure, which is nearly optimal. A mixing prop-
+erty of the field of the invariant measure is then achieved. We next obtain
+rates of convergence for the homogenization of the Dirichlet problem, which
+are generically optimal for 푑 ≥ 5. Afterwards, we quantify the ergodicity
+of the environmental process for both the continuous-time and discrete-time
+random walks, and as a consequence, we get explicit convergence rates for
+the quenched central limit theorem of the balanced random walk.
+Contents
+1
+Introduction
+2
+1.1
+Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+2
+1.2
+Main assumptions . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+1.3
+Earlier results in the literature . . . . . . . . . . . . . . . . . . . .
+5
+1.4
+Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+2010 Mathematics subject classification.
+35J15 35J25 35K10 35K20 60G50 60J65 60K37
+74Q20 76M50.
+Key words and phrases.
+random walks in a balanced random environment; non-divergence
+form difference operators; invariant measures; quantitative stochastic homogenization; quantitative
+large-scale average; optimal convergence rates .
+∗The work of XG is supported by Simons Foundation through Collaboration Grant for Mathe-
+maticians #852943.
+†HT is supported in part by NSF CAREER grant DMS-1843320 and a Vilas Faculty Early-Career
+Investigator Award.
+1
+
+2
+Large scale 퐶0,1 and 퐶1,1 estimates
+11
+2.1
+Some regularity properties of deterministic functions . . . . . . .
+12
+2.2
+Large scale regularity . . . . . . . . . . . . . . . . . . . . . . . .
+14
+3
+Mixing properties of the invariant measure
+17
+3.1
+A sensitivity estimate of the invariant measure . . . . . . . . . . .
+18
+3.2
+Rate of convergence for the average of the invariant measure: Proof
+of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+21
+3.3
+Correlation structure of the field of the invariant measure . . . . .
+24
+4
+Homogenization of the Dirichlet problem
+25
+4.1
+Homogenization of the approximate corrector . . . . . . . . . . .
+25
+4.2
+Rate of homogenization for the Dirichlet problem (19)
+. . . . . .
+27
+5
+Quantification of the diffusive behavior
+29
+5.1
+Quantification of the ergodicity of the environmental process: Proof
+of Theorem 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+29
+5.2
+A Berry-Esseen estimate for the QCLT: Proof of Corollary 10
+. .
+32
+A Appendix
+33
+A.1
+Proof of Proposition 16 . . . . . . . . . . . . . . . . . . . . . . .
+33
+A.2
+Verification of (68) . . . . . . . . . . . . . . . . . . . . . . . . .
+34
+A.3
+Verification of (70) . . . . . . . . . . . . . . . . . . . . . . . . .
+35
+A.4
+Verification of (78) . . . . . . . . . . . . . . . . . . . . . . . . .
+35
+1
+Introduction
+In this paper, we consider random walks in a uniformly elliptic, balanced, i.i.d. ran-
+dom environment in ℤ푑 for 푑 ≥ 2. Our main goals are two-fold. Firstly, we derive
+a quantitative large-scale average of the invariant measure, which is nearly optimal,
+in Theorem 5. A mixing property of the field of the invariant measure is achieved.
+Secondly, we obtain rates of convergence for the homogenization of the Dirichlet
+problem in Theorem 7. When 푑 ≥ 5, the convergence rate is 푂(푅−1), which is
+generically optimal. Afterwards, we quantify the ergodicity of the environmental
+process for both the continuous- and discrete-time random walks in Theorem 8, and
+as a consequence, we get explicit convergence rates for the quenched central limit
+theorem (QCLT) of the balanced random walk.
+1.1
+Settings
+Let 핊푑×푑 denote the set of 푑 × 푑 positive-definite diagonal matrices. A map
+휔 ∶ ℤ푑 → 핊푑×푑
+2
+
+is called an environment. We denote the set of all environments by Ω. Let ℙ be a
+probability measure on Ω so that
+{휔(푥) = diag[휔1(푥), … , 휔푑(푥)], 푥 ∈ ℤ푑}
+are i.i.d. under ℙ. Expectation with respect to ℙ is denoted by 피 or 퐸ℙ.
+Let {푒1, … , 푒푑} be the canonical basis for ℝ푑. For any function 푢 ∶ ℤ푑 → ℝ
+and 휔 ∈ Ω, define the non-divergence form difference operator
+tr(휔(푥)∇2푢) =
+푑
+∑
+푖=1
+휔푖(푥)[푢(푥 + 푒푖) + 푢(푥 − 푒푖) − 2푢(푥)],
+(1)
+where ∇2 = diag[∇2
+1, … , ∇2
+푑], and ∇2
+푖 푢(푥) = 푢(푥 + 푒푖) + 푢(푥 − 푒푖) − 2푢(푥).
+For 푟 > 0, 푦 ∈ ℝ푑 we let
+픹푟(푦) = {푥 ∈ ℝ푑 ∶ |푥 − 푦| < 푟} ,
+퐵푟(푦) = 픹푟(푦) ∩ ℤ푑
+denote the continuous and discrete balls with center 푦 and radius 푟, respectively.
+When 푦 = 0, we also write 픹푟 = 픹푟(0) and 퐵푟 = 퐵푟(0). For any 퐵 ⊂ ℤ푑, its
+discrete boundary is defined as
+휕퐵 ∶=
+{
+푧 ∈ ℤ푑 ⧵ 퐵 ∶ dist(푧, 푥) = 1 for some 푥 ∈ 퐵
+}
+.
+Let ̄퐵 = 퐵 ∪ 휕퐵. By abuse of notations, whenever confusion does not occur, we
+also use 휕퐴 and ̄퐴 to denote the usual continuous boundary and closure of 퐴 ⊂ ℝ푑,
+respectively.
+For 푥 ∈ ℤ푑, a spatial shift 휃푥 ∶ Ω → Ω is defined by
+(휃푥휔)(⋅) = 휔(푥 + ⋅).
+In a random environment 휔 ∈ Ω, we consider the discrete elliptic Dirichlet problem
+⎧
+⎪
+⎨
+⎪⎩
+1
+2tr(휔∇2푢(푥)) =
+1
+푅2 푓
+(
+푥
+푅
+)
+휓(휃푥휔)
+푥 ∈ 퐵푅,
+푢(푥) = 푔
+(
+푥
+|푥|
+)
+푥 ∈ 휕퐵푅,
+(2)
+where 푓 ∈ ℝ픹1, 푔 ∈ ℝ휕픹1 are functions with good regularity properties and
+휓 ∈ ℝΩ is bounded and satisfies suitable measurability condition. Stochastic ho-
+mogenization studies (for ℙ-almost all 휔) the convergence of 푢 to the solution ̄푢 of
+a deterministic effective equation
+{ 1
+2tr( ̄푎퐷2 ̄푢) = 푓 ̄휓
+in 픹1,
+̄푢 = 푔
+on 휕픹1,
+(3)
+as 푅 → ∞. Here 퐷2 ̄푢 denotes the Hessian matrix of ̄푢 and ̄푎 = ̄푎(ℙ) ∈ 핊푑×푑 and
+̄휓 = ̄휓(ℙ, 휓) ∈ ℝ are deterministic and do not depend on the realization of the
+random environment (see the statement of Theorem C for formulas for ̄푎 and ̄휓).
+3
+
+The difference equation (2) is used to describe random walks in a random en-
+vironment (RWRE) in ℤ푑. To be specific, we set
+휔(푥, 푥 ± 푒푖) ∶=
+휔푖(푥)
+2tr휔(푥)
+for 푖 = 1, … 푑,
+(4)
+and 휔(푥, 푦) = 0 if |푥 − 푦| ≠ 1. Namely, we normalize 휔 to get a transition prob-
+ability. We remark that the configuration of {휔(푥, 푦) ∶ 푥, 푦 ∈ ℤ푑} is also called a
+balanced environment in the literature [40, 33, 10].
+Definition 1. For each fixed 휔 ∈ Ω, the random walk (푋푛)푛≥0 in the environment
+휔 with 푋0 = 푥 is a Markov chain in ℤ푑 with transition probability 푃 푥
+휔 specified by
+푃 푥
+휔
+(푋푛+1 = 푧|푋푛 = 푦) = 휔(푦, 푧).
+(5)
+The expectation with respect to 푃 푥
+휔 is written as 퐸푥
+휔. When the starting point of
+the random walk is 0, we sometimes omit the superscript and simply write 푃 0
+휔, 퐸0
+휔
+as 푃휔 and 퐸휔, respectively. Notice that for random walks (푋푛) in an environment
+휔,
+̄휔푖 = 휃푋푖휔 ∈ Ω,
+푖 ≥ 0,
+(6)
+is also a Markov chain, called the environment viewed from the particle process.
+By abuse of notation, we enlarge our probability space so that 푃휔 still denotes the
+joint law of the random walks and ( ̄휔푖)푖≥0.
+We also consider the continuous-time RWRE (푌푡) on ℤ푑.
+Definition 2. Let (푌푡)푡≥0 be the Markov process on ℤ푑 with generator
+퐿휔푢(푥) =
+∑
+푦
+휔(푥, 푦)[푢(푦) − 푢(푥)] =
+1
+2tr휔(푥)tr(휔(푥)∇2푢).
+(7)
+By abuse of notation, we also denote by 푃 푥
+휔 the quenched law of (푌푡). If there
+is no ambiguity from the context, we also write, for 푥, 푦 ∈ ℤ푑, 푛 ∈ ℤ, 푡 ∈ ℝ, the
+transition kernels of the discrete and continuous time walks as
+푝휔
+푛 (푥, 푦) = 푃 푥
+휔(푋푛 = 푦),
+and
+푝휔
+푡 (푥, 푦) = 푃 푥
+휔(푌푡 = 푦),
+respectively.
+1.2
+Main assumptions
+Throughout the paper, the following assumptions are always in force.
+(A1)
+{
+휔(푥), 푥 ∈ ℤ푑}
+are i.i.d. under the probability measure ℙ.
+(A2)
+휔
+tr휔 ≥ 2휅I for ℙ-almost every 휔 and some constant 휅 ∈ (0, 1
+2푑 ].
+(A3) 휓 is a measurable function of the environment with the property that {휓(휃푥휔) ∶
+푥 ∈ ℤ푑} are i.i.d. under ℙ.
+4
+
+In the paper, we use 푐, 퐶 to denote positive constants which may change from line
+to line but only depend on the dimension 푑 and the ellipticity constant 휅 unless
+otherwise stated. We write 퐴 ≲ 퐵 if 퐴 ≤ 퐶퐵, and 퐴 ≍ 퐵 if 퐴 ≲ 퐵 and 퐴 ≳ 퐵.
+We also use notations 퐴 ≲푗 퐵, 퐴 ≍푗 퐵 to indicate that the multiplicative constant
+depends on the variable 푗 other than (푑, 휅).
+1.3
+Earlier results in the literature
+We first recall the following quenched central limit theorem (QCLT) proved by
+Lawler [40], which is a discrete version of Papanicolaou, Varadhan [44].
+Theorem A. Assume (A2) and that law ℙ of the environment is ergodic under
+spatial shifts {휃푥 ∶ 푥 ∈ ℤ푑}. Then
+(i) There exists a probability measure ℚ ≈ ℙ such that ( ̄휔푖)푖≥0 is an ergodic (with
+respect to time shifts) sequence under law ℚ × 푃휔.
+(ii) For ℙ-almost every 휔, the rescaled path 푋푛2푡∕푛 converges weakly (under law
+푃휔) to a Brownian motion with covariance matrix ̄푎 = 퐸ℚ[휔∕tr휔] > 0.
+QCLT for the balanced RWRE in static environments under weaker ellipticity
+assumptions can be found at [33, 10]. For dynamic balanced random environment,
+QCLT was established in [23] and finer results concerning the local limit theorem
+and heat kernel estimates was obtained at [22]. When the RWRE is allowed to
+make long jumps, non-CLT stable limits of the balanced random walk is considered
+in [20, 21]. We refer to the lecture notes [13, 48, 12, 24, 38] for QCLT results in
+different models of RWRE.
+We are moreover interested in characterizing the invariant measure ℚ. Denote
+the Radon-Nikodym derivative of ℚ with respect to ℙ as
+휌(휔) = dℚ∕dℙ.
+(8)
+For any 푥 ∈ ℤ푑 and finite set 퐴 ⊂ ℤ푑, we define
+휌휔(푥) ∶= 휌(휃푥휔)
+and
+휌휔(퐴) =
+∑
+푥∈퐴
+휌휔(푥).
+As an important feature of the non-divergence form model, 휌휔 does not have
+deterministic (nonzero) upper and lower bounds. Moreover, the heat kernel 푝휔
+푡 (⋅, ⋅)
+is not expected to have deterministic Gaussian bounds.
+For 푟 ≥ 0, 푡 > 0, define a function
+픥(푟, 푡) =
+푟2
+푟 ∨ 푡 + 푟 log(푟
+푡 ∨ 1),
+푟 ≥ 0, 푡 > 0.
+(9)
+The following result was obtained by Guo, Tran [31].
+Theorem B. Assume (A1), (A2), and 푑 ≥ 2. Let 푠 = 푠(푑, 휅) = 2 + 1
+2휅 − 푑 ≥ 2.
+For any 휀 ∈ (0, 1), there exists a random variable ℋ(휔) = ℋ(휔, 푑, 휅, 휀) > 0 with
+피[exp(푐ℋ푑−휀)] < ∞ such that the following properties hold.
+5
+
+(a) For ℙ-almost all 휔,
+푐ℋ−푠 ≤ 휌(휔) ≤ 퐶ℋ푑−1.
+(b) Recall the function 픥 in (9). For any 푟 ≥ 1 and ℙ-almost all 휔,
+푐ℋ−푠 ≤ 푟푑휌휔(0)
+휌휔(퐵푟) ≤ 퐶ℋ푑−1.
+(c) For any 푥 ∈ ℤ푑, 푡 > 0, and ℙ-almost all 휔,
+푝휔
+푡 (푥, 0) ≤ 퐶ℋ푑−1(1 + 푡)−푑∕2푒−푐픥(|푥|,푡),
+푝휔
+푡 (푥, 0) ≥ 푐ℋ−푠(1 + 푡)−푑∕2푒−퐶|푥|2∕푡.
+Remark 3. In the PDE setting, positive and negative algebraic moment bounds and
+volume doubling property of 휌 were proved by Bauman [7]. 퐿푝 The positive mo-
+ment bound in (73) with 푞 =
+푑
+푑−1 was obtained by Lawler [40]. The 퐿푝 integrability
+of the heat kernel moment was proved by Fabes and Stroock [26]. Deterministic
+heat kernel bounds in terms of 휌 was shown by Escauriaza [25] in the PDE setting,
+and by Mustapha [42] for discrete time balanced random walks. In the more general
+dynamic ergodic balanced environment setting, the bounds
+푐휌휔(0)
+휌휔(퐵√
+푡)푒−퐶|푥|2∕푡 ≤ 푝휔
+푡 (푥, 0) ≤ 퐶휌휔(0)
+휌휔(퐵√
+푡)푒−푐픥(|푥|,푡)
+(10)
+were proved by Deuschel, Guo [22, Theorem 11]. Recently, Armstrong, Fehrman,
+Lin [2] obtain an algebraic rate of convergence for the heat kernels.
+A function 휓 ∶ Ω → ℝ is said to be local if it is measurable and depends
+only on the environment {휔(푥) ∶ 푥 ∈ 푆} in a finite set 푆 ⊂ ℤ푑. We now state a
+quantitative homogenization result in Guo, Peterson, Tran [29, Theorem 1.5], which
+can be considered as a discrete version of Armstrong, Smart [4, Theorem 1.2].
+Proposition C. Assume (A1), (A2), and that the 휓 is a local function. Recall the
+measure ℚ in Theorem A. Suppose 푔 ∈ 퐶훼(휕픹1), 푓 ∈ 퐶훼(픹1) for some 훼 ∈ (0, 1],
+and 휓 is a measurable function of 휔(0) with ‖휓∕tr휔‖∞ < ∞. Let ̄푢 be the solution
+of the Dirichlet problem
+{ 1
+2tr( ̄푎퐷2 ̄푢) = 푓 ̄휓
+in 픹1,
+̄푢 = 푔
+on 휕픹1,
+with ̄푎 = 퐸ℚ[휔∕tr휔] > 0 being a positive-definite matrix and ̄휓 = 퐸ℚ[휓∕tr휔].
+For any 휀 ∈ (0, 1), there exists a random variable ℋ(휔) = ℋ(휔, 푑, 휅, 휀) > 0
+with 피[exp(푐ℋ푑−휀)] < ∞ and a constant 훽 = 훽(푑, 휅, 휀) ∈ (0, 1) such that for any
+푦 ∈ 퐵3푅, the solution 푢 of
+{ 1
+2tr(휔∇2푢(푥)) =
+1
+푅2 푓(푥−푦
+푅 )휓(휃푥−푦휔)
+푥 ∈ 퐵푅(푦),
+푢(푥) = 푔( 푥−푦
+|푥−푦|)
+푥 ∈ 휕퐵푅(푦)
+(11)
+6
+
+satisfies, with 퐴1 = ‖푓‖퐶0,훼(픹1)‖
+휓
+tr(휔)‖∞ + [푔]퐶0,훼(휕픹1),
+max
+푥∈퐵푅(푦)
+|||푢(푥) − ̄푢(푥−푦
+푅 )|||≲ 퐴1(1 + (ℋ
+푅 )1−휀∕푑)푅−훼훽.
+(12)
+When the balanced environment is allowed to be non-elliptic and genuinely 푑-
+dimensional, (weak) quantitative results and Harnack inequalities for non-divergence
+form difference operators are obtained by Berger, Cohen, Deuschel, Guo [9], and
+Berger, Criens [11] for 휔-harmonic and 휔-caloric functions, respectively.
+Let us also give a brief overview of the quantitative homogenization of non-
+divergence form operators in the continuous PDE setting. Yurinski derived a second
+moment estimate of the homogenization error in [47] for linear elliptic case. Caf-
+farelli, Souganidis [18] proved a logarithmic convergence rate for the fully nonlinear
+case. Afterwards, Armstrong, Smart [4], and Lin, Smart [41] achieved an algebraic
+convergence rate for fully nonlinear elliptic equations, and fully nonlinear parabolic
+equations, respectively. Armstrong, Lin [3] obtained quantitative estimates for the
+approximate corrector problems.
+For 푑 ≥ 2 and any finite set 퐴 ⊂ ℤ푑, denote the exit time from 퐴 by
+휏(퐴) = 휏(퐴; 푋) = inf{푛 ≥ 0 ∶ 푋푛 ∉ 퐴}.
+(13)
+Definition 4. For 푅 ≥ 1, 휔 ∈ Ω, 푥 ∈ ℤ푑, 푆 ⊂ ℤ푑, the Green function 퐺푅(⋅, ⋅) in
+the ball 퐵푅 for the balanced random walk is defined by
+퐺푅(푥, 푆) = 퐺휔
+푅(푥, 푆) ∶= 퐸푥
+휔
+[
+∫
+휏(퐵푅)
+0
+1푌푡∈푆d푡],
+푥 ∈ ̄퐵푅.
+We also write 퐺푅(푥, 푦) ∶= 퐺휔
+푅(푥, {푦}) and 퐺푅(푥) ∶= 퐺푅(푥, 0). When 푑 ≥ 3, for
+any finite set 푆 ⊂ ℤ푑, the Green function on the whole space can be defined as
+퐺휔(푥, 푆) = ∫
+∞
+0
+푝휔
+푡 (푥, 푆)d푡 < ∞.
+When 푑 = 2, for any 푥, 푦 ⊂ ℤ푑, the potential kernel is defined as
+퐴(푥, 푦) = 퐴휔(푥, 푦) = ∫
+∞
+0
+[푝휔
+푡 (푦, 푦) − 푝휔
+푡 (푥, 푦)]d푡,
+푥 ∈ ℤ2.
+(14)
+The bounds for the Green functions and the potential kernel were proved in Guo,
+Tran [31], which was based on the idea of Armstrong, Lin [3, Proposition 4.1].
+Theorem D. Assume (A1), (A2). For 휀 > 0, let 푠 > 0, ℋ = ℋ(휔, 푑, 휅, 휀) > 0 be
+as in Theorem B. For 푟 > 0, let
+푈(푟) ∶=
+{ − log 푟
+푑 = 2,
+푟2−푑
+푑 ≥ 3.
+(15)
+7
+
+Then ℙ-almost surely, for all 푥 ∈ 퐵푅,
+ℋ−푠[푈(|푥| + 1) − 푈(푅 + 2)] ≲ 퐺휔
+푅(푥, 0) ≲ ℋ푑−1[푈(|푥| + 1) − 푈(푅 + 2)].
+As consequences, ℙ-almost surely, for all 푥 ∈ ℤ푑,
+ℋ−푠 log(|푥| + 1) ≲ 퐴휔(푥, 0) ≲ ℋ log(|푥| + 1), when 푑 = 2;
+ℋ−푠(1 + |푥|)2−푑 ≲ 퐺휔(푥, 0) ≲ ℋ푑−1(1 + |푥|)2−푑, when 푑 ≥ 3.
+Recall the continuous time RWRE (푌푡)푡≥0 in Definition 2. Define the semi-
+group 푃푡, 푡 ≥ 0, on ℝΩ by
+푃푡휁(휔) = 퐸0
+휔[휁(휃푌푡휔)] =
+∑
+푧
+푝휔
+푡 (0, 푧)휁(휃푧휔).
+(16)
+The following theorem from Guo, Tran [31] estimates the optimal speed of decor-
+relation of the environmental process ̄휔푡 from the original environment.
+Theorem E. Assume (A1), (A2), and 푑 ≥ 3. For any local measurable function
+휁 ∶ Ω → ℝ with ‖휁‖∞ ≤ 1 and 푡 ≥ 0, we have
+Varℚ(푃푡휁) ≤ 퐶(1 + 푡)−푑∕2;
+(17)
+‖푃푡휁‖1 + ‖푃푡휁 − 피[푃푡휁]‖푝 ≤ 퐶푝(1 + 푡)−푑∕4
+for all 푝 ∈ (0, 2).
+(18)
+1.4
+Main results
+The field {휌휔(푥) ∶ 푥 ∈ ℤ푑} of the invariant measure, which governs the long term
+behavior of the diffusion and which determines the effective PDE, plays a central
+role in the theory of homogenization of non-divergence form equations.
+We first obtain a rate of convergence of the average 휌휔(퐵푅)∕|퐵푅| of the invari-
+ant measure to 1 as 푅 → ∞.
+Theorem 5. Assume (A1), (A2). For any 푑 ≥ 2, 푝 ∈ (0, 2
+3), 푡 > 0 and 푅 ≥ 2,
+푃
+(|||
+휌휔(퐵푅)
+|퐵푅| − 1|||≥ 푡푅−푑∕2 log 푅
+)
+≤ 퐶푝 exp(−푐푡푝).
+Note that the rate 푅−푑∕2 log 푅 is very close to the size 푅−푑∕2 of the diffusive
+scaling. In other words, to some extent the field (휌휔(푥))푥∈ℤ푑 behaves quite simi-
+larly to i.i.d. random variables. Hence, we expect the rate 푅−푑∕2 log 푅 obtained
+here to be either optimal or nearly optimal. For non-divergence form PDEs, the
+volume-doubling property for the measure 휌휔(⋅) was proved by Bauman [7]. An
+algebraic convergence rate 푅−훾 for some 훾 ∈ (0, 1) was proved recently by Arm-
+strong, Fehrman, Lin [2, Theorem 1.4].
+In the course of obtaining our homogenization results in this paper, sensitivity
+estimates together with an Efron-Stein type inequality are used to control fluctu-
+ations of a random field around its mean. This method was used in the stochastic
+8
+
+homogenization of divergence-form operators, e.g., [43, 28]. To facilitate this strat-
+egy, obtaining sensitivity estimates (with respect to the change of the environment)
+is crucial, and 퐶1,1 estimates for the random equation is necessary, cf. e.g., [28, 3].
+To obtain 퐶1,1 regularity for the heterogeneous equation, we follow the idea of
+Armstrong, Lin [3] who generalized the compactness argument of Avellaneda, Lin
+[5] to the random non-divergence form setting.
+The key observation in the proof of Theorem 5 is explained as follows. Al-
+though the invariant measure 휌휔(푥) does not have an explicit expression, it can be
+interpreted as the long term frequency of visits to location 푥. Hence, modifying
+the local value of the environment is related to the Green function of the RWRE.
+Guided by this intuition, we will obtain a formula for the sensitivity estimate of the
+invariant measure in terms of the Green function.
+As indicated in Theorem 5, the field {휌휔(푥) ∶ 푥 ∈ ℤ푑} is expected to have
+weak enough correlation so that the behavior of its mean fluctuation over 퐵푅 re-
+sembles (up to a logarithmic factor) that of i.i.d. random variables. The following
+proposition reveals the mixing property of the field of the invariant measure.
+Proposition 6. Assume (A1), (A2). For any 푥, 푦 ∈ ℤ푑 with 푥 ≠ 푦,
+||Covℙ(휌(푥), 휌(푦))|| ≲
+{ |푥 − 푦|−1(1 + log |푥 − 푦|),
+푑 = 2
+|푥 − 푦|−푑∕2,
+푑 ≥ 3.
+This is perhaps the first characterization of the correlation structure of the field
+of the invariant measure (with algebraic mixing rates) in a balanced environment.
+Next, we derive rates of convergence for the homogenization of the Dirichlet
+problem.
+Theorem 7. Assume (A1), (A2). Consider the Dirichlet problem
+{
+퐿휔푢 =
+1
+푅2 푓( 푥
+푅),
+푥 ∈ 퐵푅,
+푢(푥) = 푔( 푥
+푅),
+푥 ∈ 휕퐵푅.
+(19)
+Assume that 푓, 푔 are both in 퐶4(ℝ푑). For any 0 < 푠 <
+푑
+3푑+2, there exists 퐶 =
+퐶(푑, 휅, 푠) such that, for 푅 ≥ 2, there exists a random variable 풵 = 풵(푅, 푠, 휔) > 0
+with 피[exp(풵푠)] < 퐶 and
+max
+푥∈퐵푅
+|푢(푥) − ̄푢( 푥
+푅)| ≲ ‖̄푢‖퐶4( ̄픹1)휏(푅)풵,
+where
+휏(푅) =
+⎧
+⎪
+⎨
+⎪⎩
+푅−2∕3,
+푑 = 2
+푅−6∕7,
+푑 = 3
+푅−1(log 푅)1∕4,
+푑 = 4
+푅−1,
+푑 ≥ 5,
+(20)
+and ̄푢 is the solution of
+{ 1
+2tr( ̄푎퐷2 ̄푢) = 푓
+in 픹1,
+̄푢 = 푔
+on 휕픹1.
+9
+
+Thus, for 푑 ≥ 5, we obtain the optimal rate of convergence for the homogeniza-
+tion of the Dirichlet problem, which is generically of scale 푅−1. This is consistent
+with the generically optimal rate 푅−1 for the periodic setting (see the classical books
+[8, 36] for the derivation, and [32, 45, 30] for discussions on the optimality of the
+rates.) It is not clear to us what the optimal rates are when 2 ≤ 푑 ≤ 4, which deserve
+further analysis.
+To prove Theorem 7, we apply the classical method of two-scale expansions
+and the quantitative homogenization of the approximation corrector (Lemma 21).
+The continuous version of Lemma 21 was proved earlier in the PDE setting by
+Armstrong, Lin [3].
+Note that the effective matrix ̄푎 = 퐸ℚ[휔∕tr휔] does not have an explicit expres-
+sion. Even though by Birkhoff’s ergodic theorem, ℚ can be approximated qualita-
+tively by
+lim
+푛→∞
+1
+푛
+푛−1
+∑
+푖=0
+퐸휔[휓(휃푋푖휔)] = 퐸ℚ[휓]
+ℙ-a.s.
+for any 퐿1 function 휓 on environments, in order to better understand the effective
+matrix ̄푎 it is important to quantify the speed of this convergence. To this end, we
+set, for 푇 ≥ 1,
+휈(푇 ) =
+⎧
+⎪
+⎨
+⎪⎩
+푇 −1∕2
+푑 = 2
+푇 −3∕4
+푑 = 3
+푇 −1(log 푇 )1∕2
+푑 = 4
+푇 −1
+푑 ≥ 5.
+(21)
+We will quantify the ergodicity of the environmental process for both the continuous-
+and discrete-time random walks in a balanced random environment.
+Theorem 8. Assume (A1), (A2), (A3). Let 휈 be as in (21). For any 0 < 푝 <
+2푑
+3푑+2,
+there exists 퐶 = 퐶(푑, 휅, 푝) such that for 푇 , 푛 ≥ 2 and any 푡 ≥ 0,
+ℙ
+(
+|||
+1
+푇 퐸휔
+[
+∫
+푇
+0
+휓(휃푌푠휔)d푠
+]
+− ̄휓|||≥ 푡휈(푇 )‖휓‖∞
+)
+≤ 퐶 exp(−푡푝
+퐶 ),
+ℙ
+(
+|||
+1
+푛퐸휔
+[ 푛−1
+∑
+푖=0
+휓(휃푋푖휔)
+]
+− ̄휓|||≥ 푡휈(푛)‖휓‖∞
+)
+≤ 퐶 exp(−푡푝
+퐶 ).
+Remark 9. Recall that Theorem E (from [31]) states that, when 푑 ≥ 3, the typical
+size of 푃푡휓 − ̄휓 is of scale 푡−푑∕4. Observe that the typical size 휈(푇 ) of the ergodic
+average in Theorem 8 satisfies (for 푇 ≥ 2)
+휈(푇 ) ≲ 1
+푇 ∫
+푇
+1
+푡−푑∕4d푡.
+(The sign ≲ can be replaced by ≍ except for 푑 = 4 when 휈(푇 ) is a
+√
+log 푇 factor
+smaller than the right side.) Hence, Theorem 8 can be regarded as the integral ver-
+sion of Theorem E which holds for all 푑 ≥ 2 and which has much better exponential
+integrability.
+10
+
+We remark that an (unknown) algebraic rate for the convergence of the ergodic
+average was obtained in [29, Theorem 1.2] in the discrete setting and recently in [2,
+Theorem 1.6] in the PDE setting.
+As a consequence of Theorem 8, we obtain explicit convergence rates for the
+QCLT of the balanced random walk.
+Corollary 10. Assume (A1), (A2). For any 0 < 푞 <
+2푑
+5(3푑+2), 푛 ≥ 2, there exists a
+random variable 풴 = 풴(휔, 푞; 푛, 휅, 푑) with 피[exp(풴푞)] ≤ 퐶 such that, ℙ-almost
+surely, for any unit vector 퓁 ∈ ℝ푑,
+sup
+푟∈ℝ
+|||푃휔
+(
+푋푛 ⋅ 퓁∕
+√
+푛 ≤ 푟
+√
+퓁푇 ̄푎퓁
+)
+− Φ(푟)|||≤ 퐶휈(푛)1∕5풴,
+where Φ(푟) = (2휋)−1∕2 ∫ 푟
+−∞ 푒−푥2∕2d푥 for 푟 ∈ ℝ.
+An algebraic rate for the QCLT was proved in [29, Theorem 1.3]. We remark
+that for the model of random walk in random conductances, algebraic rates similar
+to ours was proved in [1] for dimensions 푑 ≥ 3.
+2
+Large scale 퐶0,1 and 퐶1,1 estimates
+In this section, we apply the ideas of Avellaneda, Lin [5, 6] in the periodic setting
+to the discrete random setting. The key idea is quite intuitive and natural: large-
+scale solutions of 퐿휔푢(푥) = 휓(휃푥휔) + 푓(푥) are well-approximated by those of the
+homogenized equation with an algebraic rate thanks to Proposition C. As the latter
+are harmonic, they possess rather nice estimates (see Proposition 16 below on the
+scaling property.) Therefore, by iterations, scalings and the triangle inequalities,
+the better regularity of the homogenized equation is inherited by the heterogeneous
+equation. It is crucial to note two points here. Firstly, in each iteration step, the
+scaling is done by using the nice estimates in Proposition 16 of the homogenized
+limit, and the triangle inequality and Proposition C are used to pass this estimate to
+the solution 푢 of the heterogeneous equation. Secondly, in the random setting, one
+can only go down to radii greater than the homogenization radius in the iterations,
+which therefore gives us only large scale estimates. The generalization of this idea
+to the random non-divergence form PDE setting was first done by Armstrong, Lin
+[3] who made the observation that an algebraic rate is sufficient for such an iteration.
+The main result in this section, Theorem 14, can be considered as a discrete
+version of Armstrong, Lin [3, Theorem 3.1,Corollary 3.4] in terms of the large scale
+퐶0,1 and 퐶1,1 regularity. We remark that for 휔-harmonic functions in a genuinely
+푑-dimensional balanced environment, a 퐶0,1−휀 regularity was achieved by Berger,
+Cohen, Deuschel, Guo [9, Corollary 1.4] using coupling arguments.
+As can be seen in the following Subsection 2.1, this sort of compactness argu-
+ment, although is applied to the random setting here, is deterministic in its core.
+11
+
+2.1
+Some regularity properties of deterministic functions
+This subsection contains the key tools for the compactness arguments used in our
+paper. It is completely deterministic and can be read independently of other parts of
+the paper. The lemmas presented here concern large scale 퐶푘,1, 푘 ≥ 0, properties
+of deterministic functions.
+For any function 푓 on a set 퐴 and 훼 ∈ (0, 1], define
+osc
+퐴 푓 ∶= sup
+푥,푦∈퐴
+|푓(푥) − 푓(푦)|,
+[푓]훼;퐴 =
+sup
+푥,푦∈퐴,푥≠푦
+|푓(푥) − 푓(푦)|
+|푥 − 푦|훼
+,
+and, if 퐴 is a finite set, for 푝 ∈ (0, ∞), we define
+‖푓‖푝;퐴 ∶=
+(
+1
+#퐴
+∑
+푥∈퐴
+|푓|푝
+)1∕푝
+,
+‖푓‖∞;퐴 = max
+푥∈퐴 |푓(푥)|.
+For any 푗 ≥ 0, let H푗 denote the set of 푗-th order polynomials, with H0 = ℝ. In
+fact, in our paper we will be only use the cases 푗 = 0, 1, 2.
+Define, for function 푓 ∶ ℝ푑 → ℝ and a bounded set 퐴 ⊂ ℝ푑, 푗 ≥ 1,
+푗
+퐴(푓) =
+inf
+푝∈H푗−1
+sup
+퐴
+|푓 − 푝| = 1
+2
+inf
+푝∈H푗−1
+osc
+퐴 (푓 − 푝).
+(22)
+푗
+퐴 satisfies the triangle inequality. Namely, 푗
+퐴(푓 ± 푔) ≤ 푗
+퐴(푓) + 푗
+퐴(푔). When
+퐴 = 퐵푅 is the discrete ball, 푅 > 0, we simply write
+푗
+푅 ∶= 푗
+퐵푅.
+Note that for 푗 ≥ 1, the above term normalized
+픻푗
+푅(푓) ∶=
+푗
+푅(푓)
+푅푗
+(23)
+is a large scale analogue of the 푗-th order derivative.
+For any 푟 > 0, 휃 ∈ (0, 1
+3), define a sequence of exponentially increasing radii
+(푟푘)푘≥0 by
+푟푘 = 푟푘(푟, 휃) ∶= 휃−푘푟,
+푘 ≥ 0.
+The following elementary lemma confirms the intuition that “the integral of the
+(푗 + 1)-th derivative is the 푗-th derivative".
+Lemma 11. For any function 푓 ∶ ℤ푑 → ℝ and 푟 > 0, 휃 ∈ (0, 1
+3), 푛 ∈ ℕ, 푗 ≥ 1,
+픻푗
+푟0(푓) ≤ 픻푗
+푟푛(푓) + 3휃−푗
+푛
+∑
+푘=0
+푟푘픻푗+1
+푟푘 (푓).
+12
+
+Proof. For any 푗-th order homogeneous polynomials 푝, 푞 ∈ H푗, 푅 > 푟 > 0, by the
+triangle inequality,
+푗
+푟(푝) ≤ 푗
+푟(푞) + 푗
+푟(푝 − 푞)
+≤ ( 푟
+푅)푗푗
+푅(푞) + 푗
+푟(푓 − 푝) + 푗
+푟(푓 − 푞)
+where in the second inequality we used the fact that 푗
+푟(푞) = ( 푟
+푅)푗푗
+푅(푞) for all 푗-th
+order homogeneous polynomial 푞. Hence, by the inequality above,
+푗
+푟(푓) ≤ 푗
+푟(푓 − 푝) + 푗
+푟(푝)
+≤ 2푗
+푟(푓 − 푝) + 푗
+푟(푓 − 푞) + ( 푟
+푅)푗푗
+푅(푞)
+≤ 2푗
+푟(푓 − 푝) + 푗
+푟(푓 − 푞) + ( 푟
+푅)푗[푗
+푅(푓) + 푗
+푅(푓 − 푞)]
+≤ 2[푗
+푟(푓 − 푝) + 푗
+푅(푓 − 푞)] + ( 푟
+푅)푗푗
+푅(푓).
+Taking infimum over all 푗-th order homogeneous polynomials 푝, 푞 ∈ H푗, we get
+푗
+푟(푓) ≤ 2[푗+1
+푟
+(푓) + 푗+1
+푅 (푓)] + ( 푟
+푅)푗푗
+푅(푓).
+Replacing 푟, 푅 by 푟푘, 푟푘+1 , and using notation (23), the above inequality yields
+픻푗
+푟푘(푓) − 픻푗
+푟푘+1(푓) ≤ 2[푟푘픻푗+1
+푟푘 (푓) + 휃−푗푟푘+1픻푗+1
+푟푘+1(푓)].
+Summing both sides over 푘 = 0, … , 푛 − 1, the lemma is proved.
+The following lemma will be crucially employed later in our derivation of large
+scale regularity estimates in Subsection 2.2.
+Lemma 12. Let 푗 ≥ 1, 푚 ∈ ℕ, 푟, 훼 > 0. Let 퐴푟 ≥ 0 be a constant depending on 푟.
+If for 푓 ∶ ℤ푑 → ℝ , 푘 = 0, … , 푚 − 1, and all 휃 ∈ (0, 1
+3),
+푗+1
+푟푘 (푓) ≲푗 휃푗+1푗+1
+푟푘+1(푓) + 푟−훼
+푘+1푗
+푟푘+1(푓) + 푟푗
+푘+1퐴푟푘+1,
+(24)
+then there exists 휃 = 휃(푗), 푁 = 푁(푗, 훼) such that, for 푁 ≤ 푟 ≤ 푅 ≤ 푟푚,
+푗
+푟(푓) ≤ 13휃−2푗 (
+푟
+푅
+)푗
+푗
+푅(푓) +
+∑
+푘≥1∶푟푘≤푅
+퐴푟푘.
+Proof. For the simplicity of notations, we suppress the dependency on 푓. Let 푛 =
+푛(푅, 휃) ≤ 푚 be such that 푟푛 ≤ 푅 < 푟푛+1. Display (24) is equivalent to
+푟푘픻푗+1
+푟푘
+≲푗 휃푟푘+1픻푗+1
+푟푘+1 + 휃−푗푟−훼
+푘+1픻푗
+푟푘+1 + 휃−푗퐴푟푘+1.
+Summing this inequality over 푘 = 0, … , 푛 − 1, we have
+푛−1
+∑
+푘=0
+푟푘픻푗+1
+푟푘
+≲푗 휃
+푛
+∑
+푘=1
+푟푘픻푗+1
+푟푘
++ 휃−푗
+푛
+∑
+푘=1
+푟−훼
+푘 픻푗
+푟푘 + 휃−푗
+푛
+∑
+푘=1
+퐴푟푘.
+(25)
+13
+
+Moreover, by Lemma 11, 픻푗
+푟푘 ≤ 픻푗
+푟푛 + 3휃−푗 ∑푛
+퓁=푘 푟퓁픻푗+1
+푟퓁 . Hence
+푛
+∑
+푘=1
+푟−훼
+푘 픻푗
+푟푘 ≤
+푛
+∑
+푘=1
+푟−훼
+푘
+(
+픻푗
+푟푛 + 3휃−푗
+푛
+∑
+퓁=푘
+푟퓁픻푗+1
+푟퓁
+)
+≤ 퐶훼푟−훼픻푗
+푟푛 + 퐶훼휃−푗푟−훼
+푛
+∑
+퓁=1
+푟퓁픻푗+1
+푟퓁 ,
+(26)
+where 퐶훼 = 1 − 3−훼. Choosing 휃 = 휃(푗) ∈ (0, 1
+3) sufficiently small, when 푟 ≥ 푁
+for some 푁 = 푁(푗, 훼), we get from (25) and (26) that
+푛−1
+∑
+푘=0
+푟푘픻푗+1
+푟푘
+≤ 1
+2
+푛
+∑
+푘=1
+푟푘픻푗+1
+푟푘
++ 픻푗
+푟푛 + 퐶푗휃−푗
+푛
+∑
+푘=1
+퐴푟푘
+which implies (Note that 푟푛픻푗+1
+푟푛
+≤ 픻푗
+푟푛)
+푛
+∑
+푘=0
+푟푘픻푗+1
+푟푘
+≤ 4픻푗
+푟푛 + 퐶푗휃−푗
+푛
+∑
+푘=1
+퐴푟푘.
+This inequality, together with Lemma 11, yields for 푟 ≥ 푁, 휃 = 휃(푗) ∈ (0, 1
+3),
+픻푗
+푟0 ≤ 13휃−푗픻푗
+푟푛 + 퐶푗휃−2푗
+푛
+∑
+푘=1
+퐴푟푘 ≤ 13휃−2푗픻푗
+푅 +
+푛
+∑
+푘=1
+퐴푟푘.
+The lemma is proved.
+Remark 13. In this subsection we consider 푓 as a function on ℤ푑 and defined H푗
+to be the set of 푗-th order polynomials just for our convenience. One may let 푓 be
+a function on ℝ푑 and redefine H푗’s to be other sub-spaces of the polynomials (e.g.,
+the set of harmonic polynomials) and Lemmas 11, 12 still hold.
+2.2
+Large scale regularity
+The goal of this section is to apply Lemma 12 to obtain 퐶0,1 and 퐶1,1 regularities
+for the heterogeneous equations in our random setting.
+Define operators ∇ = (∇푖)1≤푖≤푑 and ∇∗ = (∇∗
+푖 )1≤푖≤푑 by
+∇푖푢(푥) = 푢(푥 + 푒푖) − 푢(푥),
+∇∗
+푖 푢(푥) = 푢(푥 − 푒푖) − 푢(푥).
+Note that ∇푖 and ∇∗
+푖 are adjoint linear operators, and ∇2
+푖 = −∇푖∇∗
+푖 .
+Theorem 14. Assume (A1), (A2), and that 휓 is a local function. Let 푅 ≥ 1. There
+exists 훼 = 훼(푑, 휅) ∈ (0, 1
+3) such that, for any any 푢 with 퐿휔푢(푥) = 휓(휃푥휔) + 푓(푥)
+on 퐵푅, 푗 ∈ {1, 2}, ℋ ≤ 푟 < 푅,
+1
+푟푗
+inf
+푝∈H푗−1
+osc
+퐵푟
+(푢 − 푝) ≲ 1
+푅푗
+inf
+푝∈H푗−1
+osc
+퐵푅
+(푢 − 푝) + 퐴푗,
+(27)
+14
+
+where the terms 퐴푗 = 퐴푗(푅, 푟) have the following bounds (for any 휎 ∈ (0, 1])
+퐴1 ≤ 푅1−훼‖휓‖∞ + 푅‖푓‖∞ and 퐴1 ≤ 푅1−훼‖휓 + 푓(0)‖∞ + 푅1+휎[푓]휎;퐵푅,
+퐴2 ≤ 푟−훼‖휓‖∞ + log(푅
+푟 )‖푓‖∞ and 퐴2 ≤ 푟−훼‖휓 + 푓(0)‖∞ + 푅휎[푓]휎;퐵푅.
+In particular, recalling the operator ∇2
+푖 in (1), for any 푅 > 1, 푗 = 1, 2,
+|∇푗푢(0)| ≲ (ℋ
+푅 )푗 (
+‖푢‖1;퐵푅 + 푅2‖휓 + 푓(0)‖∞ + 푅2+휎[푓]휎;퐵푅.
+)
+(28)
+As a consequence of (28), any 휔-harmonic function on ℤ푑 with sublinear growth
+is a constant. That is, if 퐿휔푢 = 0 on ℤ푑 and max퐵푅 |푢| = 표(푅) for all 푅 > 0, then
+푢 is constant. To prove Theorem 14, it suffices to prove the following lemma.
+Lemma 15. There exists 훾 = 훾(푑, 휅) such that, for 푅 ≥ ℋ, 휃 ∈ (0, 1
+3), 1 ≤ 푗 ≤ 3
+and any 푢 with 퐿휔푢(푥) = 휓(휃푥휔) + 푓(푥) for 푥 ∈ 퐵푅, we have
+푗
+휃푅(푢) ≲ 푅−훾훽2
+푅(푢) + 휃푗푗
+푅(푢) + 푅2−훾훽‖휓‖∞ + 푅2‖푓‖푑;퐵푅.
+The proof of Lemma 15 uses the following fact of deterministic harmonic func-
+tions. For completeness, we include its proof in Section A.1 of the Appendix.
+Proposition 16. Recall the notation in (22). Let 푐0 be a constant. Let 푣 be a function
+satisfying tr ̄푎퐷2푣 = 푐0 in ̄픹푅. Then, for 휃 ∈ (0, 1
+3), 푗 ∈ {1, 2, 3} and 푅 ≥ 1,
+푗
+픹휃푅(푣) ≤ 퐶휃푗푗
+픹푅∕2(푣).
+(29)
+We also have
+푗
+픹휃푅(푣) ≲ 휃푗
+푅 ( sup
+휕픹2푅∕3
+|푣| + 푅2|푐0|) + 휃푗푗
+2푅∕3(푣).
+(30)
+Remark 17. Property (29) for deterministic harmonic functions (푐0 = 0) was used
+in [3, Lemma 3.3] to obtain regularities of the heterogeneous solution in the PDE
+setting. Comparing to [3, Corollary 3.4], here by allowing 푐0 ≠ 0 we will gain a
+tiny improvement for the coefficient of ‖휓 +푓(0)‖∞ in the 퐶0,1 estimate by an 푅−훼
+factor (cf. Theorem 14). Note that in the discrete setting, we will need (30) as well
+because of discretization. It would be also clear later in Section 3 that the log 푅
+factor in the bound of 퐴2 will help us achieve the log 푅 factor in Theorem 5.
+Proof of Lemma 15. By the Hölder estimate of Krylov-Safonov, there exists 훾 =
+훾(푑, 휅) > 0 such that, for 푟 ∈ (0, 푅),
+osc
+퐵푟
+푢 ≲
+(
+푟
+푅
+)훾
+(osc
+퐵푅
+푢 + 푅2‖휓 + 푓‖푑;퐵푅).
+(31)
+Note that this allows us to extend 푢 to be a function ̃푢 ∈ 퐶훾(ℝ푑) with [̃푢]훾;ℝ푑 =
+[푢]훾; ̄퐵2푅∕3. Indeed, define the function ̃푢 as
+̃푢(푥) =
+min
+푦∈ ̄퐵2푅∕3
+{
+푢(푦) + |푥 − 푦|휎[푢]휎; ̄퐵2푅∕3
+}
+.
+15
+
+It is straightforward to check that ̃푢 = 푢 in ̄퐵2푅∕3 and [̃푢]훾;ℝ푑 ≤ [푢]훾; ̄퐵2푅∕3. By (31),
+[̃푢]훾;ℝ푑 = [푢]훾; ̄퐵2푅∕3 ≲ 푅−훾(max
+퐵푅
+|푢| + 푅2‖휓 + 푓‖푑;퐵푅).
+(32)
+Let ̄푣 ∶ ̄픹2∕3 → ℝ be the solution of
+{
+1
+2tr( ̄푎퐷2 ̄푣) = 푅2 ̄휓
+in 픹2∕3
+̄푣(푥) = ̃푢(푅푥)
+for 푥 ∈ 휕픹2∕3.
+First, write 퐴 ∶= 푅2−훾훽‖휓‖∞ + 푅2‖푓‖푑;퐵푅. We will show that, for 푅 ≥ ℋ,
+max
+푥∈퐵2푅∕3
+|푢(푥) − ̄푣( 푥
+푅)| ≲ 푅−훾훽 max
+퐵푅
+|푢| + 퐴.
+(33)
+To this end, let 푢1 ∶ ̄퐵2푅∕3 → ℝ be the solution of
+{
+퐿휔푢1 = 휓(휃푥휔)
+in 퐵2푅∕3
+푢1(푥) = ̃푢(2푅푥
+3|푥| )
+푥 ∈ 휕퐵2푅∕3.
+By Proposition C and (32), when 푅 ≥ ℋ, noting that [̃푢(푅⋅)]훾;ℝ푑 ≤ 푅훾[̃푢(⋅)]훾;ℝ푑,
+max
+푥∈퐵2푅∕3
+|푢1(푥) − ̄푣( 푥
+푅)| ≲ 푅−훾훽([̃푢(푅⋅)]훾;ℝ푑 + 푅2‖휓‖∞)
+≲ 푅−훾훽(max
+퐵푅
+|푢| + 푅2‖휓‖∞ + 푅2‖푓‖푑;퐵푅).
+Moreover, by the ABP maximum principle,
+max
+퐵2푅∕3
+|푢 − 푢1| ≤
+max
+푥∈휕퐵2푅∕3
+|푢(푥) − ̃푢(2푅푥
+3|푥| )| + 퐶푅2‖푓‖푑;퐵푅
+≲ [̃푢]훾;ℝ푑 + 푅2‖푓‖푑;퐵푅
+(32)
+≲ 푅−훾(max
+퐵푅
+|푢| + 푅2‖휓‖∞) + 푅2‖푓‖푑;퐵푅.
+Combining the two inequalities above, display (33) is proved.
+By the triangle inequality and Proposition 16, for 1 ≤ 푗 ≤ 3,
+푗
+휃푅(푢) ≤ max
+퐵푅∕2
+|푢 − ̄푣( ⋅
+푅)| + 푗
+휃푅( ̄푣( ⋅
+푅))
+≤ max
+퐵푅∕2
+|푢 − ̄푣( ⋅
+푅)| + 퐶휃푗
+푅 ( sup
+휕픹2∕3
+| ̄푣| + 푅2| ̄휓|) + 퐶휃푗푗
+2푅∕3( ̄푣( ⋅
+푅))
+≲ max
+퐵2푅∕3
+|푢 − ̄푣( ⋅
+푅)| + 휃푗
+푅 ( sup
+휕픹2∕3
+| ̄푣| + 푅2| ̄휓|) + 휃푗푗
+2푅∕3(푢).
+(34)
+Since sup휕픹2∕3 | ̄푣| = sup휕픹2푅∕3 |̃푢| ≤ max퐵2푅∕3 |푢| + [̃푢]훾; ̄퐵2푅∕3 ≤ max퐵푅 |푢| + 퐴, by
+(33) and (34), we have, for 1 ≤ 푗 ≤ 3,
+푗
+휃푅(푢) ≲ 푅−훾훽 max
+퐵푅
+|푢| + 퐴 + 휃푗푗
+푅(푢).
+16
+
+Finally, note that since every 푝 ∈ H1 is 휔-harmonic, (푢 − 푝) still solves 퐿휔(푢 −
+푝) = 휓(휃푥휔) + 푓(푥) for 푥 ∈ 퐵푅. Therefore, substituting 푢 by (푢 − 푝) in the above
+inequality and optimizing over 푝 ∈ H1, the lemma follows.
+Proof of Theorem 14: By Lemma 15 and Lemma 12, there exists 휃 = 휃(푑, 휅) ∈
+(0, 1
+3) such that (27) holds with the terms 퐴푗, 푗 ∈ {1, 2} satisfying
+퐴푗 =
+∑
+푘≥1∶푟푘≤푅
+푟2−훼−푗
+푘
+‖휓‖∞ + 푟2−푗
+푘
+‖푓‖푑;퐵푟푘.
+Note that ‖푓 − 푓(0)‖푑;퐵푟 ≲ 푟휎[푓]휎;퐵푟 for all 휎 ∈ (0, 1]. The bounds of 퐴1, 퐴2 in
+the theorem follow immediately.
+To prove (28), note that |∇푢(0)| ≤ osc ̄퐵1 푢 and that for any 퓁 ∈ H1, |∇2푢(0)| =
+|∇2(푢 − 퓁)| ≲ osc ̄퐵1(푢 − 퓁). Hence, by (27), we get
+|∇푗푢(0)| ≲ (ℋ
+푅 )푗
+(
+osc
+퐵푅∕2
+푢 + 푅2‖휓 + 푓(0)‖∞ + 푅2+휎[푓]휎;퐵푅∕2
+)
+.
+By the Harnack inequality and the ABP inequality, we have
+osc
+퐵푅∕2
+푢 ≲ ‖푢 − 푢퐵푅‖1;퐵푅 + 푅2‖휓 + 푓‖푑;퐵푅.
+(35)
+Display (28) follows by using again ‖푓−푓(0)‖푑;퐵푅 ≲ 푅휎[푓]휎;퐵푅 for 휎 ∈ (0, 1].
+3
+Mixing properties of the invariant measure
+The goal of this section is to investigate the mixing properties of the field {휌휔(푥) ∶
+푥 ∈ ℤ푑} of the invariant measure. We will obtain a rate of convergence (Theorem
+5) of the average of the invariant measure over balls 퐵푅. We will also quantify the
+correlation of the field (Proposition 6).
+The Efron-Stein inequality (38) of Boucheron, Bousquet, and Massart [14] will
+be used in our derivation of quantitative estimates.
+Let 휔′(푥), 푥 ∈ ℤ푑, be i.i.d. copies of 휔(푥), 푥 ∈ ℤ푑. For any 푦 ∈ ℤ푑, let 휔′
+푦 ∈ Ω
+be the environment such that
+휔′
+푦(푥) =
+{
+휔(푥)
+if 푥 ≠ 푦,
+휔′(푦)
+if 푥 = 푦.
+That is, 휔′
+푦 is a modification of 휔 only at location 푦. For any measurable function
+푍 of the environment 휔, we write, for 푦 ∈ ℤ푑,
+푍′
+푦 = 푍(휔′
+푦),
+휕′
+푦푍(휔) = 푍′
+푦 − 푍,
+(36)
+and set
+푉 (푍) =
+∑
+푦∈ℤ푑
+(휕′
+푦푍)2.
+(37)
+17
+
+With abuse of notations, we enlarge the probability space and still use ℙ to denote
+the distribution of both 휔, 휔′.
+The 퐿푝 version of Efron-Stein inequality in [14, Theorem 3] states that, for
+푞 ≥ 2,
+피[|푍 − 피푍|푞] ≤ 퐶푞푞∕2피[푉 푞∕2].
+(38)
+3.1
+A sensitivity estimate of the invariant measure
+The main contribution of this subsection is a formula for the “vertical" derivative
+of the invariant measure 휌.
+Definition 18. For 푡 > 0, we let 푉 (푡, 휔) = ∑
+푥∈ℤ푑 푝휔
+푡 (푥, 0). Let ℚ푡 be the probability
+measure on Ω defined by
+ℚ푡(d휔) = 푉 (푡, 휔)ℙ(d휔).
+We remark that by Theorem B,
+푉 (푡, 휔) ≲ ℋ푑−1
+for ℙ-a.e. 휔
+(39)
+and so ℚ푡 is well-defined. Note that for any bounded measurable function 휁 on Ω,
+we have 퐸ℚ푡[휁] = 피[푃푡휁]. In other words, ℚ푡 is the distribution of the environment
+viewed from the particle at time 푡. It is natural to expect that ℚ푡 → ℚ as 푡 → ∞.
+For any function 푢 of the environment, we denote by 푢휔 the corresponding func-
+tion on ℤ푑 defined by 푢휔(푥) ∶= 푢(휃푥휔).
+Lemma 19. As 푡 → ∞, ℚ푡 converges weakly to ℚ.
+Proof. Since {ℚ푡} is a sequence of probability measures on the compact space Ω,
+it has a weak convergent subsequence {ℚ푡푘} which has a weak limit ℚ∞.
+To prove that ℚ∞ is an invariant measure for the Markov chain (휃푌푡휔), it suffices
+to show that for any bounded measurable function 푓 on Ω,
+퐸ℚ∞[퐿휔푓휔(0)] = 0.
+Indeed, by the translation invariance of the measure ℙ, for any 푒 with |푒| = 1,
+피[휔(0, 푒)푉 (푡, 휔)푓(휃푒휔)] = 피[휔(−푒, 0)푉 (푡, 휃−푒휔)푓(휔)]
+= 피[휌휔(0)휔∗(0, −푒) ̃푉 (푡, 휃−푒휔)푓(휔)],
+(40)
+where 휔∗(푥, 푦) ∶= 휌휔(푦)휔(푦, 푥)∕휌휔(푥) denotes the adjoint of 휔, cf. e.g., [22], and
+̃푉 (푡, 휔) ∶= 푉 (푡, 휔)∕휌(휔). Noting that ∑
+푦 휔∗(푥, 푦) = 1, we have
+퐸ℚ푡[퐿휔푓휔(0)] = 피[푉 (푡, 휔)
+∑
+푒
+휔(0, 푒)[푓(휃푒휔) − 푓(휔)]]
+(40)
+= 피[휌휔(0)
+∑
+푒
+휔∗(0, 푒)[ ̃푉 (푡, 휃푒휔) − ̃푉 (푡, 휔)]푓(휔)]
+= 퐸ℚ[푓(휔)퐿휔∗ ̃푉휔(푡, 0)],
+(41)
+18
+
+where ̃푉휔(푡, 푥) ∶= ̃푉 (푡, 휃푥휔), and 퐿휔∗ only acts on the spatial (ℤ푑) coordinate of
+the function ̃푉휔 ∶ ℝ × ℤ푑 → ℝ of space and time. Observe that ̃푉휔 solves the
+parabolic equation
+(휕푡 − 퐿휔∗) ̃푉휔 = 0
+in (0, ∞) × ℤ푑.
+By the Hölder estimate [22, Corollary 7] and the Harnack inequality [22, Theorem
+6] for the operator (휕푡 − 퐿휔∗), there exists 훾 = 훾(푑, 휅) > 0 such that, for 푡 > 1,
+max
+푒∶|푒|=1 | ̃푉휔(푡, 푒) − ̃푉휔(푡, 0)| ≲ 푡−훾
+sup
+(푠,푥)∈(0.5푡,푡)×퐵√
+푡
+̃푉휔(푠, 푥)
+≲ 푡−훾 ̃푉휔(2푡, 0)
+(42)
+(39)
+≲ 푡−훾휌−1ℋ푑−1.
+Thus, by (41), |Eℚ푡[퐿휔푓휔(0)]| ≲ 푡−훾피[ℋ푑−1]‖푓‖∞ ≲ 푡−훾‖푓‖∞ . In particular, for
+any bounded measurable function 푓 on Ω,
+퐸ℚ∞[퐿휔푓휔(0)] = lim
+푘→∞ 퐸ℚ푡푘[퐿휔푓휔(0)] = 0
+which implies that ℚ∞ is an invariant measure for the Markov chain (휃푌푡휔). More-
+over, for any bounded measurable function 푓 ∶ Ω → ℝ and 푝 > 0,
+퐸ℚ푡[푓] = 피[푉휔(푡, 0)푓(휔)] ≲ 피[ℋ푑−1푓] ≲푝 ‖푓‖퐿푝(ℙ),
+and so 퐸ℚ∞[푓] ≲푝 ‖푓‖퐿푝(ℙ) which implies ℚ ≪ ℙ. Therefore, by the same argu-
+ment as in [33, (4)], we have ℚ∞ = ℚ.
+Before stating the formula for 휕′
+푦휌 in the following proposition, we remark that
+although the global Green function 퐺휔(푥, 푦) is only defined for 푑 ≥ 3, the second
+order difference ∇2
+푖;1퐺(푥, 푦) can be defined for all dimensions, where ∇2
+푖;1 is ∇2
+푖
+applied to the first ℤ푑 coordinate. That is, for any fixed 푦, ∇2
+푖;1퐺(⋅, 푦) ∶= ∇2
+푖 퐺(⋅, 푦).
+Indeed, recalling 퐴(푥, 푦) in (14), we can set
+∇2
+푖;1퐺(푥, 푦) ∶= −∇2
+푖;1퐴(푥, 푦)
+when 푑 = 2.
+Since 퐺(⋅, ⋅) is not defined in Definition 4 for 푑 = 2, for the convenience of nota-
+tions, throughout this section we denote
+퐺(푥, 푦) ∶= −퐴(푥, 푦), and 퐺(푥, 푆) = −
+∑
+푦∈푆
+퐴(푥, 푦)
+when 푑 = 2.
+(43)
+Proposition 20. For any 푥, 푦 ∈ ℤ푑, ℙ-almost surely,
+휕′
+푦휌휔(푥) = 휌휔(푦)
+푑
+∑
+푖=1
+(휕′
+푦휔)(푦, 푒푖)∇2
+푖;1퐺휔′
+푦(푦, 푥).
+19
+
+We will use the fact that for any measurable functions 푓, 푔 on Ω,
+피[(휕′
+푦푓)푔] = 피[푓(휕′
+푦푔)],
+(44)
+휕′
+푦(푓푔) = (휕′
+푦푓)푔 + 푓 ′
+푦(휕′
+푦푔) = (휕′
+푦푓)푔′
+푦 + 푓(휕′
+푦푔).
+(45)
+Proof. It suffices to consider the case 푥 = 0. The formula for general 푥 will follow
+from the fact that 휕′
+푦휌휔(푥) = 휕′
+푦−푥휌휃푥휔(0). We divide the proof into several steps.
+Step 1. First, we will show a formula for 휕′
+푦푉 (푡, 휔):
+휕′
+푦푉 (푡, 휔) = ∫
+푡
+0
+푉휔(푡 − 푠, 푦)
+푑
+∑
+푖=1
+(휕′
+푦휔)(푦, 푒푖)∇2
+푖 푝
+휔′
+푦
+푠 (푦, 0)d푠,
+(46)
+where 푉휔(푠, 푦) = 푉 (푠, 휃푦휔), and 푉 ′
+푦 (푠, 푦) = 푉휔′
+푦(푠, 푦).
+Indeed, notice that 푢(푥, 푡) = 푝휔
+푡 (푥, 0) satisfies 푢(푥, 0) =
+1푥=0 and
+(휕푡 − 퐿휔)푢(푥, 푡) = 0
+for (푥, 푡) ∈ ℤ푑 × (0, ∞).
+(47)
+By the equation above and the product rule (45), we have
+−휕푡[휕′
+푦푢(푥, 푡)] = 휕′
+푦[퐿휔푢(푥, 푡)] =
+푑
+∑
+푖=1
+(휕′
+푦휔)(푥, 푒푖)∇2
+푖 푢′
+푦(푥, 푡) + 퐿휔(휕′
+푦푢)(푥, 푡).
+Hence, for every fixed 푦 ∈ ℤ푑, 휕′
+푦푢(푥, 푡) solves the heat equation
+{
+(휕푡 − 퐿휔)휕′
+푦푢(푥, 푡) = ∑푑
+푖=1(휕′
+푦휔)(푥, 푒푖)∇2
+푖 푢′
+푦(푥, 푡)
+for (푥, 푡) ∈ ℤ푑 × (0, ∞),
+휕′
+푦푢(푥, 0) = 0
+for 푥 ∈ ℤ푑
+whose solution can be represented by Duhamel’s formula
+휕′
+푦푢(푥, 푡) =
+∑
+푧
+푑
+∑
+푖=1 ∫
+푡
+0
+푝휔
+푡−푠(푥, 푧)(휕′
+푦휔)(푧, 푒푖)∇2
+푖 푢′
+푦(푧, 푠)d푠
+=
+푑
+∑
+푖=1 ∫
+푡
+0
+푝휔
+푡−푠(푥, 푦)(휕′
+푦휔)(푦, 푒푖)∇2
+푖 푢′
+푦(푦, 푠)d푠
+where we used the fact that 휕′
+푦휔(푧, 푒) = 0 if 푧 ≠ 푦. Recall that 푢(푥, 푡) = 푝휔
+푡 (푥, 0).
+Summing the above equality over all 푥 ∈ ℤ푑, we obtain formula (46).
+Step 2. We claim that the integrand in (46) has the following bound: ∀푠 ∈ (0, 푡),
+|||푉휔(푡 − 푠, 푦)
+푑
+∑
+푖=1
+(휕′
+푦휔)(푦, 푒푖)∇2
+푖 푝
+휔′
+푦
+푠 (푦, 0)|||≲ (ℋ푦ℋ′
+푦)푑−1(1 + 푠)−훾−0.5푑.
+(48)
+20
+
+Indeed, by (47) and applying the Harnack inequality (Corollary A.2) for the operator
+(휕푡 − 퐿휔) in a similar manner as in (42), we have
+|||푉휔(푡 − 푠, 푦)
+푑
+∑
+푖=1
+(휕′
+푦휔)(푦, 푒푖)∇2
+푖 푝
+휔′
+푦
+푠 (푦, 0)|||
+(39)
+≲ ℋ푑−1
+푦
+osc
+̄퐵1(푦) 푝
+휔′
+푦
+푠 (⋅, 0) ≲ ℋ푑−1
+푦
+푠−훾푝
+휔′
+푦
+2푠 (푦, 0)
+for 푠 > 1. Hence (48) follows from Theorem B when 푠 > 1. When 푠 ≤ 1, (48) is a
+trivial consequence of (39) since |∇2
+푖 푝
+휔′
+푦
+푠 (푦, 0)| ≤ 2. Display (48) is proved.
+Step 3. For any bounded measurable function 푓 on Ω, by Lemma 19 and (44),
+피[(휕′
+푦휌)푓] = 피[휌(휕′
+푦푓)] = lim
+푡→∞ 피[푉 (푡, 휔)(휕′
+푦푓)] = lim
+푡→∞ 피[(휕′
+푦푉 (푡, 휔))푓].
+Furthermore, by (46), (48), and the dominated convergence theorem, we get
+피[(휕′
+푦휌)푓] = ∫
+∞
+0
+lim
+푡→∞ 피
+[
+푉휔(푡 − 푠, 푦)
+푑
+∑
+푖=1
+(휕′
+푦휔)(푦, 푒푖)∇2
+푖 푝
+휔′
+푦
+푠 (푦, 0)1푡>푠푓
+]
+d푠
+퐿푒푚푚푎 19
+=
+∫
+∞
+0
+피
+[
+휌푓
+푑
+∑
+푖=1
+(휕′
+푦휔)(푦, 푒푖)∇2
+푖 푝
+휔′
+푦
+푠 (푦, 0)
+]
+d푠
+= 피
+[
+휌푓
+푑
+∑
+푖=1
+(휕′
+푦휔)(푦, 푒푖)∇2
+푖 퐺휔′
+푦(푦, 0)
+]
+.
+Proposition 20 follows.
+3.2
+Rate of convergence for the average of the invariant measure: Proof
+of Theorem 5
+Now we will proceed to prove one of the main theorems in this paper, Theorem 5.
+It will be clear in the proof that the log 푅 term in the 퐶1,1 bound of Theorem 14
+is important for us to obtain the logarithmic term in Theorem 5.
+Proof of Theorem 5. We divide the proof into several steps.
+Step 1. Let 푢 ∶ ℝ+ → ℝ+ be the function
+푢(푟) =
+{ log(푟 + 1)
+when 푑 = 2
+(푟 + 1)2−푑
+when 푑 ≥ 3.
+(49)
+We will show that for any 휀 > 0, 푅 ≥ 2, there exists a random variable ℋ∗(휔) =
+ℋ∗(푅, 휔; 푑, 휅, 휀) > 0 with 피[exp(푐ℋ∗푑−휀)] < 퐶 such that, ℙ-a.s.,
+osc
+퐵(|푦|+푅)∕2(푦) 퐺휔(⋅, 퐵푅) ≲
+{ ℋ∗푑−1푢(|푦|)푅푑
+if |푦| > 4푅
+ℋ∗푑−1푅2 log 푅
+if |푦| ≤ 4푅.
+(50)
+21
+
+Indeed, by Theorem D, for 푧 ∈ ℤ푑,
+|퐺(푧, 퐵푅)| ≲
+∑
+푥∈퐵푅
+ℋ푑−1
+푥
+푢(|푧 − 푥|).
+(51)
+When |푦| > 4푅, 푢(|푧 − 푥|) ≍ 푢(|푦|) for all 푥 ∈ 퐵푅, 푧 ∈ 퐵(|푦|+푅)∕2(푦), and so
+|퐺(푧, 퐵푅)|
+(51)
+≲
+∑
+푥∈퐵푅
+ℋ푑−1
+푥
+푢(|푦|) ≲ ℋ∗푑−1
+1
+푢(|푦|)푅푑,
+(52)
+where ℋ∗
+1 = (
+1
+|퐵푅|
+∑
+푥∈퐵푅 ℋ푑−1
+푥
+)1∕(푑−1).
+When |푦| ≤ 4푅 and 푑 = 2, for all 푧 ∈ 퐵(|푦|+푅)∕2(푦), we have 푢(|푧−푥|) ≲ log 푅
+∀푥 ∈ 퐵푅, and so
+|퐺(푧, 퐵푅)|
+(51)
+≲ log 푅
+∑
+퐵푅
+ℋ푑−1
+푥
+= ℋ∗푑−1
+1
+푅2 log 푅.
+(53)
+When |푦| ≤ 4푅 and 푑 ≥ 3, for all 푧 ∈ 퐵(|푦|+푅)∕2(푦), (51) yields
+|퐺(푧, 퐵푅)| ≲ [ℋ∗
+2 + (log 푅)1∕(푑−1)]푑−1 ∑
+푥∈퐵4푅
+푢(|푥|)
+≤ (ℋ∗푑−1
+2
++ log 푅)푅2 ≲ ℋ∗푑−1
+2
+푅2 log 푅,
+(54)
+where ℋ∗
+2 = [max푥∈퐵푅 ℋ푥 − (log 푅)1∕(푑−1)]+. Recall ℋ = ℋ(휔, 푑, 휅, 휀) in Theo-
+rem B. Note that for 푡 > 1 and 푝 = 푑 − 휀 > 푑 − 1,
+푃 (ℋ∗
+2 > 푡) ≤
+∑
+푥∈퐵푅
+푃 (ℋ푥 > 푡 + (log 푅)1∕(푑−1))
+≲ 푅푑 exp[−푐(푡 + (log 푅)1∕(푑−1))푝]
+≲ exp(−푐푡푝)
+where we used Chebyshev’s inequality in the second inequality. Hence 피[exp(−푐ℋ푑−휀
+2
+)] ≤
+퐶. Note also that, by Jensen’s inequality, 피[exp(−푐ℋ푑−휀
+1
+)] ≤ 퐶.
+Setting ℋ∗ = ℋ∗
+1 + ℋ∗
+2 , (50) follows from (52), (53), and (54).
+Step 2. Next, we will show that
+|∇2퐺휔(푦, 퐵푅)| ≲
+{ ℋ2
+푦 ℋ∗푑−1|푦|−2푢(|푦|)푅푑
+if |푦| > 4푅
+ℋ2
+푦 ℋ∗푑−1 log 푅
+if |푦| ≤ 4푅,
+(55)
+where the operator ∇2 is only applied to the first ℤ푑 coordinate of 퐺(⋅, ⋅).
+When |푦| > 4푅, by Theorem 14 and (50),
+|∇2퐺(푦, 퐵푅)| ≲
+ℋ2
+푦
+|푦|2
+osc
+퐵|푦|∕2(푦) 퐺(⋅, 퐵푅) ≲ ℋ2
+푦 ℋ∗푑−1
+1
+|푦|−2푢(|푦|)푅푑.
+22
+
+When |푦| ≤ 4푅, applying Theorem 14 (with 휓 = 0, 푓 = −1퐵푅) again, we get
+|∇2퐺(푦, 퐵푅)| ≲
+ℋ2
+푦
+푅2
+osc
+퐵푅∕2(푦) 퐺(⋅, 퐵푅) + ℋ2
+푦 log 푅
+(50)
+≲ ℋ2
+푦 ℋ∗푑−1 log 푅.
+Step 3. By Proposition 20, Theorem B, and (55),
+|||휕′
+푦
+휌휔(퐵푅)
+|퐵푅|
+|||≲ 푅−푑휌휔(푦)|∇2퐺휔′
+푦(푦, 퐵푅)| ≲ 풥2푑
+푦 (휔, 휔′)푤(|푦|),
+(56)
+where 풥푦(휔, 휔′) ∶= [ℋ푑−1
+푦
+(휔)ℋ2
+푦 (휔′
+푦)ℋ∗푑−1(휔′
+푦)]1∕(2푑), and
+푤(푟) =
+{ 푟−2푢(푟)
+if 푟 > 4푅
+푅−푑 log 푅
+if 푟 ≤ 4푅.
+Note that 피[exp(푐풥푑−휀
+푦
+)] < 퐶, and
+∑
+푥∈ℤ푑
+푤2(|푥|) ≍ 푅−푑(log 푅)2
+We let 푊 (푅) = 푅−푑∕2 log 푅 so that 푊 (푅)2 ≍ ∑
+푥∈ℤ푑 푤2(|푥|). Set
+푍(휔) ∶= 휌휔(퐵푅)∕|퐵푅|
+푊 (푅)
+.
+Step 4. By (37), (56), and Jensen’s inequality, for any 푞 ≥ 2,
+푉 (푍)푞∕2 ≲
+(∑
+푦 풥2푑
+푦 푤(|푦|)2)
+∑
+푧 푤(|푧|)2
+)푞∕2
+≤
+∑
+푦
+푤(|푦|)2
+∑
+푧 푤(|푧|)2 풥푑푞
+푦 .
+Taking expectations on both sides and using translation-invariance of ℙ, we get
+피[푉 푞∕2] ≲ 피[풥푑푞
+0 ].
+Thus, by (38),
+피[|푍 − 피푍|푞] ≤ 퐶푞푞∕2피[풥푑푞
+0 ],
+∀푞 ≥ 2.
+(57)
+As a fact, for any 훼 ∈ [0, 1), there exists 푐 = 푐(훼) > 0 such that, for all 푥 > 0,
+∞
+∑
+푛=1
+푐푛
+푛!푥푛푛훼푛 ≤ exp(푥1∕(1−훼)).
+(58)
+Indeed, when 푥 > 0, putting 푐 = 푒−훼∕2 and using inequality 푛푛
+푛! ≤ 푒푛,
+∞
+∑
+푛=1
+푐푛
+푛!푥푛푛훼푛 ≤
+∞
+∑
+푛=1
+2−푛
+푥푛
+(푛!)1−훼 =
+∞
+∑
+푛=1
+2−푛(푥푛∕(1−훼)
+푛!
+)1−훼 ≤ exp(푥1∕(1−훼)),
+where we used 푦푛
+푛! ≤ 푒푦 for 푦 ≥ 0 in the last inequality. Thus, recalling 피[exp(푐풥푑−휀
+0
+)] <
+퐶 and letting 푝 = (3
+2 +
+휀
+푑−휀)−1, displays (57) and (58) yield
+피[exp(푐|푍 − 피푍|푝)] ≲ 피
+[ ∞
+∑
+푛=0
+푐푛
+푛!풥푑푛푝
+0
+(푛푝)푛푝∕2
+]
+≲ 피[exp(푐풥푑−휀
+0
+)] < 퐶.
+Note that 피푍 =
+1
+푊 (푅). The theorem follows by Chebyshev’s inequality.
+23
+
+3.3
+Correlation structure of the field of the invariant measure
+In this subsection we will investigate the mixing property of the field by showing
+the rate of decay of its correlations. Intuitively, since 휌휔(푥) is determined by the
+long term frequency of visits of the RWRE to 푥, the influence of environments at
+remote locations will be small.
+Our proof uses a localization of the invariant measure 휌휔(푥) to a finite ball. For
+any 푥 ∈ ℤ푑, 푟 > 0, we introduce the notation
+휌푟(푥) = 휌푟,휔(푥) ∶= 피[휌휔(푥)|휔(푦) ∶ 푦 ∈ 퐵푟(푥)]
+so that 휌푟(푥) is only a function of environments within the ball 퐵푟(푥).
+Proof of Proposition 6. As usual, we divide the proof into two steps.
+Step 1. We will show that, for 푟 ≥ 2,
+‖휌(0) − 휌푟(0)‖퐿2(ℙ) ≲
+{ 푟−1 log 푟,
+푑 = 2
+푟−푑∕2,
+푑 ≥ 3.
+(59)
+Recall the function 푢(푟) defined in (49). By applying the Efron-Stein inequality to
+every fixed realization of the environment within 퐵푟, we get
+‖휌 − 휌푟‖2
+퐿2(ℙ) ≲ 피[
+∑
+푦∉퐵푟
+(휕′
+푦휌)2]
+푃푟표푝표푠푖푡푖표푛 20
+≲
+피[
+∑
+푦∉퐵푟
+(휌(푦)|∇2퐺휔′
+푦(푦, 0)|)2]
+≲ 피[
+∑
+푦∉퐵푟
+1
+|푦|4 (ℋ푦ℋ(휔′
+푦))2푑−2ℋ4
+푦 (휔′
+푦) 푢(|푦|)2],
+where in the last inequality we used Theorem B, Theorem D, and Theorem 14.
+Since for 푟 ≥ 2,
+∑
+푦∉퐵푟
+|푦|−4푢(|푦|)2 ≍ ∫
+∞
+푟
+푠−4푢(푠)2푠푑−1d푠 ≲
+{ 푟−2(log 푟)2,
+푑 = 2
+푟−푑,
+푑 ≥ 3,
+inequality (59) follows.
+Step 2. We will use (59) to estimate Covℙ(휌(푥), 휌(푦)). By the translation invariance
+of ℙ, it suffices to consider the case 푦 = 0. Then
+Covℙ(휌(0), 휌(푥))
+= 피
+[
+휌(0)(휌(푥) − 휌|푥|∕2(푥))
+]
++ 피
+[
+휌(0)(휌|푥|∕2(푥) − 1)
+]
+= 피 [휌(0)(휌(푥) − 휌|푥|∕2(푥))] + 피 [(휌(0) − 휌|푥|∕2(0))(휌|푥|∕2(푥) − 1)]
+where in the last equality we used the fact that 휌|푥|∕2(0) and 휌|푥|∕2(푥) are independent
+under ℙ, and that 피[휌|푥|∕2(푥)] = 피[휌] = 1.
+Hence, by Hölder’s inequality and the moment bound (cf. Theorem B) of 휌, we
+have
+|Covℙ(휌(0), 휌(푥))| ≲ ‖휌 − 휌|푥|∕2‖퐿2(ℙ).
+The proposition is proved by recalling inequality (59).
+24
+
+4
+Homogenization of the Dirichlet problem
+In this section, 휓 is always assumed to be a local function.
+4.1
+Homogenization of the approximate corrector
+We consider the function ̂휙 ∶ ℤ푑 → ℝ defined as
+̂휙(푥) = ̂휙(푥; 휓, 푅, 휔) = − ∫
+∞
+0
+푒−푡∕푅2퐸푥
+휔[휓(휃푌푡휔)]d푡.
+(60)
+where 푅 ≥ 1, and 휓 is measurable function of 휔(0) with 퐸ℚ[휓] = 0. Notice that
+̂휙 is stationary, i.e., ̂휙(푥; 휓, 푅, 휔) = ̂휙(0; 휓, 푅, 휃푥휔). Moreover, ̂휙 is a solution of
+퐿휔 ̂휙(푥) =
+1
+푅2 ̂휙(푥) + 휓(휃푥휔),
+푥 ∈ ℤ푑.
+(61)
+Clearly, by the definition of ̂휙 in (60), for any 휔 ∈ Ω,
+sup
+푥∈ℤ푑 | ̂휙(푥)| ≤ 푅2‖휓‖∞.
+(62)
+and so ‖ 1
+푅2 ̂휙(푥) + 휓(휃푥휔)‖∞ ≤ 2‖휓‖∞. By (62) and the Hölder estimate (31),
+[ ̂휙]훾;퐵푅∕2 ≲ 푅−훾[max
+퐵푅
+| ̂휙| + 푅2‖푅−2 ̂휙 + 휓‖푑;퐵푅] ≲ 푅2−훾‖휓‖∞.
+Hence, for any 2 ≤ 퐷 ≤ 푅, applying (28) to 푓 = ̂휙∕푅2 and 휎 = 훾 in 퐵퐷, we get
+|∇ ̂휙(0)| ≲ ℋ(퐷‖휓‖∞ + 1
+퐷‖ ̂휙‖1;퐵퐷),
+(63)
+|∇2 ̂휙(0)| ≲ ℋ2‖휓‖∞.
+(64)
+The goal of this subsection is to establish the optimal rate of convergence of the
+approximate corrector. To this end, set, for 푅 ≥ 2,
+휇(푅) ∶=
+⎧
+⎪
+⎨
+⎪⎩
+푅
+푑 = 2
+푅1∕2
+푑 = 3
+(log 푅)1∕2
+푑 = 4
+1
+푑 ≥ 5.
+(65)
+Lemma 21. Assume that 휓(휔) = 휓(휔(0)) is a bounded function of 휔(0). For any
+0 < 푝 <
+2푑
+3푑+2, there exists 퐶 = 퐶(푑, 휅, 푝) such that for 푡 ≥ 0, 푅 ≥ 2, with 휇(푅) as
+defined in (65) and ̂휙(푥) = ̂휙(푥; 휓, 푅, 휔) as in (60),
+ℙ
+(
+| ̂휙(0)| ≥ 푡휇(푅)‖휓‖∞
+)
+≤ 퐶 exp(− 1
+퐶 푡푝).
+25
+
+The continuous version of Lemma 21 was proved earlier by Armstrong, Lin [3].
+Our result in two dimensions (푑 = 2) is slightly better than that in [3].
+We now obtain Lemma 21 using the concentration inequality (38). To this end,
+we regard ̂휙 as a function of the environment and write, for 푦 ∈ ℤ푑,
+̂휙′
+푦(푥) ∶= ̂휙(푥; 휓, 푅, 휔′
+푦),
+휕′
+푦 ̂휙 = ̂휙′
+푦 − ̂휙.
+(66)
+We will need a bound for 휕′
+푦 ̂휙(0). Note that 푤(푥) = 휕′
+푦 ̂휙(푥) satisfies, for 푥, 푦 ∈ ℤ푑,
+퐿휔′
+푦푤(푥) = 푅−2푤(푥) + [푅−2 ̂휙 + 휓(휔′(푦)) − tr(휔′∇2 ̂휙)]1푦=푥,
+which yields
+푤(푥) = − [푅−2 ̂휙(푦) + 휓(휔′(푦)) − tr(휔′∇2 ̂휙)(푦)]
+∫
+∞
+0
+푒−푡∕푅2푝
+휔′
+푦
+푡 (푥, 푦)d푡.
+(67)
+This equality, together with (64), (62) and Theorem B(c), implies
+|휕′
+푦 ̂휙(0)| = |푤(0)| ≲ ℋ2
+푦 ‖휓‖∞ ∫
+∞
+0
+푒−푡∕푅2푝
+휔′
+푦
+푡 (0, 푦)d푡
+≲ ℋ2
+푦 ℋ′
+푦
+푑−1‖휓‖∞ ∫
+∞
+0
+(1 + 푡)−푑∕2 exp
+[
+− 푡
+푅2 − 푐픥(|푦|, 푡)
+]
+d푡
+≲ ℋ2
+푦 ℋ′
+푦
+푑−1‖휓‖∞푣(|푦|),
+(68)
+where ℋ푦 = ℋ(휃푦휔), ℋ′
+푦 = ℋ(휃푦휔′
+푦) and, with 푐2 = 푐2(휅, 푑) > 0 denoting an
+appropriate constant,
+푣(푟) =
+{
+푒−푐2푟∕푅 [
+1 + log(
+푅
+(푟+1)∧푅)
+]
+푑 = 2
+푒−푐2푟∕푅(푟 + 1)2−푑
+푑 ≥ 3.
+(69)
+Recall 휇(푅) in (65). Notice that
+∑
+푦∈ℤ푑
+푣(|푦|)2 ≲ ∫
+∞
+0
+푣(푟)2푟푑−1d푟 ≲ 휇(푅)2.
+(70)
+The verifications of inequalities (68) and (70) are included in the Appendix.
+Proof of Lemma 21: For 푦 ∈ ℤ푑, set ℋ푦 ∶= ℋ(휃푦휔), and
+푍(휔) ∶=
+̂휙(0)
+‖휓‖∞휇(푅).
+(71)
+By (37), (68), (70), and Jensen’s inequality, for any 푞 ≥ 2,
+푉 (푍)푞∕2 ≲
+(∑
+푦(ℋ4
+푦 ℋ′
+푦
+2(푑−1))푣(|푦|)2)
+∑
+푧 푣(|푧|)2
+)푞∕2
+≤
+∑
+푦
+푣(|푦|)2
+∑
+푧 푣(|푧|)2 ℋ2푞
+푦 ℋ′
+푦
+(푑−1)푞.
+26
+
+Taking expectations on both sides and using translation-invariance of ℙ, we get
+피[푉 푞∕2] ≲ 피[ℋ2푞ℋ′푞(푑−1)] ≲ 피[ℋ(1+푑)푞],
+where we used Hölder’s inequality in the second inequality. Thus, by (38),
+피[|푍 − 피푍|푞] ≤ 퐶푞푞∕2피[ℋ(1+푑)푞],
+∀푞 ≥ 2.
+(72)
+Thus, recalling 피[exp(푐ℋ푑−휀)] < 퐶 in Theorem B and letting 푝 = (3
+2 + 1+휀
+푑−휀)−1,
+displays (72) and (58) yield
+피[exp(푐|푍 − 피푍|푝)] ≲ 피
+[ ∞
+∑
+푛=0
+푐푛
+푛!ℋ(1+푑)푛푝(푛푝)푛푝∕2
+]
+≲ 피[exp(푐ℋ푑−휀)] < 퐶.
+In particular, 피[|푍 − 피푍|2] < 퐶.
+To prove Lemma 21, it suffices to show that
+피[exp(푐|푍|푝)] = 피
+[
+exp (푐|
+̂휙(0)
+휇(푅)‖휓‖∞ |푝)]
+< 퐶.
+(73)
+It suffices to show that |피푍| < 퐶. Since ℚ is an invariant measure for (휃푌푡휔)푡≥0,
+we have 퐸ℚ퐸0
+휔[휓(휃푌푡휔)] = 퐸ℚ[휓] = 0 for all 푡 ≥ 0. Hence, by (60), we know
+퐸ℚ[ ̂휙(0)] = 0
+and so 퐸ℚ[푍] = 0. Further, by Hölder’s inequality and Theorem B,
+|피푍| = |퐸ℚ[푍 − 피푍]| ≤ 퐸ℚ[|푍 − 피푍|] ≤ ‖휌‖퐿2(ℙ)‖푍 − 피푍‖퐿2(ℙ) ≤ 퐶
+Therefore, we obtain (73). Lemma 21 follows by Chebyshev’s inequality.
+4.2
+Rate of homogenization for the Dirichlet problem (19)
+We need the following notations. Let 푟 = 푟(푅) be a function of 푅 defined as
+푟 =
+⎧
+⎪
+⎨
+⎪⎩
+푅2∕3
+푑 = 2
+푅4∕7
+푑 = 3
+푅1∕2(log 푅)1∕8
+푑 = 4
+푅1∕2
+푑 ≥ 5.
+(74)
+Recall ̂휙 in (60). For 푘 = 1, … , 푑, let
+푣푘(푥) = ̂휙(푥; 휔푘 − ̄푎푘, 푟, 휔),
+푥 ∈ ℤ푑.
+(75)
+For 푅 ≥ 2 and 0 < 푝 <
+2푑
+3푑+2, set Λ = {휔푘 − ̄푎푘 ∶ 푘 = 1, … , 푑}, and define
+풴 = 풴(휔; 푅) = 1 + 푐푝
+max
+휉∈Λ,|푒|≤1 | ̂휙(0; 휉, 푟, 휃푒휔)|∕[‖휉‖∞휇(푟)],
+27
+
+so that, by Lemma 21, for all 푅 ≥ 2, 피[exp(풴푝)] ≤ 퐶 and
+max
+휉∈Λ,|푒|≤1 | ̂휙(0; 휉, 푟, 휃푒휔)| ≲푝 풴휇(푟).
+(76)
+We write 풴푥 = 풴푥(휔; 푅) ∶= 풴(휃푥휔; 푅).
+In what follows we will apply the classical method of two-scale expansions to
+quantify the rate of the homogenization for the Dirichlet problem (19). The proof
+is similar to that in the periodic setting (see [32, 45, 30] for example). The only
+difference is that we use the approximate corrector 푣푘 here instead of the actual
+corrector in the periodic setting.
+Proof of Theorem 7: We can replace the function 푔 in (19) by ̄푢, because doing this
+only introduces an error of size 퐶푅−1‖푢‖퐶4 to |푢(푥) − ̄푢( 푥
+푅)|.
+Consider
+푤(푥) = 푢(푥) − ̄푢( 푥
+푅) + 1
+푅2 푣푘(푥)휕푘푘 ̄푢( 푥
+푅),
+푥 ∈ ̄퐵푅.
+(77)
+Here we follow the convention of summation over repeated indices. Then, ∀푥 ∈
+퐵푅,
+|퐿휔푤(푥)| =|||
+1
+푅2 푓( 푥
+푅) − 퐿휔[̄푢( 푥
+푅)] + 1
+푅2 (푣푘
+푟2 + 휔푘 − ̄푎푘)휕푘푘 ̄푢( 푥
+푅) + 1
+푅2 푣푘퐿휔[휕푘푘 ̄푢( 푥
+푅)]
++ 1
+푅2
+∑
+푦∼푥
+휔(푥, 푦)[휕푘푘 ̄푢( 푦
+푅2 ) − 휕푘푘 ̄푢( 푥
+푅2 )][푣푘(푦) − 푣푘(푥)]|||
+=|||
+1
+2푅−3휔푖(푥)휕푘푘푖 ̄푢( 푥
+푅)[푣푘(푥 + 푒푖) − 푣푘(푥 − 푒푖)] + (푅푟)−2휕푘푘 ̄푢( 푥
+푅)푣푘(푥)
++ 푅−4‖̄푢‖퐶4|푣푘(푥)|푂(1)|||
+≲ ‖̄푢‖퐶4
+(
+푅−3|푣푘(푥 + 푒푖) − 푣푘(푥 − 푒푖)| + (푅푟)−2|푣푘| + 푅−4|푣푘|
+)
+.
+(78)
+See Appendix A.4 for verification of the first part of (78). Set 퐷 = 휇(푟)1∕2 ≤ 푟. By
+(63) and (76),
+osc
+̄퐵1(푥) 푣푘 ≲ ℋ푥(퐷 + ‖푣푘‖1;퐵퐷(푥)) ≲ ℋ푥풴∗
+푥
+√
+휇(푟),
+(79)
+where
+풴∗
+푥 =
+1
+#퐵퐷
+∑
+푧∈퐵퐷(푥)
+풴푧.
+(80)
+By (78), (79) and (64), we get that for 푥 ∈ 퐵푅,
+|퐿휔푤| ≲ ‖̄푢‖퐶4[ℋ푥풴∗
+푥푅−3√
+휇(푟) + (푅푟)−2휇(푟)풴푥]
+≲ ‖̄푢‖퐶4푅−2휏(푅)(ℋ푥풴∗
+푥 + 풴푥).
+28
+
+Recall 푟(푅), 휏(푅) in (74), (20). Hence, by (77) and the ABP inequality,
+max
+퐵푅
+|푢(푥) − ̄푢( 푥
+푅)| ≲ max
+퐵푅
+|푤| + 푅−2 max
+푥∈퐵푅
+|푣푘(푥)휕푘푘 ̄푢( 푥
+푅)|
+≲ ‖̄푢‖퐶4
+[
+휏(푅)
+( 1
+#퐵푅
+∑
+푥∈퐵푅
+(ℋ푥풴∗
+푥 + 풴푥)푑)1∕푑 + 푅−2휇(푟) max
+푦∈ ̄퐵푅
+풴푦
+]
+≲ ‖̄푢‖퐶4
+[
+휏(푅)퐴1 + 푅−2휇(푟)퐴2 + 푅−2휇(푟)(log 푅)1∕(2푠)]
+≲ ‖̄푢‖퐶4휏(푅)(퐴1 + 퐴2),
+(81)
+where
+퐴1 =
+(
+1
+#퐵푅
+∑
+푥∈퐵푅
+(ℋ푥풴∗
+푥 + 풴푥)푑
+)1∕푑
+,
+퐴2 =
+(
+max
+푦∈ ̄퐵푅
+풴푦 − (2푑 log 푅)1∕(2푠)
+)
++
+.
+For 푞 ≥ 푑, by the translation-invariance of ℙ,
+피[퐴푞
+1] ≲ 피[(ℋ풴∗
+0 + 풴)푞] ≲ 피[풳2푞 + 풴∗2푞
+0
++ 풴푞] ≲ 피[풳2푞 + 풴2푞],
+which implies, for 푠 ∈ (0,
+푑
+3푑+2),
+피[exp(푐퐴푠
+1)] ≲ 피[exp(풳2푠) + exp(풴2푠)] ≤ 퐶.
+(82)
+Moreover, for 푡 > 0, by a union bound and Chebyshev’s inequality,
+ℙ(퐴2 ≥ 푡) ≲ 푅푑ℙ(풴 − [2푑 log 푅]1∕(2푠) ≥ 푡)
+≤ # ̄퐵푅피
+[
+exp
+(
+풴2푠 − 1
+2푡2푠 − 푑 log 푅
+)]
+≲ 푒−푡2푠∕2.
+Thus 피[exp(푐퐴푠
+2)] < 퐶. This, together with (82) and (81), yields
+max
+퐵푅
+|푢(푥) − ̄푢( 푥
+푅)| ≲ ‖̄푢‖퐶4휏(푅)풵,
+with 풵 ∶= 푐(퐴1 + 퐴2) satisfying 피[exp(풵푠)] ≤ 퐶. Our proof is complete.
+5
+Quantification of the diffusive behavior
+5.1
+Quantification of the ergodicity of the environmental process: Proof
+of Theorem 8
+In this section we will derive the optimal rates of convergence (as 푡 → ∞) of the er-
+godic average 1
+푡 퐸휔[∫ 푡
+0 휓( ̄휔푠)d푠], where ̄휔푠 denotes the process of the environment
+viewed from the particle:
+̄휔푠 ∶= 휃푌푠휔.
+With Lemma 21, it may be tempting to compare the approximate corrector ̂휙
+in (60) to the corrector within a finite ball 퐵푅, i.e., the solution 푢 to the Dirichlet
+29
+
+problem 퐿휔푢 = 휓휔 in 퐵푅 with 푢 = 0 on 휕퐵푅. However, such comparison involves
+controlling the boundary error max휕퐵푅 ̂휙 which would result in an extra log 푅 fac-
+tor. In what follows, we will follow the argument of Kipnis and Varadhan [37] to
+approximate 퐸휔[∫ 푇
+0 휓(휃푌푠휔)d푠] with a martingale using the approximate corrector.
+Proof of Theorem 8. Without loss of generality, assume ‖휓‖∞ = 1 and ̄휓 = 0.
+First, we will construct a martingale (for both continuous and discrete time
+cases) using the approximate corrector.
+For any fixed 푇 > 1, let 휙 ∶ Ω → ℝ denote the function
+휙(휔) = 휙휓,푇(휔) ∶= ̂휙(0; 휓,
+√
+푇 , 휔),
+where ̂휙 is as in (60). Then, for a.s. 휔 ∈ Ω, the process (푀푡)푡≥0 defined by
+푀푡 ∶ = 휙(휃푌푡휔) − 휙(휃푌0휔) − ∫
+푡
+0
+퐿휔휙(휃푌푠휔)d푠
+(61)
+= 휙( ̄휔푡) − 휙( ̄휔0) − ∫
+푡
+0
+[ 1
+푇 휙( ̄휔푠) + 휓( ̄휔푠)]d푠
+(83)
+is a 푃휔-martingale with respect to the filtration ℱ푡 = 휎(푌푠 ∶ 푠 ≤ 푡). Similarly, for
+discrete-time RWRE, we have that
+푁푛 ∶= 휙( ̄휔푛) − 휙( ̄휔0) −
+푛−1
+∑
+푖=0
+[ 1
+푇 휙( ̄휔푖) + 휓( ̄휔푖)]
+is a 푃휔-martingale with respect to the filtration ℱ푛 = 휎(푋푖 ∶ 푖 ≤ 푛).
+Next, we will derive an exponential moment bounds for ∫ 푡
+0 푃푠휓d푠 and ∑푛
+푖=0 푃푖휓,
+where the operator 푃푠 is as in (16). We will only provide a proof for the continuous-
+time case, because the argument for the discrete-time setting is exactly the same.
+Since 퐸휔[푀푠] = 퐸휔[푀0] = 0, taking expectations in (83), we get
+∫
+푡
+0
+푃푠휓d푠 = 푃푡휙 − 휙 − 1
+푇 ∫
+푡
+0
+푃푠휙d푠.
+(84)
+Since the process ( ̄휔푠) is a stationary sequence under the measure ℚ×푃휔, we have,
+by Jensen’s inequality, for any 푡 ≥ 0, 푞 ≥ 1,
+‖푃푡휙‖푞
+퐿푞(ℚ) = 퐸ℚ[|퐸휔휙( ̄휔푡)|푞] ≤ 퐸ℚ×푃휔[|휙( ̄휔푡)|푞] = 퐸ℚ[|휙|푞].
+Hence, taking the 퐿푞(ℚ)-norms on both sides of (84), we get
+‖∫
+푇
+0
+푃푠휓d푠‖퐿푞(ℚ) ≤ 3‖휙‖퐿푞(ℚ),
+∀푞 ≥ 1
+30
+
+which implies
+퐸ℚ
+[
+exp
+(
+푐||| ∫
+푇
+0
+푃푠휓d푠
+/
+휇(
+√
+푇 )|||
+푝)]
+≤ 퐸ℚ
+[
+exp
+(
+푐|||3휙∕휇(
+√
+푇 )|||
+푝)]
+≤ ‖휌‖퐿2(ℙ)퐸ℙ
+[
+exp
+(
+0.5푐|||3휙∕휇(
+√
+푇 )|||
+푝)]1∕2 (73)
+≤ 퐶,
+where we used Hölder’s inequality in the second inequality.
+Note that 휈(푇 ) = 푇 −1휇(
+√
+푇 ) as defined in (21). The theorem follows from the
+above moment bound and Chebyshev’s inequality.
+As a consequence of Theorem 8, we can show the existence and uniqueness of
+a stationary corrector in 푑 ≥ 5.
+Corollary 22. Assume (A1), (A2). When 푑 ≥ 5, for any bounded local measurable
+function 휁 ∶ Ω → ℝ, there exists 휙 ∶ Ω → ℝ with the following properties.
+(a) The function 휙휔(푥) ∶= 휙(휃푥휔) solves
+퐿휔휙(푥) = 휁(휃푥휔) − 퐸ℚ[휁],
+for all 푥 ∈ ℤ푑;
+(85)
+(b) Up to an additive constant, 휙 is the unique function that satisfies (85) and
+피[exp(푐|휙|푝)] < ∞
+for all 0 < 푝 < (3
+2 + 1
+푑 )−1.
+(86)
+In the continuous PDE setting, the existence of the stationary corrector in 푑 ≥ 5
+stochastic integrability (86) with 푝 = 1
+2 was proved in [3, Theorem 7.1]. See also
+[31, Corollary 7] for a proof of the existence part (a) in the discrete setting.
+Proof. Without loss of generality, assume 퐸ℚ[휁] = 0.
+When 푑 ≥ 5, we let 휙휔(푥) = 퐸푥
+휔[∫ ∞
+0
+휁( ̄휔푠)d푠]. The existence (both as a.s.
+and 퐿푝(ℙ)-limits for all 푝 > 0) and stochastic integrability (86) of such a function
+follow immediately from Theorem 8. It clearly solves (85). It remains to show the
+uniqueness up to an additive constant.
+Suppose there is another stationary corrector ̃휙 that satisfies (85). Then, for
+푘 ∈ ℕ and for 퐶 > 0 sufficiently large, by Chebyshev’s inequality,
+ℙ(max
+푥∈퐵푘
+|휙(푥) − ̃휙(푥)| ≥ 퐶 log 푘) ≲ 푘푑ℙ(|휙(0) − ̃휙(0)| ≥ 퐶 log 푘) ≲ 푘−2.
+By Borel-Cantelli’s lemma, ℙ-almost surely, we have
+lim
+푘→∞ max
+퐵푘
+|휙 − ̃휙|∕ log 푘 < ∞.
+Since 휙 − ̃휙 is 휔-harmonic on ℤ푑 with sublinear growth, by Theorem 14, it is a
+constant.
+31
+
+5.2
+A Berry-Esseen estimate for the QCLT: Proof of Corollary 10
+To prove Corollary 10 we will apply the Berry-Esseen estimates for martingales by
+Heyde and Brown [35]. Here we will use the version in [34, Theorem 2] which is
+also applicable to the continuous-time setting.
+Proof of Corollary 10. For any unit vector 퓁 ∈ ℝ푑, let 휓0(휔) = 퓁푇 휔(0)
+tr휔(0)퓁, 휓 =
+휓0 − 퐸ℚ[휓0]. Following the notations in [34], we set
+푁푛,2 ∶ = 퐸휔
+[
+|||
+푛−1
+∑
+푘=0
+퐸휔
+[ 1
+√
+푛
+(
+(푋푘+1 − 푋푘) ⋅ 퓁
+)2|ℱ푘
+]
+− 퓁푇 ̄푎퓁|||
+2
+]
+= 1
+푛2 퐸휔
+[
+( 푛−1
+∑
+푘=0
+휓( ̄휔푘)
+)2
+]
+,
+퐿푛,2 ∶=
+푛−1
+∑
+푘=0
+퐸휔[| 1
+√
+푛(푋푘+1 − 푋푘) ⋅ 퓁|4] = 1
+푛2 퐸휔
+[푛−1
+∑
+푘=0
+휓0( ̄휔푘)
+]
+.
+The term 푁푛,2 can be further written as
+푛2푁푛,2 = 2
+푛−1
+∑
+푖=0
+퐸휔
+[
+휓( ̄휔푖)
+푛−푖−1
+∑
+푗=0
+휓( ̄휔푖+푗)
+]
+= 2
+푛−1
+∑
+푖=0
+퐸휔
+[
+휓( ̄휔푖)퐸푋푖
+휔
+[푛−푖−1
+∑
+푗=0
+휓( ̄휔푗)
+]]
+.
+Hence, for any 푞 ≥ 1, using the fact that ( ̄휔푖) is a stationary sequence under ℚ×푃휔,
+we get (note ‖휓0‖∞ ≲ 1)
+‖푁푛,2‖퐿푞(ℚ) ≲ 1
+푛2
+푛−1
+∑
+푖=0
+‖
+푛−푖−1
+∑
+푗=0
+푃푗휓‖퐿푞(ℚ×푃휔)
+which, by Jensen’s inequality and the fact 1
+푛2
+∑푛
+푘=1 휇(
+√
+푘) ≍ 휈(푛), implies that for
+any 0 < 푝 <
+2푑
+3푑+2,
+퐸ℚ
+[
+exp
+(
+푐|푁푛,2∕휈(푛)|푝)]
+≲
+1
+푛2휈(푛)
+푛
+∑
+푘=1
+휇(
+√
+푘)퐸ℚ
+[
+exp
+(
+푐|||
+푘−1
+∑
+푗=0
+푃푗휓
+/
+휇(
+√
+푘)|||
+푝)
+]
+푇 ℎ푒표푟푒푚 8
+≤
+퐶.
+Thus, using the moment bound of 휌−1 in Theorem B, by Hölder’s inequality,
+퐸ℙ
+[
+exp
+(
+0.5푐|푁푛,2∕휈(푛)|푝)]
+≤ ‖휌−1∕2‖퐿2(ℙ)퐸ℚ
+[
+exp
+(
+푐|푁푛,2∕휈(푛)|푝)]1∕2 ≤ 퐶.
+By Theorem 8 we already know that 퐸ℙ[exp (푐|푛퐿푛,2|푝)] ≤ 퐶. Therefore, we
+conclude that there exists a random variable 풴5 with 퐸ℙ[exp(풴5푝)] < ∞ such that
+퐿푛,2 + 푁푛,2 ≤ 퐶휈(푛)풴5.
+The corollary follows by applying [34, Theorem 2].
+32
+
+A
+Appendix
+Define the parabolic operator ℒ휔 as
+ℒ휔푢(푥, 푡) =
+∑
+푦∶푦∼푥
+휔(푥, 푦)[푢(푦, 푡) − 푢(푥, 푡)] − 휕푡푢(푥, 푡)
+for every function 푢 ∶ ℤ푑 × ℝ → ℝ which is differentiable in 푡. The following
+results are used in the paper.
+Theorem A.1. ([22, Theorem 17]) Assume
+휔
+tr휔 > 2휅퐼 for some 휅 > 0. Any
+non-negative function 푢 with ℒ휔푢 = 0 in 퐵2푅 × (0, 4푅2) for 푅 > 0 satisfies
+sup
+퐵푅×(푅2,2푅2)
+푢 ≤ 퐶
+inf
+퐵푅×(3푅2,4푅2) 푢.
+As a consequence, we have the following Hölder regularity for 푢.
+Corollary A.2. Assume
+휔
+tr휔 > 2휅퐼 for some 휅 > 0. There exists 훾 = 훾(푑, 휅) ∈
+(0, 1) such that any non-negative function 푢 with ℒ휔푢 = 0 in 퐵푅(푥0)×(푡0 −푅2, 푡0),
+for some (푥0, 푡0) ∈ ℤ푑 × ℝ and 푅 > 0, satisfies
+|푢( ̂푥) − 푢( ̂푦)| ≤ 퐶
+( 푟
+푅
+)훾
+sup
+퐵푅(푥0)×(푡0−푅2,푡0)
+푢
+for all ̂푥, ̂푦 ∈ 퐵푟(푥0) × (푡0 − 푟2, 푡0) and 푟 ∈ (0, 푅).
+A.1
+Proof of Proposition 16
+Proof. Let 푝 ∈ H푗 be the 푗-th order Taylor polynomial (around 0) of 푣. Then
+sup
+픹휃푅
+|푣 − 푝| ≤ 퐶(휃푅)푗+1 sup
+픹푅∕3
+|퐷푗+1푣|.
+This gives 푗+1
+픹휃푅(푣) ≲ (휃푅)푗+1 sup픹푅∕3 |퐷푗+1푣|. Furthermore, for any 푞 ∈ H푗,
+푗 ≤ 2, note that 퐷(푣−푞) is an ̄푎-harmonic function. Hence, by [27, Theorem 2.10],
+sup
+픹푅∕3
+|퐷푗+1푣| = sup
+픹푅∕3
+|퐷푗+1(푣 − 푞)|
+≤ 퐶
+푅푗 sup
+픹5푅∕12
+|퐷(푣 − 푞)|
+= 퐶
+푅푗
+sup
+푥∈픹5푅∕12
+| ⨏픹푅∕12(푥)
+퐷(푣 − 푞)| ≤
+퐶
+푅푗+1 sup
+픹푅∕2
+|푣 − 푞|
+for 푗 ≤ 2. Hence, taking infimum over 푞 ∈ H푗, we get 푗+1
+픹휃푅(푣) ≲ 휃푗+1푗+1
+픹푅∕2(푣) for
+푗 ≤ 2. The first statement is proved.
+33
+
+To prove the second statement, observe that for any 푥 ∈ 픹푅∕2, there are 2푑
+points 푦푖 ∈ ̄퐵푅∕2, 푖 ∈ Λ = {1, … , 2푑}, such that |푦푖 − 푥| ≤ 1 and 푥 is a convex
+combination of the 푦푖’s. That is, 푥 = ∑
+푖∈Λ 훼푖푦푖 for some 훼푖 ≥ 0 with ∑
+푖∈Λ 훼푖 = 1.
+Let 푝 ∈ H푗, 푗 ≤ 2, be such that max퐵2푅∕3 |푣 − 푝| ≤ 2푗+1
+2푅∕3(푣) and denote the
+Hessian matrix of 푝 by 푀푝. Then, for 푥 ∈ 픹푅∕2, 푗 ≤ 2,
+|푣(푥) − 푝(푥)| ≤ [푣]1;픹푅∕2+1 +
+∑
+푖∈Λ
+훼푖|푣(푦푖) − 푝(푦푖)|+|||푝(푥) −
+∑
+푖∈Λ
+훼푖푝(푦푖)|||
+≤ [푣]1;픹푅∕2+1 + max
+̄퐵푅∕2
+|푣 − 푝| + 퐶푅|푀푝|.
+(87)
+Further, using the fact (see [27, Cor.6.3]) that
+푅[푣]1;픹푅∕2+1 ≲ sup
+픹2푅∕3
+|푣| + 푅2|푐0| ≲ sup
+휕픹2푅∕3
+|푣| + 푅2|푐0|
+and (Note that the following bound is not needed for the case 푗 = 1 where 푀푝 ≡ 0.)
+푅2|푀푝| ≲ max
+푦∈퐵푅∕2
+|푝(푦)+푝(−푦)−2푝(0)| ≲ max
+퐵푅∕2
+|푣−푝|+max
+퐵푅∕2
+|푣| ≲ 3
+2푅∕3(푣)+max
+퐵푅∕2
+|푣|,
+display (87) implies, for 푗 ≤ 2,
+푗+1
+픹푅∕2(푣) ≲ 1
+푅 sup
+휕픹2푅∕3
+|푣| + 푅|푐0| + 푗+1
+2푅∕3(푣).
+The second claim follows.
+A.2
+Verification of (68)
+In this subsection we will verify the inequality
+∫
+∞
+0
+(1 + 푡)−푑∕2 exp
+[
+− 푡
+푅2 − 푐픥(|푦|, 푡)
+]
+d푡 ≲ 푣(|푦| + 1),
+∀푦 ∈ ℤ푑.
+We break the integral on the left side of the above inequality as
+∫
+∞
+0
+= ∫
+|푦|∕2
+0
++ ∫
+|푦|2
+|푦|∕2
++ ∫
+∞
+|푦|2 =∶ I + II + III.
+It suffices to consider the case |푦| ≥ 1. First, with 푐2 > 0 sufficiently small,
+I = ∫
+|푦|∕2
+0
+(1 + 푡)−푑∕2 exp
+(
+− 푡
+푅2 − 푐|푦| log |푦|
+푡
+)
+d푡 ≤ |푦|푒−푐|푦| ≲ 푣(|푦|).
+Moreover, noting that −
+푡
+2푅2 − 푐 |푦|2
+푡
+≲ −|푦|
+푅 ,
+II = ∫
+|푦|2
+|푦|∕2
+(1 + 푡)−푑∕2 exp
+(
+− 푡
+푅2 − 푐 |푦|2
+푡
+)
+d푡
+≲ 푒−푐|푦|∕푅
+∫
+|푦|2
+0
+푡−푑∕2푒−푐|푦|2∕푡d푡
+≲ 푒−푐|푦|∕푅|푦|2−푑
+∫
+∞
+1
+푠푑∕2−2푒−푐푠d푠 ≲ 푣(|푦|).
+34
+
+Similarly, for 푑 = 2,
+III ≲ 푒−푐|푦|∕푅
+∫
+∞
+|푦|2 (1 + 푡)−푑∕2 exp
+(
+−
+푡
+2푅2
+)
+d푡
+≲ 푒−푐|푦|∕푅
+∫
+∞
+|푦|2∕푅2 푠−1푒−푠∕2d푠
+≲ 푒−푐|푦|∕푅
+[
+1 + ∫
+∞
+0
+푠−1
+1{|푦|2∕푅2≤푠≤1}d푠
+]
+≲ 푣(푦|).
+For 푑 ≥ 3, we have
+III ≲ 푒−푐|푦|∕푅
+∫
+∞
+|푦|2 (1 + 푡)−푑∕2d푡 ≲ 푣(|푦|).
+Therefore, the above bounds of I, II, III imply inequality(68).
+A.3
+Verification of (70)
+When 푑 = 2,
+∫
+∞
+0
+푒−푐푟∕푅푣(푟)2푟푑−1d푟 ≲ ∫
+∞
+0
+푒−푐푟∕푅[1 + (log
+푅
+(푟+1)∧푅)2]푟d푟
+≲ 푅2 + ∫
+푅
+1
+푒−푐푟∕푅 (
+log 푅
+푟
+)2
+푟d푟 ≲ 푅2.
+When 푑 = 3,
+∫
+∞
+0
+푒−푐푟∕푅푣(푟)2푟푑−1d푟 ≲ ∫
+∞
+0
+푒−푐푟∕푅 ≲ 푅.
+When 푑 = 4,
+∫
+∞
+0
+푒−푐푟∕푅푣(푟)2푟푑−1d푟 ≲ ∫
+푅
+0
+(1 + 푟)−1d푟 + ∫
+∞
+푅
+푒−푐푟∕푅푅−1d푟 ≲ log 푅.
+When 푑 ≥ 5,
+∫
+∞
+0
+푒−푐푟∕푅푣(푟)2푟푑−1d푟 ≲ ∫
+∞
+0
+(1 + 푟)−2d푟 = 1.
+A.4
+Verification of (78)
+We verify the first part of (78). For functions 푢, 푣 on ℤ푑,
+∇2
+푒(푢푣)(푥) = 푢(푥 + 푒)푣(푥 + 푒) + 푢(푥 − 푒)푣(푥 − 푒) − 2푢(푥)푣(푥)
+= 푣(푥)∇2
+푒푢(푥) + 푢(푥 + 푒)[푣(푥 + 푒) − 푣(푥)] + 푢(푥 − 푒)[푣(푥 − 푒) − 푣(푥)].
+35
+
+From this we have the expression
+퐿휔(푢푣) = 푢퐿휔푣 + 푣퐿휔푢 +
+∑
+푦∶푦∼푥
+휔(푥, 푦)[푢(푦) − 푢(푥)][푣(푦) − 푣(푥)].
+In particular, if 푢 is 퐶2 in ℝ푑, then, doing Taylor expansion to 푢,
+∇2
+푒(푢푣)(푥) = 푣(푥)∇2
+푒푢(푥) + [푢(푥) + 퐷푒푢(푥) + 1
+2퐷2
+푒푢(푦)][푣(푥 + 푒) − 푣(푥)]
++ [푢(푥) − 퐷푒푢(푥) + 1
+2퐷2
+푒푢(푧)][푣(푥 − 푒) − 푣(푥)]
+= 푣(푥)∇2
+푒푢(푥) + 푢(푥)∇2
+푒푣(푥) + 퐷푒푢(푥)[푣(푥 + 푒) − 푣(푥 − 푒)]
++ 1
+2[퐷2
+푒푢(푦)푎+(푥) + 퐷2
+푒푢(푧)푎−(푥)]∇2
+푒푣(푥)
+≤ 푣(푥)∇2
+푒푢(푥) + 푢(푥)∇2
+푒푣(푥) + 퐷푒푢(푥)[푣(푥 + 푒) − 푣(푥 − 푒)]
++ ‖푢‖퐶2∇2
+푒푣(푥).
+with 푎±(푥) = [푣(푥 ± 푒) − 푣(푥)]∕∇2
+푒푣(푥), and 푦, 푧 points within the line segment
+[푥 − 푒, 푥 + 푒]. Note that 푎+ + 푎− = 1.
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+39
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+E-mail address, Xiaoqin Guo: guoxq@ucmail.uc.edu
+E-mail address, Hung Vinh Tran: hung@math.wisc.edu
+40
+
diff --git a/FtAzT4oBgHgl3EQfUfxu/content/tmp_files/load_file.txt b/FtAzT4oBgHgl3EQfUfxu/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf,len=1415
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+page_content='PR]  3 Jan 2023  Optimal convergence rates in stochastic  homogenization in a balanced random environment  Xiaoqin Guo ∗1 and Hung V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Tran †2  1 Department of Mathematical Sciences, University of Cincinnati , 2815  Commons Way, Cincinnati, OH 45221, USA  2 Department of Mathematics, University of Wisconsin Madison, 480  Lincoln Drive, Madison, WI 53706, USA  January 4, 2023  Abstract  We consider random walks in a uniformly elliptic, balanced, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' ran-  dom environment in ℤ푑 for 푑 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' We first derive a quantitative law of large  numbers for the invariant measure, which is nearly optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' A mixing prop-  erty of the field of the invariant measure is then achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' We next obtain  rates of convergence for the homogenization of the Dirichlet problem, which  are generically optimal for 푑 ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Afterwards, we quantify the ergodicity  of the environmental process for both the continuous-time and discrete-time  random walks, and as a consequence, we get explicit convergence rates for  the quenched central limit theorem of the balanced random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Contents  1  Introduction  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1  Settings .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='2  Main assumptions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='  4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='3  Earlier results in the literature .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  8  2010 Mathematics subject classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  35J15 35J25 35K10 35K20 60G50 60J65 60K37  74Q20 76M50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  random walks in a balanced random environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' non-divergence  form difference operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' invariant measures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' quantitative stochastic homogenization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' quantitative  large-scale average;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' optimal convergence rates .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  ∗The work of XG is supported by Simons Foundation through Collaboration Grant for Mathe-  maticians #852943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  †HT is supported in part by NSF CAREER grant DMS-1843320 and a Vilas Faculty Early-Career  Investigator Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  1  2  Large scale 퐶0,1 and 퐶1,1 estimates  11  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1  Some regularity properties of deterministic functions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='  12  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='2  Large scale regularity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='  14  3  Mixing properties of the invariant measure  17  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1  A sensitivity estimate of the invariant measure .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='  18  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='2  Rate of convergence for the average of the invariant measure: Proof  of Theorem 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='  21  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='3  Correlation structure of the field of the invariant measure .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='  24  4  Homogenization of the Dirichlet problem  25  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1  Homogenization of the approximate corrector .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='  25  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='2  Rate of homogenization for the Dirichlet problem (19)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='  29  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='2  A Berry-Esseen estimate for the QCLT: Proof of Corollary 10  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='  32  A Appendix  33  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1  Proof of Proposition 16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='  33  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='2  Verification of (68) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='  34  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='3  Verification of (70) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='  35  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='4  Verification of (78) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='  35  1  Introduction  In this paper, we consider random walks in a uniformly elliptic, balanced, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' ran-  dom environment in ℤ푑 for 푑 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Our main goals are two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Firstly, we derive  a quantitative large-scale average of the invariant measure, which is nearly optimal,  in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' A mixing property of the field of the invariant measure is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Secondly, we obtain rates of convergence for the homogenization of the Dirichlet  problem in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' When 푑 ≥ 5, the convergence rate is 푂(푅−1), which is  generically optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Afterwards, we quantify the ergodicity of the environmental  process for both the continuous- and discrete-time random walks in Theorem 8, and  as a consequence, we get explicit convergence rates for the quenched central limit  theorem (QCLT) of the balanced random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1  Settings  Let 핊푑×푑 denote the set of 푑 × 푑 positive-definite diagonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' A map  휔 ∶ ℤ푑 → 핊푑×푑  2  is called an environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' We denote the set of all environments by Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Let ℙ be a  probability measure on Ω so that  {휔(푥) = diag[휔1(푥), … , 휔푑(푥)], 푥 ∈ ℤ푑}  are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' under ℙ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Expectation with respect to ℙ is denoted by 피 or 퐸ℙ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Let {푒1, … , 푒푑} be the canonical basis for ℝ푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For any function 푢 ∶ ℤ푑 → ℝ  and 휔 ∈ Ω, define the non-divergence form difference operator  tr(휔(푥)∇2푢) =  푑  ∑  푖=1  휔푖(푥)[푢(푥 + 푒푖) + 푢(푥 − 푒푖) − 2푢(푥)],  (1)  where ∇2 = diag[∇2  1, … , ∇2  푑], and ∇2  푖 푢(푥) = 푢(푥 + 푒푖) + 푢(푥 − 푒푖) − 2푢(푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  For 푟 > 0, 푦 ∈ ℝ푑 we let  픹푟(푦) = {푥 ∈ ℝ푑 ∶ |푥 − 푦| < 푟} ,  퐵푟(푦) = 픹푟(푦) ∩ ℤ푑  denote the continuous and discrete balls with center 푦 and radius 푟, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  When 푦 = 0, we also write 픹푟 = 픹푟(0) and 퐵푟 = 퐵푟(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For any 퐵 ⊂ ℤ푑, its  discrete boundary is defined as  휕퐵 ∶=  {  푧 ∈ ℤ푑 ⧵ 퐵 ∶ dist(푧, 푥) = 1 for some 푥 ∈ 퐵  }  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Let ̄퐵 = 퐵 ∪ 휕퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' By abuse of notations, whenever confusion does not occur, we  also use 휕퐴 and ̄퐴 to denote the usual continuous boundary and closure of 퐴 ⊂ ℝ푑,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  For 푥 ∈ ℤ푑, a spatial shift 휃푥 ∶ Ω → Ω is defined by  (휃푥휔)(⋅) = 휔(푥 + ⋅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  In a random environment 휔 ∈ Ω, we consider the discrete elliptic Dirichlet problem  ⎧  ⎪  ⎨  ⎪⎩  1  2tr(휔∇2푢(푥)) =  1  푅2 푓  (  푥  푅  )  휓(휃푥휔)  푥 ∈ 퐵푅,  푢(푥) = 푔  (  푥  |푥|  )  푥 ∈ 휕퐵푅,  (2)  where 푓 ∈ ℝ픹1, 푔 ∈ ℝ휕픹1 are functions with good regularity properties and  휓 ∈ ℝΩ is bounded and satisfies suitable measurability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Stochastic ho-  mogenization studies (for ℙ-almost all 휔) the convergence of 푢 to the solution ̄푢 of  a deterministic effective equation  { 1  2tr( ̄푎퐷2 ̄푢) = 푓 ̄휓  in 픹1,  ̄푢 = 푔  on 휕픹1,  (3)  as 푅 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Here 퐷2 ̄푢 denotes the Hessian matrix of ̄푢 and ̄푎 = ̄푎(ℙ) ∈ 핊푑×푑 and  ̄휓 = ̄휓(ℙ, 휓) ∈ ℝ are deterministic and do not depend on the realization of the  random environment (see the statement of Theorem C for formulas for ̄푎 and ̄휓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  3  The difference equation (2) is used to describe random walks in a random en-  vironment (RWRE) in ℤ푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' To be specific, we set  휔(푥, 푥 ± 푒푖) ∶=  휔푖(푥)  2tr휔(푥)  for 푖 = 1, … 푑,  (4)  and 휔(푥, 푦) = 0 if |푥 − 푦| ≠ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Namely, we normalize 휔 to get a transition prob-  ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' We remark that the configuration of {휔(푥, 푦) ∶ 푥, 푦 ∈ ℤ푑} is also called a  balanced environment in the literature [40, 33, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For each fixed 휔 ∈ Ω, the random walk (푋푛)푛≥0 in the environment  휔 with 푋0 = 푥 is a Markov chain in ℤ푑 with transition probability 푃 푥  휔 specified by  푃 푥  휔  (푋푛+1 = 푧|푋푛 = 푦) = 휔(푦, 푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (5)  The expectation with respect to 푃 푥  휔 is written as 퐸푥  휔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' When the starting point of  the random walk is 0, we sometimes omit the superscript and simply write 푃 0  휔, 퐸0  휔  as 푃휔 and 퐸휔, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Notice that for random walks (푋푛) in an environment  휔,  ̄휔푖 = 휃푋푖휔 ∈ Ω,  푖 ≥ 0,  (6)  is also a Markov chain, called the environment viewed from the particle process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  By abuse of notation, we enlarge our probability space so that 푃휔 still denotes the  joint law of the random walks and ( ̄휔푖)푖≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  We also consider the continuous-time RWRE (푌푡) on ℤ푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Let (푌푡)푡≥0 be the Markov process on ℤ푑 with generator  퐿휔푢(푥) =  ∑  푦  휔(푥, 푦)[푢(푦) − 푢(푥)] =  1  2tr휔(푥)tr(휔(푥)∇2푢).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (7)  By abuse of notation, we also denote by 푃 푥  휔 the quenched law of (푌푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' If there  is no ambiguity from the context, we also write, for 푥, 푦 ∈ ℤ푑, 푛 ∈ ℤ, 푡 ∈ ℝ, the  transition kernels of the discrete and continuous time walks as  푝휔  푛 (푥, 푦) = 푃 푥  휔(푋푛 = 푦),  and  푝휔  푡 (푥, 푦) = 푃 푥  휔(푌푡 = 푦),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='2  Main assumptions  Throughout the paper, the following assumptions are always in force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (A1)  {  휔(푥), 푥 ∈ ℤ푑}  are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' under the probability measure ℙ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (A2)  휔  tr휔 ≥ 2휅I for ℙ-almost every 휔 and some constant 휅 ∈ (0, 1  2푑 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (A3) 휓 is a measurable function of the environment with the property that {휓(휃푥휔) ∶  푥 ∈ ℤ푑} are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' under ℙ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  4  In the paper, we use 푐, 퐶 to denote positive constants which may change from line  to line but only depend on the dimension 푑 and the ellipticity constant 휅 unless  otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' We write 퐴 ≲ 퐵 if 퐴 ≤ 퐶퐵, and 퐴 ≍ 퐵 if 퐴 ≲ 퐵 and 퐴 ≳ 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  We also use notations 퐴 ≲푗 퐵, 퐴 ≍푗 퐵 to indicate that the multiplicative constant  depends on the variable 푗 other than (푑, 휅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='3  Earlier results in the literature  We first recall the following quenched central limit theorem (QCLT) proved by  Lawler [40], which is a discrete version of Papanicolaou, Varadhan [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Assume (A2) and that law ℙ of the environment is ergodic under  spatial shifts {휃푥 ∶ 푥 ∈ ℤ푑}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Then  (i) There exists a probability measure ℚ ≈ ℙ such that ( ̄휔푖)푖≥0 is an ergodic (with  respect to time shifts) sequence under law ℚ × 푃휔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (ii) For ℙ-almost every 휔, the rescaled path 푋푛2푡∕푛 converges weakly (under law  푃휔) to a Brownian motion with covariance matrix ̄푎 = 퐸ℚ[휔∕tr휔] > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  QCLT for the balanced RWRE in static environments under weaker ellipticity  assumptions can be found at [33, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For dynamic balanced random environment,  QCLT was established in [23] and finer results concerning the local limit theorem  and heat kernel estimates was obtained at [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' When the RWRE is allowed to  make long jumps, non-CLT stable limits of the balanced random walk is considered  in [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' We refer to the lecture notes [13, 48, 12, 24, 38] for QCLT results in  different models of RWRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  We are moreover interested in characterizing the invariant measure ℚ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Denote  the Radon-Nikodym derivative of ℚ with respect to ℙ as  휌(휔) = dℚ∕dℙ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (8)  For any 푥 ∈ ℤ푑 and finite set 퐴 ⊂ ℤ푑, we define  휌휔(푥) ∶= 휌(휃푥휔)  and  휌휔(퐴) =  ∑  푥∈퐴  휌휔(푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  As an important feature of the non-divergence form model, 휌휔 does not have  deterministic (nonzero) upper and lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Moreover, the heat kernel 푝휔  푡 (⋅, ⋅)  is not expected to have deterministic Gaussian bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  For 푟 ≥ 0, 푡 > 0, define a function  픥(푟, 푡) =  푟2  푟 ∨ 푡 + 푟 log(푟  푡 ∨ 1),  푟 ≥ 0, 푡 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (9)  The following result was obtained by Guo, Tran [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Assume (A1), (A2), and 푑 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Let 푠 = 푠(푑, 휅) = 2 + 1  2휅 − 푑 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  For any 휀 ∈ (0, 1), there exists a random variable ℋ(휔) = ℋ(휔, 푑, 휅, 휀) > 0 with  피[exp(푐ℋ푑−휀)] < ∞ such that the following properties hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  5  (a) For ℙ-almost all 휔,  푐ℋ−푠 ≤ 휌(휔) ≤ 퐶ℋ푑−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (b) Recall the function 픥 in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For any 푟 ≥ 1 and ℙ-almost all 휔,  푐ℋ−푠 ≤ 푟푑휌휔(0)  휌휔(퐵푟) ≤ 퐶ℋ푑−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (c) For any 푥 ∈ ℤ푑, 푡 > 0, and ℙ-almost all 휔,  푝휔  푡 (푥, 0) ≤ 퐶ℋ푑−1(1 + 푡)−푑∕2푒−푐픥(|푥|,푡),  푝휔  푡 (푥, 0) ≥ 푐ℋ−푠(1 + 푡)−푑∕2푒−퐶|푥|2∕푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' In the PDE setting, positive and negative algebraic moment bounds and  volume doubling property of 휌 were proved by Bauman [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 퐿푝 The positive mo-  ment bound in (73) with 푞 =  푑  푑−1 was obtained by Lawler [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' The 퐿푝 integrability  of the heat kernel moment was proved by Fabes and Stroock [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Deterministic  heat kernel bounds in terms of 휌 was shown by Escauriaza [25] in the PDE setting,  and by Mustapha [42] for discrete time balanced random walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' In the more general  dynamic ergodic balanced environment setting, the bounds  푐휌휔(0)  휌휔(퐵√  푡)푒−퐶|푥|2∕푡 ≤ 푝휔  푡 (푥, 0) ≤ 퐶휌휔(0)  휌휔(퐵√  푡)푒−푐픥(|푥|,푡)  (10)  were proved by Deuschel, Guo [22, Theorem 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Recently, Armstrong, Fehrman,  Lin [2] obtain an algebraic rate of convergence for the heat kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  A function 휓 ∶ Ω → ℝ is said to be local if it is measurable and depends  only on the environment {휔(푥) ∶ 푥 ∈ 푆} in a finite set 푆 ⊂ ℤ푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' We now state a  quantitative homogenization result in Guo, Peterson, Tran [29, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='5], which  can be considered as a discrete version of Armstrong, Smart [4, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Assume (A1), (A2), and that the 휓 is a local function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Recall the  measure ℚ in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Suppose 푔 ∈ 퐶훼(휕픹1), 푓 ∈ 퐶훼(픹1) for some 훼 ∈ (0, 1],  and 휓 is a measurable function of 휔(0) with ‖휓∕tr휔‖∞ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Let ̄푢 be the solution  of the Dirichlet problem  { 1  2tr( ̄푎퐷2 ̄푢) = 푓 ̄휓  in 픹1,  ̄푢 = 푔  on 휕픹1,  with ̄푎 = 퐸ℚ[휔∕tr휔] > 0 being a positive-definite matrix and ̄휓 = 퐸ℚ[휓∕tr휔].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  For any 휀 ∈ (0, 1), there exists a random variable ℋ(휔) = ℋ(휔, 푑, 휅, 휀) > 0  with 피[exp(푐ℋ푑−휀)] < ∞ and a constant 훽 = 훽(푑, 휅, 휀) ∈ (0, 1) such that for any  푦 ∈ 퐵3푅, the solution 푢 of  { 1  2tr(휔∇2푢(푥)) =  1  푅2 푓(푥−푦  푅 )휓(휃푥−푦휔)  푥 ∈ 퐵푅(푦),  푢(푥) = 푔( 푥−푦  |푥−푦|)  푥 ∈ 휕퐵푅(푦)  (11)  6  satisfies, with 퐴1 = ‖푓‖퐶0,훼(픹1)‖  휓  tr(휔)‖∞ + [푔]퐶0,훼(휕픹1),  max  푥∈퐵푅(푦)  |||푢(푥) − ̄푢(푥−푦  푅 )|||≲ 퐴1(1 + (ℋ  푅 )1−휀∕푑)푅−훼훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (12)  When the balanced environment is allowed to be non-elliptic and genuinely 푑-  dimensional, (weak) quantitative results and Harnack inequalities for non-divergence  form difference operators are obtained by Berger, Cohen, Deuschel, Guo [9], and  Berger, Criens [11] for 휔-harmonic and 휔-caloric functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Let us also give a brief overview of the quantitative homogenization of non-  divergence form operators in the continuous PDE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Yurinski derived a second  moment estimate of the homogenization error in [47] for linear elliptic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Caf-  farelli, Souganidis [18] proved a logarithmic convergence rate for the fully nonlinear  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Afterwards, Armstrong, Smart [4], and Lin, Smart [41] achieved an algebraic  convergence rate for fully nonlinear elliptic equations, and fully nonlinear parabolic  equations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Armstrong, Lin [3] obtained quantitative estimates for the  approximate corrector problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  For 푑 ≥ 2 and any finite set 퐴 ⊂ ℤ푑, denote the exit time from 퐴 by  휏(퐴) = 휏(퐴;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 푋) = inf{푛 ≥ 0 ∶ 푋푛 ∉ 퐴}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (13)  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For 푅 ≥ 1, 휔 ∈ Ω, 푥 ∈ ℤ푑, 푆 ⊂ ℤ푑, the Green function 퐺푅(⋅, ⋅) in  the ball 퐵푅 for the balanced random walk is defined by  퐺푅(푥, 푆) = 퐺휔  푅(푥, 푆) ∶= 퐸푥  휔  [  ∫  휏(퐵푅)  0  1푌푡∈푆d푡],  푥 ∈ ̄퐵푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  We also write 퐺푅(푥, 푦) ∶= 퐺휔  푅(푥, {푦}) and 퐺푅(푥) ∶= 퐺푅(푥, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' When 푑 ≥ 3, for  any finite set 푆 ⊂ ℤ푑, the Green function on the whole space can be defined as  퐺휔(푥, 푆) = ∫  ∞  0  푝휔  푡 (푥, 푆)d푡 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  When 푑 = 2, for any 푥, 푦 ⊂ ℤ푑, the potential kernel is defined as  퐴(푥, 푦) = 퐴휔(푥, 푦) = ∫  ∞  0  [푝휔  푡 (푦, 푦) − 푝휔  푡 (푥, 푦)]d푡,  푥 ∈ ℤ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (14)  The bounds for the Green functions and the potential kernel were proved in Guo,  Tran [31], which was based on the idea of Armstrong, Lin [3, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Assume (A1), (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For 휀 > 0, let 푠 > 0, ℋ = ℋ(휔, 푑, 휅, 휀) > 0 be  as in Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For 푟 > 0, let  푈(푟) ∶=  { − log 푟  푑 = 2,  푟2−푑  푑 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (15)  7  Then ℙ-almost surely, for all 푥 ∈ 퐵푅,  ℋ−푠[푈(|푥| + 1) − 푈(푅 + 2)] ≲ 퐺휔  푅(푥, 0) ≲ ℋ푑−1[푈(|푥| + 1) − 푈(푅 + 2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  As consequences, ℙ-almost surely, for all 푥 ∈ ℤ푑,  ℋ−푠 log(|푥| + 1) ≲ 퐴휔(푥, 0) ≲ ℋ log(|푥| + 1), when 푑 = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  ℋ−푠(1 + |푥|)2−푑 ≲ 퐺휔(푥, 0) ≲ ℋ푑−1(1 + |푥|)2−푑, when 푑 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Recall the continuous time RWRE (푌푡)푡≥0 in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Define the semi-  group 푃푡, 푡 ≥ 0, on ℝΩ by  푃푡휁(휔) = 퐸0  휔[휁(휃푌푡휔)] =  ∑  푧  푝휔  푡 (0, 푧)휁(휃푧휔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (16)  The following theorem from Guo, Tran [31] estimates the optimal speed of decor-  relation of the environmental process ̄휔푡 from the original environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Assume (A1), (A2), and 푑 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For any local measurable function  휁 ∶ Ω → ℝ with ‖휁‖∞ ≤ 1 and 푡 ≥ 0, we have  Varℚ(푃푡휁) ≤ 퐶(1 + 푡)−푑∕2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (17)  ‖푃푡휁‖1 + ‖푃푡휁 − 피[푃푡휁]‖푝 ≤ 퐶푝(1 + 푡)−푑∕4  for all 푝 ∈ (0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (18)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='4  Main results  The field {휌휔(푥) ∶ 푥 ∈ ℤ푑} of the invariant measure, which governs the long term  behavior of the diffusion and which determines the effective PDE, plays a central  role in the theory of homogenization of non-divergence form equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  We first obtain a rate of convergence of the average 휌휔(퐵푅)∕|퐵푅| of the invari-  ant measure to 1 as 푅 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Assume (A1), (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For any 푑 ≥ 2, 푝 ∈ (0, 2  3), 푡 > 0 and 푅 ≥ 2,  푃  (|||  휌휔(퐵푅)  |퐵푅| − 1|||≥ 푡푅−푑∕2 log 푅  )  ≤ 퐶푝 exp(−푐푡푝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Note that the rate 푅−푑∕2 log 푅 is very close to the size 푅−푑∕2 of the diffusive  scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' In other words, to some extent the field (휌휔(푥))푥∈ℤ푑 behaves quite simi-  larly to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Hence, we expect the rate 푅−푑∕2 log 푅 obtained  here to be either optimal or nearly optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For non-divergence form PDEs, the  volume-doubling property for the measure 휌휔(⋅) was proved by Bauman [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' An  algebraic convergence rate 푅−훾 for some 훾 ∈ (0, 1) was proved recently by Arm-  strong, Fehrman, Lin [2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  In the course of obtaining our homogenization results in this paper, sensitivity  estimates together with an Efron-Stein type inequality are used to control fluctu-  ations of a random field around its mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' This method was used in the stochastic  8  homogenization of divergence-form operators, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=', [43, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' To facilitate this strat-  egy, obtaining sensitivity estimates (with respect to the change of the environment)  is crucial, and 퐶1,1 estimates for the random equation is necessary, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=', [28, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  To obtain 퐶1,1 regularity for the heterogeneous equation, we follow the idea of  Armstrong, Lin [3] who generalized the compactness argument of Avellaneda, Lin  [5] to the random non-divergence form setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  The key observation in the proof of Theorem 5 is explained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Al-  though the invariant measure 휌휔(푥) does not have an explicit expression, it can be  interpreted as the long term frequency of visits to location 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Hence, modifying  the local value of the environment is related to the Green function of the RWRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Guided by this intuition, we will obtain a formula for the sensitivity estimate of the  invariant measure in terms of the Green function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  As indicated in Theorem 5, the field {휌휔(푥) ∶ 푥 ∈ ℤ푑} is expected to have  weak enough correlation so that the behavior of its mean fluctuation over 퐵푅 re-  sembles (up to a logarithmic factor) that of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' The following  proposition reveals the mixing property of the field of the invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Assume (A1), (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For any 푥, 푦 ∈ ℤ푑 with 푥 ≠ 푦,  ||Covℙ(휌(푥), 휌(푦))|| ≲  { |푥 − 푦|−1(1 + log |푥 − 푦|),  푑 = 2  |푥 − 푦|−푑∕2,  푑 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  This is perhaps the first characterization of the correlation structure of the field  of the invariant measure (with algebraic mixing rates) in a balanced environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Next, we derive rates of convergence for the homogenization of the Dirichlet  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Assume (A1), (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Consider the Dirichlet problem  {  퐿휔푢 =  1  푅2 푓( 푥  푅),  푥 ∈ 퐵푅,  푢(푥) = 푔( 푥  푅),  푥 ∈ 휕퐵푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (19)  Assume that 푓, 푔 are both in 퐶4(ℝ푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For any 0 < 푠 <  푑  3푑+2, there exists 퐶 =  퐶(푑, 휅, 푠) such that, for 푅 ≥ 2, there exists a random variable 풵 = 풵(푅, 푠, 휔) > 0  with 피[exp(풵푠)] < 퐶 and  max  푥∈퐵푅  |푢(푥) − ̄푢( 푥  푅)| ≲ ‖̄푢‖퐶4( ̄픹1)휏(푅)풵,  where  휏(푅) =  ⎧  ⎪  ⎨  ⎪⎩  푅−2∕3,  푑 = 2  푅−6∕7,  푑 = 3  푅−1(log 푅)1∕4,  푑 = 4  푅−1,  푑 ≥ 5,  (20)  and ̄푢 is the solution of  { 1  2tr( ̄푎퐷2 ̄푢) = 푓  in 픹1,  ̄푢 = 푔  on 휕픹1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  9  Thus, for 푑 ≥ 5, we obtain the optimal rate of convergence for the homogeniza-  tion of the Dirichlet problem, which is generically of scale 푅−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' This is consistent  with the generically optimal rate 푅−1 for the periodic setting (see the classical books  [8, 36] for the derivation, and [32, 45, 30] for discussions on the optimality of the  rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=') It is not clear to us what the optimal rates are when 2 ≤ 푑 ≤ 4, which deserve  further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  To prove Theorem 7, we apply the classical method of two-scale expansions  and the quantitative homogenization of the approximation corrector (Lemma 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  The continuous version of Lemma 21 was proved earlier in the PDE setting by  Armstrong, Lin [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Note that the effective matrix ̄푎 = 퐸ℚ[휔∕tr휔] does not have an explicit expres-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Even though by Birkhoff’s ergodic theorem, ℚ can be approximated qualita-  tively by  lim  푛→∞  1  푛  푛−1  ∑  푖=0  퐸휔[휓(휃푋푖휔)] = 퐸ℚ[휓]  ℙ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  for any 퐿1 function 휓 on environments, in order to better understand the effective  matrix ̄푎 it is important to quantify the speed of this convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' To this end, we  set, for 푇 ≥ 1,  휈(푇 ) =  ⎧  ⎪  ⎨  ⎪⎩  푇 −1∕2  푑 = 2  푇 −3∕4  푑 = 3  푇 −1(log 푇 )1∕2  푑 = 4  푇 −1  푑 ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (21)  We will quantify the ergodicity of the environmental process for both the continuous-  and discrete-time random walks in a balanced random environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Assume (A1), (A2), (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Let 휈 be as in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For any 0 < 푝 <  2푑  3푑+2,  there exists 퐶 = 퐶(푑, 휅, 푝) such that for 푇 , 푛 ≥ 2 and any 푡 ≥ 0,  ℙ  (  |||  1  푇 퐸휔  [  ∫  푇  0  휓(휃푌푠휔)d푠  ]  − ̄휓|||≥ 푡휈(푇 )‖휓‖∞  )  ≤ 퐶 exp(−푡푝  퐶 ),  ℙ  (  |||  1  푛퐸휔  [ 푛−1  ∑  푖=0  휓(휃푋푖휔)  ]  − ̄휓|||≥ 푡휈(푛)‖휓‖∞  )  ≤ 퐶 exp(−푡푝  퐶 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Recall that Theorem E (from [31]) states that, when 푑 ≥ 3, the typical  size of 푃푡휓 − ̄휓 is of scale 푡−푑∕4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Observe that the typical size 휈(푇 ) of the ergodic  average in Theorem 8 satisfies (for 푇 ≥ 2)  휈(푇 ) ≲ 1  푇 ∫  푇  1  푡−푑∕4d푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (The sign ≲ can be replaced by ≍ except for 푑 = 4 when 휈(푇 ) is a  √  log 푇 factor  smaller than the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=') Hence, Theorem 8 can be regarded as the integral ver-  sion of Theorem E which holds for all 푑 ≥ 2 and which has much better exponential  integrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  10  We remark that an (unknown) algebraic rate for the convergence of the ergodic  average was obtained in [29, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='2] in the discrete setting and recently in [2,  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='6] in the PDE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  As a consequence of Theorem 8, we obtain explicit convergence rates for the  QCLT of the balanced random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Assume (A1), (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For any 0 < 푞 <  2푑  5(3푑+2), 푛 ≥ 2, there exists a  random variable 풴 = 풴(휔, 푞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 푛, 휅, 푑) with 피[exp(풴푞)] ≤ 퐶 such that, ℙ-almost  surely, for any unit vector 퓁 ∈ ℝ푑,  sup  푟∈ℝ  |||푃휔  (  푋푛 ⋅ 퓁∕  √  푛 ≤ 푟  √  퓁푇 ̄푎퓁  )  − Φ(푟)|||≤ 퐶휈(푛)1∕5풴,  where Φ(푟) = (2휋)−1∕2 ∫ 푟  −∞ 푒−푥2∕2d푥 for 푟 ∈ ℝ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  An algebraic rate for the QCLT was proved in [29, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' We remark  that for the model of random walk in random conductances, algebraic rates similar  to ours was proved in [1] for dimensions 푑 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  2  Large scale 퐶0,1 and 퐶1,1 estimates  In this section, we apply the ideas of Avellaneda, Lin [5, 6] in the periodic setting  to the discrete random setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' The key idea is quite intuitive and natural: large-  scale solutions of 퐿휔푢(푥) = 휓(휃푥휔) + 푓(푥) are well-approximated by those of the  homogenized equation with an algebraic rate thanks to Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' As the latter  are harmonic, they possess rather nice estimates (see Proposition 16 below on the  scaling property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=') Therefore, by iterations, scalings and the triangle inequalities,  the better regularity of the homogenized equation is inherited by the heterogeneous  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' It is crucial to note two points here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Firstly, in each iteration step, the  scaling is done by using the nice estimates in Proposition 16 of the homogenized  limit, and the triangle inequality and Proposition C are used to pass this estimate to  the solution 푢 of the heterogeneous equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Secondly, in the random setting, one  can only go down to radii greater than the homogenization radius in the iterations,  which therefore gives us only large scale estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' The generalization of this idea  to the random non-divergence form PDE setting was first done by Armstrong, Lin  [3] who made the observation that an algebraic rate is sufficient for such an iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  The main result in this section, Theorem 14, can be considered as a discrete  version of Armstrong, Lin [3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1,Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='4] in terms of the large scale  퐶0,1 and 퐶1,1 regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' We remark that for 휔-harmonic functions in a genuinely  푑-dimensional balanced environment, a 퐶0,1−휀 regularity was achieved by Berger,  Cohen, Deuschel, Guo [9, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='4] using coupling arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  As can be seen in the following Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1, this sort of compactness argu-  ment, although is applied to the random setting here, is deterministic in its core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  11  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1  Some regularity properties of deterministic functions  This subsection contains the key tools for the compactness arguments used in our  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' It is completely deterministic and can be read independently of other parts of  the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' The lemmas presented here concern large scale 퐶푘,1, 푘 ≥ 0, properties  of deterministic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  For any function 푓 on a set 퐴 and 훼 ∈ (0, 1], define  osc  퐴 푓 ∶= sup  푥,푦∈퐴  |푓(푥) − 푓(푦)|,  [푓]훼;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐴 =  sup  푥,푦∈퐴,푥≠푦  |푓(푥) − 푓(푦)|  |푥 − 푦|훼  ,  and, if 퐴 is a finite set, for 푝 ∈ (0, ∞), we define  ‖푓‖푝;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐴 ∶=  (  1  #퐴  ∑  푥∈퐴  |푓|푝  )1∕푝  ,  ‖푓‖∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐴 = max  푥∈퐴 |푓(푥)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  For any 푗 ≥ 0, let H푗 denote the set of 푗-th order polynomials, with H0 = ℝ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' In  fact, in our paper we will be only use the cases 푗 = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Define, for function 푓 ∶ ℝ푑 → ℝ and a bounded set 퐴 ⊂ ℝ푑, 푗 ≥ 1,  \ue230푗  퐴(푓) =  inf  푝∈H푗−1  sup  퐴  |푓 − 푝| = 1  2  inf  푝∈H푗−1  osc  퐴 (푓 − 푝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (22)  \ue230푗  퐴 satisfies the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Namely, \ue230푗  퐴(푓 ± 푔) ≤ \ue230푗  퐴(푓) + \ue230푗  퐴(푔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' When  퐴 = 퐵푅 is the discrete ball, 푅 > 0, we simply write  \ue230푗  푅 ∶= \ue230푗  퐵푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Note that for 푗 ≥ 1, the above term normalized  픻푗  푅(푓) ∶=  \ue230푗  푅(푓)  푅푗  (23)  is a large scale analogue of the 푗-th order derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  For any 푟 > 0, 휃 ∈ (0, 1  3), define a sequence of exponentially increasing radii  (푟푘)푘≥0 by  푟푘 = 푟푘(푟, 휃) ∶= 휃−푘푟,  푘 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  The following elementary lemma confirms the intuition that “the integral of the  (푗 + 1)-th derivative is the 푗-th derivative".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For any function 푓 ∶ ℤ푑 → ℝ and 푟 > 0, 휃 ∈ (0, 1  3), 푛 ∈ ℕ, 푗 ≥ 1,  픻푗  푟0(푓) ≤ 픻푗  푟푛(푓) + 3휃−푗  푛  ∑  푘=0  푟푘픻푗+1  푟푘 (푓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  12  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For any 푗-th order homogeneous polynomials 푝, 푞 ∈ H푗, 푅 > 푟 > 0, by the  triangle inequality,  \ue230푗  푟(푝) ≤ \ue230푗  푟(푞) + \ue230푗  푟(푝 − 푞)  ≤ ( 푟  푅)푗\ue230푗  푅(푞) + \ue230푗  푟(푓 − 푝) + \ue230푗  푟(푓 − 푞)  where in the second inequality we used the fact that \ue230푗  푟(푞) = ( 푟  푅)푗\ue230푗  푅(푞) for all 푗-th  order homogeneous polynomial 푞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Hence, by the inequality above,  \ue230푗  푟(푓) ≤ \ue230푗  푟(푓 − 푝) + \ue230푗  푟(푝)  ≤ 2\ue230푗  푟(푓 − 푝) + \ue230푗  푟(푓 − 푞) + ( 푟  푅)푗\ue230푗  푅(푞)  ≤ 2\ue230푗  푟(푓 − 푝) + \ue230푗  푟(푓 − 푞) + ( 푟  푅)푗[\ue230푗  푅(푓) + \ue230푗  푅(푓 − 푞)]  ≤ 2[\ue230푗  푟(푓 − 푝) + \ue230푗  푅(푓 − 푞)] + ( 푟  푅)푗\ue230푗  푅(푓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Taking infimum over all 푗-th order homogeneous polynomials 푝, 푞 ∈ H푗, we get  \ue230푗  푟(푓) ≤ 2[\ue230푗+1  푟  (푓) + \ue230푗+1  푅 (푓)] + ( 푟  푅)푗\ue230푗  푅(푓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Replacing 푟, 푅 by 푟푘, 푟푘+1 , and using notation (23), the above inequality yields  픻푗  푟푘(푓) − 픻푗  푟푘+1(푓) ≤ 2[푟푘픻푗+1  푟푘 (푓) + 휃−푗푟푘+1픻푗+1  푟푘+1(푓)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Summing both sides over 푘 = 0, … , 푛 − 1, the lemma is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  The following lemma will be crucially employed later in our derivation of large  scale regularity estimates in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Let 푗 ≥ 1, 푚 ∈ ℕ, 푟, 훼 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Let 퐴푟 ≥ 0 be a constant depending on 푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  If for 푓 ∶ ℤ푑 → ℝ , 푘 = 0, … , 푚 − 1, and all 휃 ∈ (0, 1  3),  \ue230푗+1  푟푘 (푓) ≲푗 휃푗+1\ue230푗+1  푟푘+1(푓) + 푟−훼  푘+1\ue230푗  푟푘+1(푓) + 푟푗  푘+1퐴푟푘+1,  (24)  then there exists 휃 = 휃(푗), 푁 = 푁(푗, 훼) such that, for 푁 ≤ 푟 ≤ 푅 ≤ 푟푚,  \ue230푗  푟(푓) ≤ 13휃−2푗 (  푟  푅  )푗  \ue230푗  푅(푓) +  ∑  푘≥1∶푟푘≤푅  퐴푟푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For the simplicity of notations, we suppress the dependency on 푓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Let 푛 =  푛(푅, 휃) ≤ 푚 be such that 푟푛 ≤ 푅 < 푟푛+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Display (24) is equivalent to  푟푘픻푗+1  푟푘  ≲푗 휃푟푘+1픻푗+1  푟푘+1 + 휃−푗푟−훼  푘+1픻푗  푟푘+1 + 휃−푗퐴푟푘+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Summing this inequality over 푘 = 0, … , 푛 − 1, we have  푛−1  ∑  푘=0  푟푘픻푗+1  푟푘  ≲푗 휃  푛  ∑  푘=1  푟푘픻푗+1  푟푘  + 휃−푗  푛  ∑  푘=1  푟−훼  푘 픻푗  푟푘 + 휃−푗  푛  ∑  푘=1  퐴푟푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (25)  13  Moreover, by Lemma 11, 픻푗  푟푘 ≤ 픻푗  푟푛 + 3휃−푗 ∑푛  퓁=푘 푟퓁픻푗+1  푟퓁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Hence  푛  ∑  푘=1  푟−훼  푘 픻푗  푟푘 ≤  푛  ∑  푘=1  푟−훼  푘  (  픻푗  푟푛 + 3휃−푗  푛  ∑  퓁=푘  푟퓁픻푗+1  푟퓁  )  ≤ 퐶훼푟−훼픻푗  푟푛 + 퐶훼휃−푗푟−훼  푛  ∑  퓁=1  푟퓁픻푗+1  푟퓁 ,  (26)  where 퐶훼 = 1 − 3−훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Choosing 휃 = 휃(푗) ∈ (0, 1  3) sufficiently small, when 푟 ≥ 푁  for some 푁 = 푁(푗, 훼), we get from (25) and (26) that  푛−1  ∑  푘=0  푟푘픻푗+1  푟푘  ≤ 1  2  푛  ∑  푘=1  푟푘픻푗+1  푟푘  + 픻푗  푟푛 + 퐶푗휃−푗  푛  ∑  푘=1  퐴푟푘  which implies (Note that 푟푛픻푗+1  푟푛  ≤ 픻푗  푟푛)  푛  ∑  푘=0  푟푘픻푗+1  푟푘  ≤ 4픻푗  푟푛 + 퐶푗휃−푗  푛  ∑  푘=1  퐴푟푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  This inequality, together with Lemma 11, yields for 푟 ≥ 푁, 휃 = 휃(푗) ∈ (0, 1  3),  픻푗  푟0 ≤ 13휃−푗픻푗  푟푛 + 퐶푗휃−2푗  푛  ∑  푘=1  퐴푟푘 ≤ 13휃−2푗픻푗  푅 +  푛  ∑  푘=1  퐴푟푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  The lemma is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Remark 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' In this subsection we consider 푓 as a function on ℤ푑 and defined H푗  to be the set of 푗-th order polynomials just for our convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' One may let 푓 be  a function on ℝ푑 and redefine H푗’s to be other sub-spaces of the polynomials (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=',  the set of harmonic polynomials) and Lemmas 11, 12 still hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='2  Large scale regularity  The goal of this section is to apply Lemma 12 to obtain 퐶0,1 and 퐶1,1 regularities  for the heterogeneous equations in our random setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Define operators ∇ = (∇푖)1≤푖≤푑 and ∇∗ = (∇∗  푖 )1≤푖≤푑 by  ∇푖푢(푥) = 푢(푥 + 푒푖) − 푢(푥),  ∇∗  푖 푢(푥) = 푢(푥 − 푒푖) − 푢(푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Note that ∇푖 and ∇∗  푖 are adjoint linear operators, and ∇2  푖 = −∇푖∇∗  푖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Assume (A1), (A2), and that 휓 is a local function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Let 푅 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' There  exists 훼 = 훼(푑, 휅) ∈ (0, 1  3) such that, for any any 푢 with 퐿휔푢(푥) = 휓(휃푥휔) + 푓(푥)  on 퐵푅, 푗 ∈ {1, 2}, ℋ ≤ 푟 < 푅,  1  푟푗  inf  푝∈H푗−1  osc  퐵푟  (푢 − 푝) ≲ 1  푅푗  inf  푝∈H푗−1  osc  퐵푅  (푢 − 푝) + 퐴푗,  (27)  14  where the terms 퐴푗 = 퐴푗(푅, 푟) have the following bounds (for any 휎 ∈ (0, 1])  퐴1 ≤ 푅1−훼‖휓‖∞ + 푅‖푓‖∞ and 퐴1 ≤ 푅1−훼‖휓 + 푓(0)‖∞ + 푅1+휎[푓]휎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅,  퐴2 ≤ 푟−훼‖휓‖∞ + log(푅  푟 )‖푓‖∞ and 퐴2 ≤ 푟−훼‖휓 + 푓(0)‖∞ + 푅휎[푓]휎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  In particular, recalling the operator ∇2  푖 in (1), for any 푅 > 1, 푗 = 1, 2,  |∇푗푢(0)| ≲ (ℋ  푅 )푗 (  ‖푢‖1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅 + 푅2‖휓 + 푓(0)‖∞ + 푅2+휎[푓]휎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  )  (28)  As a consequence of (28), any 휔-harmonic function on ℤ푑 with sublinear growth  is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' That is, if 퐿휔푢 = 0 on ℤ푑 and max퐵푅 |푢| = 표(푅) for all 푅 > 0, then  푢 is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' To prove Theorem 14, it suffices to prove the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' There exists 훾 = 훾(푑, 휅) such that, for 푅 ≥ ℋ, 휃 ∈ (0, 1  3), 1 ≤ 푗 ≤ 3  and any 푢 with 퐿휔푢(푥) = 휓(휃푥휔) + 푓(푥) for 푥 ∈ 퐵푅, we have  \ue230푗  휃푅(푢) ≲ 푅−훾훽\ue2302  푅(푢) + 휃푗\ue230푗  푅(푢) + 푅2−훾훽‖휓‖∞ + 푅2‖푓‖푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  The proof of Lemma 15 uses the following fact of deterministic harmonic func-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For completeness, we include its proof in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Recall the notation in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Let 푐0 be a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Let 푣 be a function  satisfying tr ̄푎퐷2푣 = 푐0 in ̄픹푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Then, for 휃 ∈ (0, 1  3), 푗 ∈ {1, 2, 3} and 푅 ≥ 1,  \ue230푗  픹휃푅(푣) ≤ 퐶휃푗\ue230푗  픹푅∕2(푣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (29)  We also have  \ue230푗  픹휃푅(푣) ≲ 휃푗  푅 ( sup  휕픹2푅∕3  |푣| + 푅2|푐0|) + 휃푗\ue230푗  2푅∕3(푣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (30)  Remark 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Property (29) for deterministic harmonic functions (푐0 = 0) was used  in [3, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='3] to obtain regularities of the heterogeneous solution in the PDE  setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Comparing to [3, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='4], here by allowing 푐0 ≠ 0 we will gain a  tiny improvement for the coefficient of ‖휓 +푓(0)‖∞ in the 퐶0,1 estimate by an 푅−훼  factor (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Theorem 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Note that in the discrete setting, we will need (30) as well  because of discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' It would be also clear later in Section 3 that the log 푅  factor in the bound of 퐴2 will help us achieve the log 푅 factor in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Proof of Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' By the Hölder estimate of Krylov-Safonov, there exists 훾 =  훾(푑, 휅) > 0 such that, for 푟 ∈ (0, 푅),  osc  퐵푟  푢 ≲  (  푟  푅  )훾  (osc  퐵푅  푢 + 푅2‖휓 + 푓‖푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (31)  Note that this allows us to extend 푢 to be a function ̃푢 ∈ 퐶훾(ℝ푑) with [̃푢]훾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='ℝ푑 =  [푢]훾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' ̄퐵2푅∕3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Indeed, define the function ̃푢 as  ̃푢(푥) =  min  푦∈ ̄퐵2푅∕3  {  푢(푦) + |푥 − 푦|휎[푢]휎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' ̄퐵2푅∕3  }  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  15  It is straightforward to check that ̃푢 = 푢 in ̄퐵2푅∕3 and [̃푢]훾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='ℝ푑 ≤ [푢]훾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' ̄퐵2푅∕3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' By (31),  [̃푢]훾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='ℝ푑 = [푢]훾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' ̄퐵2푅∕3 ≲ 푅−훾(max  퐵푅  |푢| + 푅2‖휓 + 푓‖푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (32)  Let ̄푣 ∶ ̄픹2∕3 → ℝ be the solution of  {  1  2tr( ̄푎퐷2 ̄푣) = 푅2 ̄휓  in 픹2∕3  ̄푣(푥) = ̃푢(푅푥)  for 푥 ∈ 휕픹2∕3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  First, write 퐴 ∶= 푅2−훾훽‖휓‖∞ + 푅2‖푓‖푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' We will show that, for 푅 ≥ ℋ,  max  푥∈퐵2푅∕3  |푢(푥) − ̄푣( 푥  푅)| ≲ 푅−훾훽 max  퐵푅  |푢| + 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (33)  To this end, let 푢1 ∶ ̄퐵2푅∕3 → ℝ be the solution of  {  퐿휔푢1 = 휓(휃푥휔)  in 퐵2푅∕3  푢1(푥) = ̃푢(2푅푥  3|푥| )  푥 ∈ 휕퐵2푅∕3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  By Proposition C and (32), when 푅 ≥ ℋ, noting that [̃푢(푅⋅)]훾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='ℝ푑 ≤ 푅훾[̃푢(⋅)]훾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='ℝ푑,  max  푥∈퐵2푅∕3  |푢1(푥) − ̄푣( 푥  푅)| ≲ 푅−훾훽([̃푢(푅⋅)]훾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='ℝ푑 + 푅2‖휓‖∞)  ≲ 푅−훾훽(max  퐵푅  |푢| + 푅2‖휓‖∞ + 푅2‖푓‖푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Moreover, by the ABP maximum principle,  max  퐵2푅∕3  |푢 − 푢1| ≤  max  푥∈휕퐵2푅∕3  |푢(푥) − ̃푢(2푅푥  3|푥| )| + 퐶푅2‖푓‖푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅  ≲ [̃푢]훾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='ℝ푑 + 푅2‖푓‖푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅  (32)  ≲ 푅−훾(max  퐵푅  |푢| + 푅2‖휓‖∞) + 푅2‖푓‖푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Combining the two inequalities above, display (33) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  By the triangle inequality and Proposition 16, for 1 ≤ 푗 ≤ 3,  \ue230푗  휃푅(푢) ≤ max  퐵푅∕2  |푢 − ̄푣( ⋅  푅)| + \ue230푗  휃푅( ̄푣( ⋅  푅))  ≤ max  퐵푅∕2  |푢 − ̄푣( ⋅  푅)| + 퐶휃푗  푅 ( sup  휕픹2∕3  | ̄푣| + 푅2| ̄휓|) + 퐶휃푗\ue230푗  2푅∕3( ̄푣( ⋅  푅))  ≲ max  퐵2푅∕3  |푢 − ̄푣( ⋅  푅)| + 휃푗  푅 ( sup  휕픹2∕3  | ̄푣| + 푅2| ̄휓|) + 휃푗\ue230푗  2푅∕3(푢).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (34)  Since sup휕픹2∕3 | ̄푣| = sup휕픹2푅∕3 |̃푢| ≤ max퐵2푅∕3 |푢| + [̃푢]훾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' ̄퐵2푅∕3 ≤ max퐵푅 |푢| + 퐴, by  (33) and (34), we have, for 1 ≤ 푗 ≤ 3,  \ue230푗  휃푅(푢) ≲ 푅−훾훽 max  퐵푅  |푢| + 퐴 + 휃푗\ue230푗  푅(푢).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  16  Finally, note that since every 푝 ∈ H1 is 휔-harmonic, (푢 − 푝) still solves 퐿휔(푢 −  푝) = 휓(휃푥휔) + 푓(푥) for 푥 ∈ 퐵푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Therefore, substituting 푢 by (푢 − 푝) in the above  inequality and optimizing over 푝 ∈ H1, the lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Proof of Theorem 14: By Lemma 15 and Lemma 12, there exists 휃 = 휃(푑, 휅) ∈  (0, 1  3) such that (27) holds with the terms 퐴푗, 푗 ∈ {1, 2} satisfying  퐴푗 =  ∑  푘≥1∶푟푘≤푅  푟2−훼−푗  푘  ‖휓‖∞ + 푟2−푗  푘  ‖푓‖푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푟푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Note that ‖푓 − 푓(0)‖푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푟 ≲ 푟휎[푓]휎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푟 for all 휎 ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' The bounds of 퐴1, 퐴2 in  the theorem follow immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  To prove (28), note that |∇푢(0)| ≤ osc ̄퐵1 푢 and that for any 퓁 ∈ H1, |∇2푢(0)| =  |∇2(푢 − 퓁)| ≲ osc ̄퐵1(푢 − 퓁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Hence, by (27), we get  |∇푗푢(0)| ≲ (ℋ  푅 )푗  (  osc  퐵푅∕2  푢 + 푅2‖휓 + 푓(0)‖∞ + 푅2+휎[푓]휎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅∕2  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  By the Harnack inequality and the ABP inequality, we have  osc  퐵푅∕2  푢 ≲ ‖푢 − 푢퐵푅‖1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅 + 푅2‖휓 + 푓‖푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (35)  Display (28) follows by using again ‖푓−푓(0)‖푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅 ≲ 푅휎[푓]휎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅 for 휎 ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  3  Mixing properties of the invariant measure  The goal of this section is to investigate the mixing properties of the field {휌휔(푥) ∶  푥 ∈ ℤ푑} of the invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' We will obtain a rate of convergence (Theorem  5) of the average of the invariant measure over balls 퐵푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' We will also quantify the  correlation of the field (Proposition 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  The Efron-Stein inequality (38) of Boucheron, Bousquet, and Massart [14] will  be used in our derivation of quantitative estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Let 휔′(푥), 푥 ∈ ℤ푑, be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' copies of 휔(푥), 푥 ∈ ℤ푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For any 푦 ∈ ℤ푑, let 휔′  푦 ∈ Ω  be the environment such that  휔′  푦(푥) =  {  휔(푥)  if 푥 ≠ 푦,  휔′(푦)  if 푥 = 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  That is, 휔′  푦 is a modification of 휔 only at location 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For any measurable function  푍 of the environment 휔, we write, for 푦 ∈ ℤ푑,  푍′  푦 = 푍(휔′  푦),  휕′  푦푍(휔) = 푍′  푦 − 푍,  (36)  and set  푉 (푍) =  ∑  푦∈ℤ푑  (휕′  푦푍)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (37)  17  With abuse of notations, we enlarge the probability space and still use ℙ to denote  the distribution of both 휔, 휔′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  The 퐿푝 version of Efron-Stein inequality in [14, Theorem 3] states that, for  푞 ≥ 2,  피[|푍 − 피푍|푞] ≤ 퐶푞푞∕2피[푉 푞∕2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (38)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1  A sensitivity estimate of the invariant measure  The main contribution of this subsection is a formula for the “vertical" derivative  of the invariant measure 휌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Definition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For 푡 > 0, we let 푉 (푡, 휔) = ∑  푥∈ℤ푑 푝휔  푡 (푥, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Let ℚ푡 be the probability  measure on Ω defined by  ℚ푡(d휔) = 푉 (푡, 휔)ℙ(d휔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  We remark that by Theorem B,  푉 (푡, 휔) ≲ ℋ푑−1  for ℙ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 휔  (39)  and so ℚ푡 is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Note that for any bounded measurable function 휁 on Ω,  we have 퐸ℚ푡[휁] = 피[푃푡휁].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' In other words, ℚ푡 is the distribution of the environment  viewed from the particle at time 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' It is natural to expect that ℚ푡 → ℚ as 푡 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  For any function 푢 of the environment, we denote by 푢휔 the corresponding func-  tion on ℤ푑 defined by 푢휔(푥) ∶= 푢(휃푥휔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' As 푡 → ∞, ℚ푡 converges weakly to ℚ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Since {ℚ푡} is a sequence of probability measures on the compact space Ω,  it has a weak convergent subsequence {ℚ푡푘} which has a weak limit ℚ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  To prove that ℚ∞ is an invariant measure for the Markov chain (휃푌푡휔), it suffices  to show that for any bounded measurable function 푓 on Ω,  퐸ℚ∞[퐿휔푓휔(0)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Indeed, by the translation invariance of the measure ℙ, for any 푒 with |푒| = 1,  피[휔(0, 푒)푉 (푡, 휔)푓(휃푒휔)] = 피[휔(−푒, 0)푉 (푡, 휃−푒휔)푓(휔)]  = 피[휌휔(0)휔∗(0, −푒) ̃푉 (푡, 휃−푒휔)푓(휔)],  (40)  where 휔∗(푥, 푦) ∶= 휌휔(푦)휔(푦, 푥)∕휌휔(푥) denotes the adjoint of 휔, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=', [22], and  ̃푉 (푡, 휔) ∶= 푉 (푡, 휔)∕휌(휔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Noting that ∑  푦 휔∗(푥, 푦) = 1, we have  퐸ℚ푡[퐿휔푓휔(0)] = 피[푉 (푡, 휔)  ∑  푒  휔(0, 푒)[푓(휃푒휔) − 푓(휔)]]  (40)  = 피[휌휔(0)  ∑  푒  휔∗(0, 푒)[ ̃푉 (푡, 휃푒휔) − ̃푉 (푡, 휔)]푓(휔)]  = 퐸ℚ[푓(휔)퐿휔∗ ̃푉휔(푡, 0)],  (41)  18  where ̃푉휔(푡, 푥) ∶= ̃푉 (푡, 휃푥휔), and 퐿휔∗ only acts on the spatial (ℤ푑) coordinate of  the function ̃푉휔 ∶ ℝ × ℤ푑 → ℝ of space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Observe that ̃푉휔 solves the  parabolic equation  (휕푡 − 퐿휔∗) ̃푉휔 = 0  in (0, ∞) × ℤ푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  By the Hölder estimate [22, Corollary 7] and the Harnack inequality [22, Theorem  6] for the operator (휕푡 − 퐿휔∗), there exists 훾 = 훾(푑, 휅) > 0 such that, for 푡 > 1,  max  푒∶|푒|=1 | ̃푉휔(푡, 푒) − ̃푉휔(푡, 0)| ≲ 푡−훾  sup  (푠,푥)∈(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='5푡,푡)×퐵√  푡  ̃푉휔(푠, 푥)  ≲ 푡−훾 ̃푉휔(2푡, 0)  (42)  (39)  ≲ 푡−훾휌−1ℋ푑−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Thus, by (41), |Eℚ푡[퐿휔푓휔(0)]| ≲ 푡−훾피[ℋ푑−1]‖푓‖∞ ≲ 푡−훾‖푓‖∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' In particular, for  any bounded measurable function 푓 on Ω,  퐸ℚ∞[퐿휔푓휔(0)] = lim  푘→∞ 퐸ℚ푡푘[퐿휔푓휔(0)] = 0  which implies that ℚ∞ is an invariant measure for the Markov chain (휃푌푡휔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' More-  over, for any bounded measurable function 푓 ∶ Ω → ℝ and 푝 > 0,  퐸ℚ푡[푓] = 피[푉휔(푡, 0)푓(휔)] ≲ 피[ℋ푑−1푓] ≲푝 ‖푓‖퐿푝(ℙ),  and so 퐸ℚ∞[푓] ≲푝 ‖푓‖퐿푝(ℙ) which implies ℚ ≪ ℙ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Therefore, by the same argu-  ment as in [33, (4)], we have ℚ∞ = ℚ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Before stating the formula for 휕′  푦휌 in the following proposition, we remark that  although the global Green function 퐺휔(푥, 푦) is only defined for 푑 ≥ 3, the second  order difference ∇2  푖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1퐺(푥, 푦) can be defined for all dimensions, where ∇2  푖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1 is ∇2  푖  applied to the first ℤ푑 coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' That is, for any fixed 푦, ∇2  푖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1퐺(⋅, 푦) ∶= ∇2  푖 퐺(⋅, 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Indeed, recalling 퐴(푥, 푦) in (14), we can set  ∇2  푖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1퐺(푥, 푦) ∶= −∇2  푖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1퐴(푥, 푦)  when 푑 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Since 퐺(⋅, ⋅) is not defined in Definition 4 for 푑 = 2, for the convenience of nota-  tions, throughout this section we denote  퐺(푥, 푦) ∶= −퐴(푥, 푦), and 퐺(푥, 푆) = −  ∑  푦∈푆  퐴(푥, 푦)  when 푑 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (43)  Proposition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For any 푥, 푦 ∈ ℤ푑, ℙ-almost surely,  휕′  푦휌휔(푥) = 휌휔(푦)  푑  ∑  푖=1  (휕′  푦휔)(푦, 푒푖)∇2  푖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1퐺휔′  푦(푦, 푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  19  We will use the fact that for any measurable functions 푓, 푔 on Ω,  피[(휕′  푦푓)푔] = 피[푓(휕′  푦푔)],  (44)  휕′  푦(푓푔) = (휕′  푦푓)푔 + 푓 ′  푦(휕′  푦푔) = (휕′  푦푓)푔′  푦 + 푓(휕′  푦푔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (45)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' It suffices to consider the case 푥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' The formula for general 푥 will follow  from the fact that 휕′  푦휌휔(푥) = 휕′  푦−푥휌휃푥휔(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' We divide the proof into several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' First, we will show a formula for 휕′  푦푉 (푡, 휔):  휕′  푦푉 (푡, 휔) = ∫  푡  0  푉휔(푡 − 푠, 푦)  푑  ∑  푖=1  (휕′  푦휔)(푦, 푒푖)∇2  푖 푝  휔′  푦  푠 (푦, 0)d푠,  (46)  where 푉휔(푠, 푦) = 푉 (푠, 휃푦휔), and 푉 ′  푦 (푠, 푦) = 푉휔′  푦(푠, 푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Indeed, notice that 푢(푥, 푡) = 푝휔  푡 (푥, 0) satisfies 푢(푥, 0) =  1푥=0 and  (휕푡 − 퐿휔)푢(푥, 푡) = 0  for (푥, 푡) ∈ ℤ푑 × (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (47)  By the equation above and the product rule (45), we have  −휕푡[휕′  푦푢(푥, 푡)] = 휕′  푦[퐿휔푢(푥, 푡)] =  푑  ∑  푖=1  (휕′  푦휔)(푥, 푒푖)∇2  푖 푢′  푦(푥, 푡) + 퐿휔(휕′  푦푢)(푥, 푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Hence, for every fixed 푦 ∈ ℤ푑, 휕′  푦푢(푥, 푡) solves the heat equation  {  (휕푡 − 퐿휔)휕′  푦푢(푥, 푡) = ∑푑  푖=1(휕′  푦휔)(푥, 푒푖)∇2  푖 푢′  푦(푥, 푡)  for (푥, 푡) ∈ ℤ푑 × (0, ∞),  휕′  푦푢(푥, 0) = 0  for 푥 ∈ ℤ푑  whose solution can be represented by Duhamel’s formula  휕′  푦푢(푥, 푡) =  ∑  푧  푑  ∑  푖=1 ∫  푡  0  푝휔  푡−푠(푥, 푧)(휕′  푦휔)(푧, 푒푖)∇2  푖 푢′  푦(푧, 푠)d푠  =  푑  ∑  푖=1 ∫  푡  0  푝휔  푡−푠(푥, 푦)(휕′  푦휔)(푦, 푒푖)∇2  푖 푢′  푦(푦, 푠)d푠  where we used the fact that 휕′  푦휔(푧, 푒) = 0 if 푧 ≠ 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Recall that 푢(푥, 푡) = 푝휔  푡 (푥, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Summing the above equality over all 푥 ∈ ℤ푑, we obtain formula (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' We claim that the integrand in (46) has the following bound: ∀푠 ∈ (0, 푡),  |||푉휔(푡 − 푠, 푦)  푑  ∑  푖=1  (휕′  푦휔)(푦, 푒푖)∇2  푖 푝  휔′  푦  푠 (푦, 0)|||≲ (ℋ푦ℋ′  푦)푑−1(1 + 푠)−훾−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='5푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (48)  20  Indeed, by (47) and applying the Harnack inequality (Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='2) for the operator  (휕푡 − 퐿휔) in a similar manner as in (42), we have  |||푉휔(푡 − 푠, 푦)  푑  ∑  푖=1  (휕′  푦휔)(푦, 푒푖)∇2  푖 푝  휔′  푦  푠 (푦, 0)|||  (39)  ≲ ℋ푑−1  푦  osc  ̄퐵1(푦) 푝  휔′  푦  푠 (⋅, 0) ≲ ℋ푑−1  푦  푠−훾푝  휔′  푦  2푠 (푦, 0)  for 푠 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Hence (48) follows from Theorem B when 푠 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' When 푠 ≤ 1, (48) is a  trivial consequence of (39) since |∇2  푖 푝  휔′  푦  푠 (푦, 0)| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Display (48) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For any bounded measurable function 푓 on Ω, by Lemma 19 and (44),  피[(휕′  푦휌)푓] = 피[휌(휕′  푦푓)] = lim  푡→∞ 피[푉 (푡, 휔)(휕′  푦푓)] = lim  푡→∞ 피[(휕′  푦푉 (푡, 휔))푓].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Furthermore, by (46), (48), and the dominated convergence theorem, we get  피[(휕′  푦휌)푓] = ∫  ∞  0  lim  푡→∞ 피  [  푉휔(푡 − 푠, 푦)  푑  ∑  푖=1  (휕′  푦휔)(푦, 푒푖)∇2  푖 푝  휔′  푦  푠 (푦, 0)1푡>푠푓  ]  d푠  퐿푒푚푚푎 19  =  ∫  ∞  0  피  [  휌푓  푑  ∑  푖=1  (휕′  푦휔)(푦, 푒푖)∇2  푖 푝  휔′  푦  푠 (푦, 0)  ]  d푠  = 피  [  휌푓  푑  ∑  푖=1  (휕′  푦휔)(푦, 푒푖)∇2  푖 퐺휔′  푦(푦, 0)  ]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Proposition 20 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='2  Rate of convergence for the average of the invariant measure: Proof  of Theorem 5  Now we will proceed to prove one of the main theorems in this paper, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  It will be clear in the proof that the log 푅 term in the 퐶1,1 bound of Theorem 14  is important for us to obtain the logarithmic term in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' We divide the proof into several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Let 푢 ∶ ℝ+ → ℝ+ be the function  푢(푟) =  { log(푟 + 1)  when 푑 = 2  (푟 + 1)2−푑  when 푑 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (49)  We will show that for any 휀 > 0, 푅 ≥ 2, there exists a random variable ℋ∗(휔) =  ℋ∗(푅, 휔;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 푑, 휅, 휀) > 0 with 피[exp(푐ℋ∗푑−휀)] < 퐶 such that, ℙ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=',  osc  퐵(|푦|+푅)∕2(푦) 퐺휔(⋅, 퐵푅) ≲  { ℋ∗푑−1푢(|푦|)푅푑  if |푦| > 4푅  ℋ∗푑−1푅2 log 푅  if |푦| ≤ 4푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (50)  21  Indeed, by Theorem D, for 푧 ∈ ℤ푑,  |퐺(푧, 퐵푅)| ≲  ∑  푥∈퐵푅  ℋ푑−1  푥  푢(|푧 − 푥|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (51)  When |푦| > 4푅, 푢(|푧 − 푥|) ≍ 푢(|푦|) for all 푥 ∈ 퐵푅, 푧 ∈ 퐵(|푦|+푅)∕2(푦), and so  |퐺(푧, 퐵푅)|  (51)  ≲  ∑  푥∈퐵푅  ℋ푑−1  푥  푢(|푦|) ≲ ℋ∗푑−1  1  푢(|푦|)푅푑,  (52)  where ℋ∗  1 = (  1  |퐵푅|  ∑  푥∈퐵푅 ℋ푑−1  푥  )1∕(푑−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  When |푦| ≤ 4푅 and 푑 = 2, for all 푧 ∈ 퐵(|푦|+푅)∕2(푦), we have 푢(|푧−푥|) ≲ log 푅  ∀푥 ∈ 퐵푅, and so  |퐺(푧, 퐵푅)|  (51)  ≲ log 푅  ∑  퐵푅  ℋ푑−1  푥  = ℋ∗푑−1  1  푅2 log 푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (53)  When |푦| ≤ 4푅 and 푑 ≥ 3, for all 푧 ∈ 퐵(|푦|+푅)∕2(푦), (51) yields  |퐺(푧, 퐵푅)| ≲ [ℋ∗  2 + (log 푅)1∕(푑−1)]푑−1 ∑  푥∈퐵4푅  푢(|푥|)  ≤ (ℋ∗푑−1  2  + log 푅)푅2 ≲ ℋ∗푑−1  2  푅2 log 푅,  (54)  where ℋ∗  2 = [max푥∈퐵푅 ℋ푥 − (log 푅)1∕(푑−1)]+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Recall ℋ = ℋ(휔, 푑, 휅, 휀) in Theo-  rem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Note that for 푡 > 1 and 푝 = 푑 − 휀 > 푑 − 1,  푃 (ℋ∗  2 > 푡) ≤  ∑  푥∈퐵푅  푃 (ℋ푥 > 푡 + (log 푅)1∕(푑−1))  ≲ 푅푑 exp[−푐(푡 + (log 푅)1∕(푑−1))푝]  ≲ exp(−푐푡푝)  where we used Chebyshev’s inequality in the second inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Hence 피[exp(−푐ℋ푑−휀  2  )] ≤  퐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Note also that, by Jensen’s inequality, 피[exp(−푐ℋ푑−휀  1  )] ≤ 퐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Setting ℋ∗ = ℋ∗  1 + ℋ∗  2 , (50) follows from (52), (53), and (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Next, we will show that  |∇2퐺휔(푦, 퐵푅)| ≲  { ℋ2  푦 ℋ∗푑−1|푦|−2푢(|푦|)푅푑  if |푦| > 4푅  ℋ2  푦 ℋ∗푑−1 log 푅  if |푦| ≤ 4푅,  (55)  where the operator ∇2 is only applied to the first ℤ푑 coordinate of 퐺(⋅, ⋅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  When |푦| > 4푅, by Theorem 14 and (50),  |∇2퐺(푦, 퐵푅)| ≲  ℋ2  푦  |푦|2  osc  퐵|푦|∕2(푦) 퐺(⋅, 퐵푅) ≲ ℋ2  푦 ℋ∗푑−1  1  |푦|−2푢(|푦|)푅푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  22  When |푦| ≤ 4푅, applying Theorem 14 (with 휓 = 0, 푓 = −1퐵푅) again, we get  |∇2퐺(푦, 퐵푅)| ≲  ℋ2  푦  푅2  osc  퐵푅∕2(푦) 퐺(⋅, 퐵푅) + ℋ2  푦 log 푅  (50)  ≲ ℋ2  푦 ℋ∗푑−1 log 푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' By Proposition 20, Theorem B, and (55),  |||휕′  푦  휌휔(퐵푅)  |퐵푅|  |||≲ 푅−푑휌휔(푦)|∇2퐺휔′  푦(푦, 퐵푅)| ≲ 풥2푑  푦 (휔, 휔′)푤(|푦|),  (56)  where 풥푦(휔, 휔′) ∶= [ℋ푑−1  푦  (휔)ℋ2  푦 (휔′  푦)ℋ∗푑−1(휔′  푦)]1∕(2푑), and  푤(푟) =  { 푟−2푢(푟)  if 푟 > 4푅  푅−푑 log 푅  if 푟 ≤ 4푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Note that 피[exp(푐풥푑−휀  푦  )] < 퐶, and  ∑  푥∈ℤ푑  푤2(|푥|) ≍ 푅−푑(log 푅)2  We let 푊 (푅) = 푅−푑∕2 log 푅 so that 푊 (푅)2 ≍ ∑  푥∈ℤ푑 푤2(|푥|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Set  푍(휔) ∶= 휌휔(퐵푅)∕|퐵푅|  푊 (푅)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' By (37), (56), and Jensen’s inequality, for any 푞 ≥ 2,  푉 (푍)푞∕2 ≲  (∑  푦 풥2푑  푦 푤(|푦|)2)  ∑  푧 푤(|푧|)2  )푞∕2  ≤  ∑  푦  푤(|푦|)2  ∑  푧 푤(|푧|)2 풥푑푞  푦 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Taking expectations on both sides and using translation-invariance of ℙ, we get  피[푉 푞∕2] ≲ 피[풥푑푞  0 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Thus, by (38),  피[|푍 − 피푍|푞] ≤ 퐶푞푞∕2피[풥푑푞  0 ],  ∀푞 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (57)  As a fact, for any 훼 ∈ [0, 1), there exists 푐 = 푐(훼) > 0 such that, for all 푥 > 0,  ∞  ∑  푛=1  푐푛  푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='푥푛푛훼푛 ≤ exp(푥1∕(1−훼)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (58)  Indeed, when 푥 > 0, putting 푐 = 푒−훼∕2 and using inequality 푛푛  푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' ≤ 푒푛,  ∞  ∑  푛=1  푐푛  푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='푥푛푛훼푛 ≤  ∞  ∑  푛=1  2−푛  푥푛  (푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' )1−훼 =  ∞  ∑  푛=1  2−푛(푥푛∕(1−훼)  푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  )1−훼 ≤ exp(푥1∕(1−훼)),  where we used 푦푛  푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' ≤ 푒푦 for 푦 ≥ 0 in the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Thus, recalling 피[exp(푐풥푑−휀  0  )] <  퐶 and letting 푝 = (3  2 +  휀  푑−휀)−1, displays (57) and (58) yield  피[exp(푐|푍 − 피푍|푝)] ≲ 피  [ ∞  ∑  푛=0  푐푛  푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='풥푑푛푝  0  (푛푝)푛푝∕2  ]  ≲ 피[exp(푐풥푑−휀  0  )] < 퐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Note that 피푍 =  1  푊 (푅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' The theorem follows by Chebyshev’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  23  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='3  Correlation structure of the field of the invariant measure  In this subsection we will investigate the mixing property of the field by showing  the rate of decay of its correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Intuitively, since 휌휔(푥) is determined by the  long term frequency of visits of the RWRE to 푥, the influence of environments at  remote locations will be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Our proof uses a localization of the invariant measure 휌휔(푥) to a finite ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For  any 푥 ∈ ℤ푑, 푟 > 0, we introduce the notation  휌푟(푥) = 휌푟,휔(푥) ∶= 피[휌휔(푥)|휔(푦) ∶ 푦 ∈ 퐵푟(푥)]  so that 휌푟(푥) is only a function of environments within the ball 퐵푟(푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' As usual, we divide the proof into two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' We will show that, for 푟 ≥ 2,  ‖휌(0) − 휌푟(0)‖퐿2(ℙ) ≲  { 푟−1 log 푟,  푑 = 2  푟−푑∕2,  푑 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (59)  Recall the function 푢(푟) defined in (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' By applying the Efron-Stein inequality to  every fixed realization of the environment within 퐵푟, we get  ‖휌 − 휌푟‖2  퐿2(ℙ) ≲ 피[  ∑  푦∉퐵푟  (휕′  푦휌)2]  푃푟표푝표푠푖푡푖표푛 20  ≲  피[  ∑  푦∉퐵푟  (휌(푦)|∇2퐺휔′  푦(푦, 0)|)2]  ≲ 피[  ∑  푦∉퐵푟  1  |푦|4 (ℋ푦ℋ(휔′  푦))2푑−2ℋ4  푦 (휔′  푦) 푢(|푦|)2],  where in the last inequality we used Theorem B, Theorem D, and Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Since for 푟 ≥ 2,  ∑  푦∉퐵푟  |푦|−4푢(|푦|)2 ≍ ∫  ∞  푟  푠−4푢(푠)2푠푑−1d푠 ≲  { 푟−2(log 푟)2,  푑 = 2  푟−푑,  푑 ≥ 3,  inequality (59) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' We will use (59) to estimate Covℙ(휌(푥), 휌(푦)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' By the translation invariance  of ℙ, it suffices to consider the case 푦 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Then  Covℙ(휌(0), 휌(푥))  = 피  [  휌(0)(휌(푥) − 휌|푥|∕2(푥))  ]  + 피  [  휌(0)(휌|푥|∕2(푥) − 1)  ]  = 피 [휌(0)(휌(푥) − 휌|푥|∕2(푥))] + 피 [(휌(0) − 휌|푥|∕2(0))(휌|푥|∕2(푥) − 1)]  where in the last equality we used the fact that 휌|푥|∕2(0) and 휌|푥|∕2(푥) are independent  under ℙ, and that 피[휌|푥|∕2(푥)] = 피[휌] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Hence, by Hölder’s inequality and the moment bound (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Theorem B) of 휌, we  have  |Covℙ(휌(0), 휌(푥))| ≲ ‖휌 − 휌|푥|∕2‖퐿2(ℙ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  The proposition is proved by recalling inequality (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  24  4  Homogenization of the Dirichlet problem  In this section, 휓 is always assumed to be a local function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1  Homogenization of the approximate corrector  We consider the function ̂휙 ∶ ℤ푑 → ℝ defined as  ̂휙(푥) = ̂휙(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 휓, 푅, 휔) = − ∫  ∞  0  푒−푡∕푅2퐸푥  휔[휓(휃푌푡휔)]d푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (60)  where 푅 ≥ 1, and 휓 is measurable function of 휔(0) with 퐸ℚ[휓] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Notice that  ̂휙 is stationary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=', ̂휙(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 휓, 푅, 휔) = ̂휙(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 휓, 푅, 휃푥휔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Moreover, ̂휙 is a solution of  퐿휔 ̂휙(푥) =  1  푅2 ̂휙(푥) + 휓(휃푥휔),  푥 ∈ ℤ푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (61)  Clearly, by the definition of ̂휙 in (60), for any 휔 ∈ Ω,  sup  푥∈ℤ푑 | ̂휙(푥)| ≤ 푅2‖휓‖∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (62)  and so ‖ 1  푅2 ̂휙(푥) + 휓(휃푥휔)‖∞ ≤ 2‖휓‖∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' By (62) and the Hölder estimate (31),  [ ̂휙]훾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅∕2 ≲ 푅−훾[max  퐵푅  | ̂휙| + 푅2‖푅−2 ̂휙 + 휓‖푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵푅] ≲ 푅2−훾‖휓‖∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Hence, for any 2 ≤ 퐷 ≤ 푅, applying (28) to 푓 = ̂휙∕푅2 and 휎 = 훾 in 퐵퐷, we get  |∇ ̂휙(0)| ≲ ℋ(퐷‖휓‖∞ + 1  퐷‖ ̂휙‖1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵퐷),  (63)  |∇2 ̂휙(0)| ≲ ℋ2‖휓‖∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (64)  The goal of this subsection is to establish the optimal rate of convergence of the  approximate corrector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' To this end, set, for 푅 ≥ 2,  휇(푅) ∶=  ⎧  ⎪  ⎨  ⎪⎩  푅  푑 = 2  푅1∕2  푑 = 3  (log 푅)1∕2  푑 = 4  1  푑 ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (65)  Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Assume that 휓(휔) = 휓(휔(0)) is a bounded function of 휔(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For any  0 < 푝 <  2푑  3푑+2, there exists 퐶 = 퐶(푑, 휅, 푝) such that for 푡 ≥ 0, 푅 ≥ 2, with 휇(푅) as  defined in (65) and ̂휙(푥) = ̂휙(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 휓, 푅, 휔) as in (60),  ℙ  (  | ̂휙(0)| ≥ 푡휇(푅)‖휓‖∞  )  ≤ 퐶 exp(− 1  퐶 푡푝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  25  The continuous version of Lemma 21 was proved earlier by Armstrong, Lin [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Our result in two dimensions (푑 = 2) is slightly better than that in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  We now obtain Lemma 21 using the concentration inequality (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' To this end,  we regard ̂휙 as a function of the environment and write, for 푦 ∈ ℤ푑,  ̂휙′  푦(푥) ∶= ̂휙(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 휓, 푅, 휔′  푦),  휕′  푦 ̂휙 = ̂휙′  푦 − ̂휙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (66)  We will need a bound for 휕′  푦 ̂휙(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Note that 푤(푥) = 휕′  푦 ̂휙(푥) satisfies, for 푥, 푦 ∈ ℤ푑,  퐿휔′  푦푤(푥) = 푅−2푤(푥) + [푅−2 ̂휙 + 휓(휔′(푦)) − tr(휔′∇2 ̂휙)]1푦=푥,  which yields  푤(푥) = − [푅−2 ̂휙(푦) + 휓(휔′(푦)) − tr(휔′∇2 ̂휙)(푦)]  ∫  ∞  0  푒−푡∕푅2푝  휔′  푦  푡 (푥, 푦)d푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (67)  This equality, together with (64), (62) and Theorem B(c), implies  |휕′  푦 ̂휙(0)| = |푤(0)| ≲ ℋ2  푦 ‖휓‖∞ ∫  ∞  0  푒−푡∕푅2푝  휔′  푦  푡 (0, 푦)d푡  ≲ ℋ2  푦 ℋ′  푦  푑−1‖휓‖∞ ∫  ∞  0  (1 + 푡)−푑∕2 exp  [  − 푡  푅2 − 푐픥(|푦|, 푡)  ]  d푡  ≲ ℋ2  푦 ℋ′  푦  푑−1‖휓‖∞푣(|푦|),  (68)  where ℋ푦 = ℋ(휃푦휔), ℋ′  푦 = ℋ(휃푦휔′  푦) and, with 푐2 = 푐2(휅, 푑) > 0 denoting an  appropriate constant,  푣(푟) =  {  푒−푐2푟∕푅 [  1 + log(  푅  (푟+1)∧푅)  ]  푑 = 2  푒−푐2푟∕푅(푟 + 1)2−푑  푑 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (69)  Recall 휇(푅) in (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Notice that  ∑  푦∈ℤ푑  푣(|푦|)2 ≲ ∫  ∞  0  푣(푟)2푟푑−1d푟 ≲ 휇(푅)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (70)  The verifications of inequalities (68) and (70) are included in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Proof of Lemma 21: For 푦 ∈ ℤ푑, set ℋ푦 ∶= ℋ(휃푦휔), and  푍(휔) ∶=  ̂휙(0)  ‖휓‖∞휇(푅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (71)  By (37), (68), (70), and Jensen’s inequality, for any 푞 ≥ 2,  푉 (푍)푞∕2 ≲  (∑  푦(ℋ4  푦 ℋ′  푦  2(푑−1))푣(|푦|)2)  ∑  푧 푣(|푧|)2  )푞∕2  ≤  ∑  푦  푣(|푦|)2  ∑  푧 푣(|푧|)2 ℋ2푞  푦 ℋ′  푦  (푑−1)푞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  26  Taking expectations on both sides and using translation-invariance of ℙ, we get  피[푉 푞∕2] ≲ 피[ℋ2푞ℋ′푞(푑−1)] ≲ 피[ℋ(1+푑)푞],  where we used Hölder’s inequality in the second inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Thus, by (38),  피[|푍 − 피푍|푞] ≤ 퐶푞푞∕2피[ℋ(1+푑)푞],  ∀푞 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (72)  Thus, recalling 피[exp(푐ℋ푑−휀)] < 퐶 in Theorem B and letting 푝 = (3  2 + 1+휀  푑−휀)−1,  displays (72) and (58) yield  피[exp(푐|푍 − 피푍|푝)] ≲ 피  [ ∞  ∑  푛=0  푐푛  푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='ℋ(1+푑)푛푝(푛푝)푛푝∕2  ]  ≲ 피[exp(푐ℋ푑−휀)] < 퐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  In particular, 피[|푍 − 피푍|2] < 퐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  To prove Lemma 21, it suffices to show that  피[exp(푐|푍|푝)] = 피  [  exp (푐|  ̂휙(0)  휇(푅)‖휓‖∞ |푝)]  < 퐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (73)  It suffices to show that |피푍| < 퐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Since ℚ is an invariant measure for (휃푌푡휔)푡≥0,  we have 퐸ℚ퐸0  휔[휓(휃푌푡휔)] = 퐸ℚ[휓] = 0 for all 푡 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Hence, by (60), we know  퐸ℚ[ ̂휙(0)] = 0  and so 퐸ℚ[푍] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Further, by Hölder’s inequality and Theorem B,  |피푍| = |퐸ℚ[푍 − 피푍]| ≤ 퐸ℚ[|푍 − 피푍|] ≤ ‖휌‖퐿2(ℙ)‖푍 − 피푍‖퐿2(ℙ) ≤ 퐶  Therefore, we obtain (73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Lemma 21 follows by Chebyshev’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='2  Rate of homogenization for the Dirichlet problem (19)  We need the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Let 푟 = 푟(푅) be a function of 푅 defined as  푟 =  ⎧  ⎪  ⎨  ⎪⎩  푅2∕3  푑 = 2  푅4∕7  푑 = 3  푅1∕2(log 푅)1∕8  푑 = 4  푅1∕2  푑 ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (74)  Recall ̂휙 in (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For 푘 = 1, … , 푑, let  푣푘(푥) = ̂휙(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 휔푘 − ̄푎푘, 푟, 휔),  푥 ∈ ℤ푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (75)  For 푅 ≥ 2 and 0 < 푝 <  2푑  3푑+2, set Λ = {휔푘 − ̄푎푘 ∶ 푘 = 1, … , 푑}, and define  풴 = 풴(휔;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 푅) = 1 + 푐푝  max  휉∈Λ,|푒|≤1 | ̂휙(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 휉, 푟, 휃푒휔)|∕[‖휉‖∞휇(푟)],  27  so that, by Lemma 21, for all 푅 ≥ 2, 피[exp(풴푝)] ≤ 퐶 and  max  휉∈Λ,|푒|≤1 | ̂휙(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 휉, 푟, 휃푒휔)| ≲푝 풴휇(푟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (76)  We write 풴푥 = 풴푥(휔;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 푅) ∶= 풴(휃푥휔;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 푅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  In what follows we will apply the classical method of two-scale expansions to  quantify the rate of the homogenization for the Dirichlet problem (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' The proof  is similar to that in the periodic setting (see [32, 45, 30] for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' The only  difference is that we use the approximate corrector 푣푘 here instead of the actual  corrector in the periodic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Proof of Theorem 7: We can replace the function 푔 in (19) by ̄푢, because doing this  only introduces an error of size 퐶푅−1‖푢‖퐶4 to |푢(푥) − ̄푢( 푥  푅)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Consider  푤(푥) = 푢(푥) − ̄푢( 푥  푅) + 1  푅2 푣푘(푥)휕푘푘 ̄푢( 푥  푅),  푥 ∈ ̄퐵푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (77)  Here we follow the convention of summation over repeated indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Then, ∀푥 ∈  퐵푅,  |퐿휔푤(푥)| =|||  1  푅2 푓( 푥  푅) − 퐿휔[̄푢( 푥  푅)] + 1  푅2 (푣푘  푟2 + 휔푘 − ̄푎푘)휕푘푘 ̄푢( 푥  푅) + 1  푅2 푣푘퐿휔[휕푘푘 ̄푢( 푥  푅)]  + 1  푅2  ∑  푦∼푥  휔(푥, 푦)[휕푘푘 ̄푢( 푦  푅2 ) − 휕푘푘 ̄푢( 푥  푅2 )][푣푘(푦) − 푣푘(푥)]|||  =|||  1  2푅−3휔푖(푥)휕푘푘푖 ̄푢( 푥  푅)[푣푘(푥 + 푒푖) − 푣푘(푥 − 푒푖)] + (푅푟)−2휕푘푘 ̄푢( 푥  푅)푣푘(푥)  + 푅−4‖̄푢‖퐶4|푣푘(푥)|푂(1)|||  ≲ ‖̄푢‖퐶4  (  푅−3|푣푘(푥 + 푒푖) − 푣푘(푥 − 푒푖)| + (푅푟)−2|푣푘| + 푅−4|푣푘|  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (78)  See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='4 for verification of the first part of (78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Set 퐷 = 휇(푟)1∕2 ≤ 푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' By  (63) and (76),  osc  ̄퐵1(푥) 푣푘 ≲ ℋ푥(퐷 + ‖푣푘‖1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='퐵퐷(푥)) ≲ ℋ푥풴∗  푥  √  휇(푟),  (79)  where  풴∗  푥 =  1  #퐵퐷  ∑  푧∈퐵퐷(푥)  풴푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (80)  By (78), (79) and (64), we get that for 푥 ∈ 퐵푅,  |퐿휔푤| ≲ ‖̄푢‖퐶4[ℋ푥풴∗  푥푅−3√  휇(푟) + (푅푟)−2휇(푟)풴푥]  ≲ ‖̄푢‖퐶4푅−2휏(푅)(ℋ푥풴∗  푥 + 풴푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  28  Recall 푟(푅), 휏(푅) in (74), (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Hence, by (77) and the ABP inequality,  max  퐵푅  |푢(푥) − ̄푢( 푥  푅)| ≲ max  퐵푅  |푤| + 푅−2 max  푥∈퐵푅  |푣푘(푥)휕푘푘 ̄푢( 푥  푅)|  ≲ ‖̄푢‖퐶4  [  휏(푅)  ( 1  #퐵푅  ∑  푥∈퐵푅  (ℋ푥풴∗  푥 + 풴푥)푑)1∕푑 + 푅−2휇(푟) max  푦∈ ̄퐵푅  풴푦  ]  ≲ ‖̄푢‖퐶4  [  휏(푅)퐴1 + 푅−2휇(푟)퐴2 + 푅−2휇(푟)(log 푅)1∕(2푠)]  ≲ ‖̄푢‖퐶4휏(푅)(퐴1 + 퐴2),  (81)  where  퐴1 =  (  1  #퐵푅  ∑  푥∈퐵푅  (ℋ푥풴∗  푥 + 풴푥)푑  )1∕푑  ,  퐴2 =  (  max  푦∈ ̄퐵푅  풴푦 − (2푑 log 푅)1∕(2푠)  )  +  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  For 푞 ≥ 푑, by the translation-invariance of ℙ,  피[퐴푞  1] ≲ 피[(ℋ풴∗  0 + 풴)푞] ≲ 피[풳2푞 + 풴∗2푞  0  + 풴푞] ≲ 피[풳2푞 + 풴2푞],  which implies, for 푠 ∈ (0,  푑  3푑+2),  피[exp(푐퐴푠  1)] ≲ 피[exp(풳2푠) + exp(풴2푠)] ≤ 퐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (82)  Moreover, for 푡 > 0, by a union bound and Chebyshev’s inequality,  ℙ(퐴2 ≥ 푡) ≲ 푅푑ℙ(풴 − [2푑 log 푅]1∕(2푠) ≥ 푡)  ≤ # ̄퐵푅피  [  exp  (  풴2푠 − 1  2푡2푠 − 푑 log 푅  )]  ≲ 푒−푡2푠∕2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Thus 피[exp(푐퐴푠  2)] < 퐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' This, together with (82) and (81), yields  max  퐵푅  |푢(푥) − ̄푢( 푥  푅)| ≲ ‖̄푢‖퐶4휏(푅)풵,  with 풵 ∶= 푐(퐴1 + 퐴2) satisfying 피[exp(풵푠)] ≤ 퐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Our proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  5  Quantification of the diffusive behavior  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1  Quantification of the ergodicity of the environmental process: Proof  of Theorem 8  In this section we will derive the optimal rates of convergence (as 푡 → ∞) of the er-  godic average 1  푡 퐸휔[∫ 푡  0 휓( ̄휔푠)d푠], where ̄휔푠 denotes the process of the environment  viewed from the particle:  ̄휔푠 ∶= 휃푌푠휔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  With Lemma 21, it may be tempting to compare the approximate corrector ̂휙  in (60) to the corrector within a finite ball 퐵푅, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=', the solution 푢 to the Dirichlet  29  problem 퐿휔푢 = 휓휔 in 퐵푅 with 푢 = 0 on 휕퐵푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' However, such comparison involves  controlling the boundary error max휕퐵푅 ̂휙 which would result in an extra log 푅 fac-  tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' In what follows, we will follow the argument of Kipnis and Varadhan [37] to  approximate 퐸휔[∫ 푇  0 휓(휃푌푠휔)d푠] with a martingale using the approximate corrector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Without loss of generality, assume ‖휓‖∞ = 1 and ̄휓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  First, we will construct a martingale (for both continuous and discrete time  cases) using the approximate corrector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  For any fixed 푇 > 1, let 휙 ∶ Ω → ℝ denote the function  휙(휔) = 휙휓,푇(휔) ∶= ̂휙(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 휓,  √  푇 , 휔),  where ̂휙 is as in (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Then, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 휔 ∈ Ω, the process (푀푡)푡≥0 defined by  푀푡 ∶ = 휙(휃푌푡휔) − 휙(휃푌0휔) − ∫  푡  0  퐿휔휙(휃푌푠휔)d푠  (61)  = 휙( ̄휔푡) − 휙( ̄휔0) − ∫  푡  0  [ 1  푇 휙( ̄휔푠) + 휓( ̄휔푠)]d푠  (83)  is a 푃휔-martingale with respect to the filtration ℱ푡 = 휎(푌푠 ∶ 푠 ≤ 푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Similarly, for  discrete-time RWRE, we have that  푁푛 ∶= 휙( ̄휔푛) − 휙( ̄휔0) −  푛−1  ∑  푖=0  [ 1  푇 휙( ̄휔푖) + 휓( ̄휔푖)]  is a 푃휔-martingale with respect to the filtration ℱ푛 = 휎(푋푖 ∶ 푖 ≤ 푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Next, we will derive an exponential moment bounds for ∫ 푡  0 푃푠휓d푠 and ∑푛  푖=0 푃푖휓,  where the operator 푃푠 is as in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' We will only provide a proof for the continuous-  time case, because the argument for the discrete-time setting is exactly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Since 퐸휔[푀푠] = 퐸휔[푀0] = 0, taking expectations in (83), we get  ∫  푡  0  푃푠휓d푠 = 푃푡휙 − 휙 − 1  푇 ∫  푡  0  푃푠휙d푠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (84)  Since the process ( ̄휔푠) is a stationary sequence under the measure ℚ×푃휔, we have,  by Jensen’s inequality, for any 푡 ≥ 0, 푞 ≥ 1,  ‖푃푡휙‖푞  퐿푞(ℚ) = 퐸ℚ[|퐸휔휙( ̄휔푡)|푞] ≤ 퐸ℚ×푃휔[|휙( ̄휔푡)|푞] = 퐸ℚ[|휙|푞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Hence, taking the 퐿푞(ℚ)-norms on both sides of (84), we get  ‖∫  푇  0  푃푠휓d푠‖퐿푞(ℚ) ≤ 3‖휙‖퐿푞(ℚ),  ∀푞 ≥ 1  30  which implies  퐸ℚ  [  exp  (  푐||| ∫  푇  0  푃푠휓d푠  /  휇(  √  푇 )|||  푝)]  ≤ 퐸ℚ  [  exp  (  푐|||3휙∕휇(  √  푇 )|||  푝)]  ≤ ‖휌‖퐿2(ℙ)퐸ℙ  [  exp  (  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='5푐|||3휙∕휇(  √  푇 )|||  푝)]1∕2 (73)  ≤ 퐶,  where we used Hölder’s inequality in the second inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Note that 휈(푇 ) = 푇 −1휇(  √  푇 ) as defined in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' The theorem follows from the  above moment bound and Chebyshev’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  As a consequence of Theorem 8, we can show the existence and uniqueness of  a stationary corrector in 푑 ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Corollary 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Assume (A1), (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' When 푑 ≥ 5, for any bounded local measurable  function 휁 ∶ Ω → ℝ, there exists 휙 ∶ Ω → ℝ with the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (a) The function 휙휔(푥) ∶= 휙(휃푥휔) solves  퐿휔휙(푥) = 휁(휃푥휔) − 퐸ℚ[휁],  for all 푥 ∈ ℤ푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (85)  (b) Up to an additive constant, 휙 is the unique function that satisfies (85) and  피[exp(푐|휙|푝)] < ∞  for all 0 < 푝 < (3  2 + 1  푑 )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (86)  In the continuous PDE setting, the existence of the stationary corrector in 푑 ≥ 5  stochastic integrability (86) with 푝 = 1  2 was proved in [3, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' See also  [31, Corollary 7] for a proof of the existence part (a) in the discrete setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Without loss of generality, assume 퐸ℚ[휁] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  When 푑 ≥ 5, we let 휙휔(푥) = 퐸푥  휔[∫ ∞  0  휁( ̄휔푠)d푠].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' The existence (both as a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  and 퐿푝(ℙ)-limits for all 푝 > 0) and stochastic integrability (86) of such a function  follow immediately from Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' It clearly solves (85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' It remains to show the  uniqueness up to an additive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Suppose there is another stationary corrector ̃휙 that satisfies (85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Then, for  푘 ∈ ℕ and for 퐶 > 0 sufficiently large, by Chebyshev’s inequality,  ℙ(max  푥∈퐵푘  |휙(푥) − ̃휙(푥)| ≥ 퐶 log 푘) ≲ 푘푑ℙ(|휙(0) − ̃휙(0)| ≥ 퐶 log 푘) ≲ 푘−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  By Borel-Cantelli’s lemma, ℙ-almost surely, we have  lim  푘→∞ max  퐵푘  |휙 − ̃휙|∕ log 푘 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Since 휙 − ̃휙 is 휔-harmonic on ℤ푑 with sublinear growth, by Theorem 14, it is a  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  31  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='2  A Berry-Esseen estimate for the QCLT: Proof of Corollary 10  To prove Corollary 10 we will apply the Berry-Esseen estimates for martingales by  Heyde and Brown [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Here we will use the version in [34, Theorem 2] which is  also applicable to the continuous-time setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Proof of Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For any unit vector 퓁 ∈ ℝ푑, let 휓0(휔) = 퓁푇 휔(0)  tr휔(0)퓁, 휓 =  휓0 − 퐸ℚ[휓0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Following the notations in [34], we set  푁푛,2 ∶ = 퐸휔  [  |||  푛−1  ∑  푘=0  퐸휔  [ 1  √  푛  (  (푋푘+1 − 푋푘) ⋅ 퓁  )2|ℱ푘  ]  − 퓁푇 ̄푎퓁|||  2  ]  = 1  푛2 퐸휔  [  ( 푛−1  ∑  푘=0  휓( ̄휔푘)  )2  ]  ,  퐿푛,2 ∶=  푛−1  ∑  푘=0  퐸휔[| 1  √  푛(푋푘+1 − 푋푘) ⋅ 퓁|4] = 1  푛2 퐸휔  [푛−1  ∑  푘=0  휓0( ̄휔푘)  ]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  The term 푁푛,2 can be further written as  푛2푁푛,2 = 2  푛−1  ∑  푖=0  퐸휔  [  휓( ̄휔푖)  푛−푖−1  ∑  푗=0  휓( ̄휔푖+푗)  ]  = 2  푛−1  ∑  푖=0  퐸휔  [  휓( ̄휔푖)퐸푋푖  휔  [푛−푖−1  ∑  푗=0  휓( ̄휔푗)  ]]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Hence, for any 푞 ≥ 1, using the fact that ( ̄휔푖) is a stationary sequence under ℚ×푃휔,  we get (note ‖휓0‖∞ ≲ 1)  ‖푁푛,2‖퐿푞(ℚ) ≲ 1  푛2  푛−1  ∑  푖=0  ‖  푛−푖−1  ∑  푗=0  푃푗휓‖퐿푞(ℚ×푃휔)  which, by Jensen’s inequality and the fact 1  푛2  ∑푛  푘=1 휇(  √  푘) ≍ 휈(푛), implies that for  any 0 < 푝 <  2푑  3푑+2,  퐸ℚ  [  exp  (  푐|푁푛,2∕휈(푛)|푝)]  ≲  1  푛2휈(푛)  푛  ∑  푘=1  휇(  √  푘)퐸ℚ  [  exp  (  푐|||  푘−1  ∑  푗=0  푃푗휓  /  휇(  √  푘)|||  푝)  ]  푇 ℎ푒표푟푒푚 8  ≤  퐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Thus, using the moment bound of 휌−1 in Theorem B, by Hölder’s inequality,  퐸ℙ  [  exp  (  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='5푐|푁푛,2∕휈(푛)|푝)]  ≤ ‖휌−1∕2‖퐿2(ℙ)퐸ℚ  [  exp  (  푐|푁푛,2∕휈(푛)|푝)]1∕2 ≤ 퐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  By Theorem 8 we already know that 퐸ℙ[exp (푐|푛퐿푛,2|푝)] ≤ 퐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Therefore, we  conclude that there exists a random variable 풴5 with 퐸ℙ[exp(풴5푝)] < ∞ such that  퐿푛,2 + 푁푛,2 ≤ 퐶휈(푛)풴5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  The corollary follows by applying [34, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  32  A  Appendix  Define the parabolic operator ℒ휔 as  ℒ휔푢(푥, 푡) =  ∑  푦∶푦∼푥  휔(푥, 푦)[푢(푦, 푡) − 푢(푥, 푡)] − 휕푡푢(푥, 푡)  for every function 푢 ∶ ℤ푑 × ℝ → ℝ which is differentiable in 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' The following  results are used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' ([22, Theorem 17]) Assume  휔  tr휔 > 2휅퐼 for some 휅 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Any  non-negative function 푢 with ℒ휔푢 = 0 in 퐵2푅 × (0, 4푅2) for 푅 > 0 satisfies  sup  퐵푅×(푅2,2푅2)  푢 ≤ 퐶  inf  퐵푅×(3푅2,4푅2) 푢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  As a consequence, we have the following Hölder regularity for 푢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Assume  휔  tr휔 > 2휅퐼 for some 휅 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' There exists 훾 = 훾(푑, 휅) ∈  (0, 1) such that any non-negative function 푢 with ℒ휔푢 = 0 in 퐵푅(푥0)×(푡0 −푅2, 푡0),  for some (푥0, 푡0) ∈ ℤ푑 × ℝ and 푅 > 0, satisfies  |푢( ̂푥) − 푢( ̂푦)| ≤ 퐶  ( 푟  푅  )훾  sup  퐵푅(푥0)×(푡0−푅2,푡0)  푢  for all ̂푥, ̂푦 ∈ 퐵푟(푥0) × (푡0 − 푟2, 푡0) and 푟 ∈ (0, 푅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='1  Proof of Proposition 16  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Let 푝 ∈ H푗 be the 푗-th order Taylor polynomial (around 0) of 푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Then  sup  픹휃푅  |푣 − 푝| ≤ 퐶(휃푅)푗+1 sup  픹푅∕3  |퐷푗+1푣|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  This gives \ue230푗+1  픹휃푅(푣) ≲ (휃푅)푗+1 sup픹푅∕3 |퐷푗+1푣|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Furthermore, for any 푞 ∈ H푗,  푗 ≤ 2, note that 퐷(푣−푞) is an ̄푎-harmonic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Hence, by [27, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='10],  sup  픹푅∕3  |퐷푗+1푣| = sup  픹푅∕3  |퐷푗+1(푣 − 푞)|  ≤ 퐶  푅푗 sup  픹5푅∕12  |퐷(푣 − 푞)|  = 퐶  푅푗  sup  푥∈픹5푅∕12  | ⨏픹푅∕12(푥)  퐷(푣 − 푞)| ≤  퐶  푅푗+1 sup  픹푅∕2  |푣 − 푞|  for 푗 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Hence, taking infimum over 푞 ∈ H푗, we get \ue230푗+1  픹휃푅(푣) ≲ 휃푗+1\ue230푗+1  픹푅∕2(푣) for  푗 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' The first statement is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  33  To prove the second statement, observe that for any 푥 ∈ 픹푅∕2, there are 2푑  points 푦푖 ∈ ̄퐵푅∕2, 푖 ∈ Λ = {1, … , 2푑}, such that |푦푖 − 푥| ≤ 1 and 푥 is a convex  combination of the 푦푖’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' That is, 푥 = ∑  푖∈Λ 훼푖푦푖 for some 훼푖 ≥ 0 with ∑  푖∈Λ 훼푖 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Let 푝 ∈ H푗, 푗 ≤ 2, be such that max퐵2푅∕3 |푣 − 푝| ≤ 2\ue230푗+1  2푅∕3(푣) and denote the  Hessian matrix of 푝 by 푀푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Then, for 푥 ∈ 픹푅∕2, 푗 ≤ 2,  |푣(푥) − 푝(푥)| ≤ [푣]1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='픹푅∕2+1 +  ∑  푖∈Λ  훼푖|푣(푦푖) − 푝(푦푖)|+|||푝(푥) −  ∑  푖∈Λ  훼푖푝(푦푖)|||  ≤ [푣]1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='픹푅∕2+1 + max  ̄퐵푅∕2  |푣 − 푝| + 퐶푅|푀푝|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  (87)  Further, using the fact (see [27, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='3]) that  푅[푣]1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='픹푅∕2+1 ≲ sup  픹2푅∕3  |푣| + 푅2|푐0| ≲ sup  휕픹2푅∕3  |푣| + 푅2|푐0|  and (Note that the following bound is not needed for the case 푗 = 1 where 푀푝 ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=')  푅2|푀푝| ≲ max  푦∈퐵푅∕2  |푝(푦)+푝(−푦)−2푝(0)| ≲ max  퐵푅∕2  |푣−푝|+max  퐵푅∕2  |푣| ≲ \ue2303  2푅∕3(푣)+max  퐵푅∕2  |푣|,  display (87) implies, for 푗 ≤ 2,  \ue230푗+1  픹푅∕2(푣) ≲ 1  푅 sup  휕픹2푅∕3  |푣| + 푅|푐0| + \ue230푗+1  2푅∕3(푣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  The second claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='2  Verification of (68)  In this subsection we will verify the inequality  ∫  ∞  0  (1 + 푡)−푑∕2 exp  [  − 푡  푅2 − 푐픥(|푦|, 푡)  ]  d푡 ≲ 푣(|푦| + 1),  ∀푦 ∈ ℤ푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  We break the integral on the left side of the above inequality as  ∫  ∞  0  = ∫  |푦|∕2  0  + ∫  |푦|2  |푦|∕2  + ∫  ∞  |푦|2 =∶ I + II + III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  It suffices to consider the case |푦| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' First, with 푐2 > 0 sufficiently small,  I = ∫  |푦|∕2  0  (1 + 푡)−푑∕2 exp  (  − 푡  푅2 − 푐|푦| log |푦|  푡  )  d푡 ≤ |푦|푒−푐|푦| ≲ 푣(|푦|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Moreover, noting that −  푡  2푅2 − 푐 |푦|2  푡  ≲ −|푦|  푅 ,  II = ∫  |푦|2  |푦|∕2  (1 + 푡)−푑∕2 exp  (  − 푡  푅2 − 푐 |푦|2  푡  )  d푡  ≲ 푒−푐|푦|∕푅  ∫  |푦|2  0  푡−푑∕2푒−푐|푦|2∕푡d푡  ≲ 푒−푐|푦|∕푅|푦|2−푑  ∫  ∞  1  푠푑∕2−2푒−푐푠d푠 ≲ 푣(|푦|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  34  Similarly, for 푑 = 2,  III ≲ 푒−푐|푦|∕푅  ∫  ∞  |푦|2 (1 + 푡)−푑∕2 exp  (  −  푡  2푅2  )  d푡  ≲ 푒−푐|푦|∕푅  ∫  ∞  |푦|2∕푅2 푠−1푒−푠∕2d푠  ≲ 푒−푐|푦|∕푅  [  1 + ∫  ∞  0  푠−1  1{|푦|2∕푅2≤푠≤1}d푠  ]  ≲ 푣(푦|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  For 푑 ≥ 3, we have  III ≲ 푒−푐|푦|∕푅  ∫  ∞  |푦|2 (1 + 푡)−푑∕2d푡 ≲ 푣(|푦|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Therefore, the above bounds of I, II, III imply inequality(68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='3  Verification of (70)  When 푑 = 2,  ∫  ∞  0  푒−푐푟∕푅푣(푟)2푟푑−1d푟 ≲ ∫  ∞  0  푒−푐푟∕푅[1 + (log  푅  (푟+1)∧푅)2]푟d푟  ≲ 푅2 + ∫  푅  1  푒−푐푟∕푅 (  log 푅  푟  )2  푟d푟 ≲ 푅2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  When 푑 = 3,  ∫  ∞  0  푒−푐푟∕푅푣(푟)2푟푑−1d푟 ≲ ∫  ∞  0  푒−푐푟∕푅 ≲ 푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  When 푑 = 4,  ∫  ∞  0  푒−푐푟∕푅푣(푟)2푟푑−1d푟 ≲ ∫  푅  0  (1 + 푟)−1d푟 + ∫  ∞  푅  푒−푐푟∕푅푅−1d푟 ≲ log 푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  When 푑 ≥ 5,  ∫  ∞  0  푒−푐푟∕푅푣(푟)2푟푑−1d푟 ≲ ∫  ∞  0  (1 + 푟)−2d푟 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='4  Verification of (78)  We verify the first part of (78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' For functions 푢, 푣 on ℤ푑,  ∇2  푒(푢푣)(푥) = 푢(푥 + 푒)푣(푥 + 푒) + 푢(푥 − 푒)푣(푥 − 푒) − 2푢(푥)푣(푥)  = 푣(푥)∇2  푒푢(푥) + 푢(푥 + 푒)[푣(푥 + 푒) − 푣(푥)] + 푢(푥 − 푒)[푣(푥 − 푒) − 푣(푥)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  35  From this we have the expression  퐿휔(푢푣) = 푢퐿휔푣 + 푣퐿휔푢 +  ∑  푦∶푦∼푥  휔(푥, 푦)[푢(푦) − 푢(푥)][푣(푦) − 푣(푥)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  In particular, if 푢 is 퐶2 in ℝ푑, then, doing Taylor expansion to 푢,  ∇2  푒(푢푣)(푥) = 푣(푥)∇2  푒푢(푥) + [푢(푥) + 퐷푒푢(푥) + 1  2퐷2  푒푢(푦)][푣(푥 + 푒) − 푣(푥)]  + [푢(푥) − 퐷푒푢(푥) + 1  2퐷2  푒푢(푧)][푣(푥 − 푒) − 푣(푥)]  = 푣(푥)∇2  푒푢(푥) + 푢(푥)∇2  푒푣(푥) + 퐷푒푢(푥)[푣(푥 + 푒) − 푣(푥 − 푒)]  + 1  2[퐷2  푒푢(푦)푎+(푥) + 퐷2  푒푢(푧)푎−(푥)]∇2  푒푣(푥)  ≤ 푣(푥)∇2  푒푢(푥) + 푢(푥)∇2  푒푣(푥) + 퐷푒푢(푥)[푣(푥 + 푒) − 푣(푥 − 푒)]  + ‖푢‖퐶2∇2  푒푣(푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  with 푎±(푥) = [푣(푥 ± 푒) − 푣(푥)]∕∇2  푒푣(푥), and 푦, 푧 points within the line segment  [푥 − 푒, 푥 + 푒].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Note that 푎+ + 푎− = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='13279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Smart, Quantitative Stochastic Homogenization of Ellip-  tic Equations in Nondivergence Form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Lin, Compactness methods in the theory of homogeniza-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Avellaneda, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Lin, Compactness methods in the theory of homogeniza-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Ark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='  [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Bensoussan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Lions, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Papanicolaou, Asymptotic analysis for periodic  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' AMS Chelsea Publishing, Providence, RI, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' xii+398 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  36  [9] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Berger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Cohen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Deuschel, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Guo, An elliptic Harnack inequality  for random walk in balanced environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 50 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='  [10] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Berger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Deuschel, A quenched invariance principle for non-elliptic  random walk in i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' balanced random environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Theory Related  Fields 158 (2014), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='  [11] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Berger, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Criens, A parabolic Harnack principle for balanced difference  equations in random environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Arch Rational Mech Anal 245, 899–947  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Biskup, Recent progress on the random conductance model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  8 (2011), 294-373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  [13] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Sznitman, Ten lectures on random media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Bousquet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Massart, Concentration Inequalities:  A  Nonasymptotic Theory of Independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Caffarelli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Caffarelli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Henri Poincaré Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Kumagai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Wang, Homogenization of symmetric  stable-like processes in stationary ergodic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Guo, Quenched local central limit theorem for random  walks in a time-dependent balanced random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Springer, New York, NY, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='5-6 (2000): 821-845.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' Reprint of the 1998 edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Classics in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Springer-Verlag,  Berlin, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' xiv+517 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Tran, Optimal convergence rates for elliptic homog-  enization problems in nondivergence-form: analysis and numerical illustra-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Multiscale Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Simul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=', 19(3):1453–1473, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  [46] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Yurinski, Averaging of second-order nondivergent equations with ran-  dom coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Sibirsk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 23(2), 176-188, 217 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  [47] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Yurinski, On the error of averaging of multidimensional diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Veroyatnost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' i Primenen 33(1), 14-24 (1988) [Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' transl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' in Theory Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' 33(1), 11-21 (1988)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  [48] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' Zeitouni, Random walks in random environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=' In Lectures on probability  theory and statistics, volume 1837 of Lecture Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content=', pages 189-312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  Springer, Berlin, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='  39  E-mail address, Xiaoqin Guo: guoxq@ucmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='edu  E-mail address, Hung Vinh Tran: hung@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
+page_content='wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfUfxu/content/2301.01267v1.pdf'}
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+page_content=' 2023  Typeset using LATEX default style in AASTeX631  Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  William Sheu  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
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+page_content=' 3 Aleksandar Cikota  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
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+page_content=' 7  1Department of Physics & Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
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+page_content=' University of San Francisco  2130 Fulton Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
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+page_content=' Chile  6Kavli Institute for the Physics and Mathematics of the Universe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' University of Tokyo  Kashiwa 277-8583,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Japan  7Institute for Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' University of Hawaii  2680 Woodlawn Drive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Honolulu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
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+page_content=' USA  ABSTRACT  The introduction of deep wide-field surveys in recent years and the adoption of machine learning  techniques have led to the discoveries of O(104) strong gravitational lensing systems and candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  However, the discovery of multiply lensed transients remains a rarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Lensed transients and especially  lensed supernovae are invaluable tools to cosmology as they allow us to constrain cosmological param-  eters via lens modeling and the measurements of their time delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' In this paper, we develop a pipeline  to perform a targeted lensed transient search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We apply this pipeline to 5807 strong lenses and candi-  dates, identified in the literature, in the DESI Legacy Imaging Surveys (Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2019) Data Release  9 (DR9) footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For each system, we analyze every exposure in all observed bands (DECam g, r,  and z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Our pipeline finds, groups, and ranks detections that are in sufficient proximity temporally  and spatially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' After the first round of inspection, for promising candidate systems, we further examine  the newly available DR10 data (with additional i and Y bands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Here we present our targeted lensed  supernova search pipeline and seven new lensed supernova candidates, including a very likely lensed  supernova — probably a Type Ia — in a system with an Einstein radius of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Keywords: Strong Lensing — Lensed Supernovae — Lensed Transient Pipeline  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' INTRODUCTION  The flat ΛCDM cosmological model is highly successful in describing our universe from the time of photon decoupling  (at a redshift z ≈ 1100) to the present time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  According to this model, our universe has a flat geometry and is  expanding at an accelerating rate (Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 1998, Perlmutter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The present day expansion rate of the  universe is known as the Hubble constant (H0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The inferred value for H0 from the Planck CMB measurements is  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5 km/s/Mpc (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' On the other hand, direct measurements of H0 by using local  distance ladders is higher by ≳ 4σ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Thus if this inconsistency is not due to systematic effects  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', Freedman 2021), then at a minimum, ΛCDM needs revision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Corresponding author: William Sheu, Xiaosheng Huang  wsheu@astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='edu, xhuang22@usfca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='edu  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='03578v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='CO]  9 Jan 2023  ID2  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The Legacy Imaging Surveys DR9 footprint, color coded by the z band observation depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We perform a targeted  search for lensed transients within the DECaLS region (dec < 32◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Exposures from the MzLS and BASS surveys (blue dashed  outline) are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  If a transient event were to occur in a strongly lensed background galaxy, it can be potentially observed multiple  times, in each of the lensed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The time delay ∆t between the different images consists of a geometric component  and a gravitational component (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', Narayan & Bartelmann 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Competitive H0 constraint can be achieved if the  time delays can be measured precisely and the lensing potential can be modelled accurately, providing an independent  method of measuring the Hubble constant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  If the observed transient is a lensed Type Ia  supernova (L-SN Ia), their standardizability can also significantly reduce the main systematic effect for lensing-based  H0 measurements: the mass sheet degeneracy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Currently, there have been seven confirmed lensed SNe: Quimby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' PS1-10afx), Kelly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' “SN  Refsdal”), Goobar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' SN 2016geu), Rodney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' “SN Requiem”), Kelly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' AT 2022riv),  Goobar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' “SN Zwicky”), and Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' C22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Out of the seven, four are lensed by a galaxy  cluster (“SN Refsdal”, “SN Requiem”, AT 2022riv, C22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Compared with galaxy-scale lenses, cluster-scale strong  lenses typically have much longer time delays and are much harder to model well to predict time delays accurately and  precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Of the remaining three which were lensed by single galaxies, two were found live (SN 2016geu, “SN Zwicky”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Both happen to be Type Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' However, both have short time delays of ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5 days, making them not useful for H0  measurements (Dhawan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Goobar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Pierel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' In this paper, we develop a targeted lensed  transient pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Applying our pipeline to the DESI Legacy Imaging Surveys (Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2019) in a retrospective  search, we have discovered seven new lensed supernovae candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Along with the discoveries of many lensed quasars  (which will be presented in a separate publication), we are confident our pipeline is capable of finding live lensed  transients for present (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', Pan-STARRS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Chambers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2016) and future surveys (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', LSST and the Roman Space  Telescope;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Ivezi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2019 and Spergel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2015 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' OBSERVATION  The DESI Legacy Imaging Surveys is composed of three surveys: the Dark Energy Camera Legacy Survey (DECaLS),  the Beijing Arizona Sky Survey (BASS), and the Mayall z-band Legacy Survey (MzLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' DECaLS is observed by the  Dark Energy Camera (DECam;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Flaugher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2015) on the 4-m Blanco telescope, which covers ∼ 9000 deg2 of the sky  in the range of −18◦ ≲ δ ≲ +32◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' BASS/MzLS are observed in the g and r bands by the 90Prime camera (Williams  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2004) on the Bok 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3-m telescope and in the z band by the Mosaic3 camera (Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2016) on the 4-m Mayall  telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Together BASS/MzLS cover the same ∼ 5000 deg2 of the northern subregion of the Legacy Surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For  this search, we exclude BASS and MzLS data as the number of exposures from each of the component surveys are  fewer, with inferior seeing in gr bands for reliable detection of transients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Data Release 9 (DR9) contains additional  75*  45"  10+  15*  I  1500  120  7-10  15"  30*  I-E  45"  60  75"Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  3  DECam data reprocessed from the Dark Energy Survey (DES;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Dark Energy Survey Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2016) for  δ ≲ −18◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This provides an additional ∼ 5000 deg2, resulting in a total footprint of ∼ 19, 000 deg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The DECam  surveys will hereafter be referred to in its entirety as DECaLS, within which we distinguish DES and non-DES regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  The pipeline deployment will mostly focus on the exposures in DECaLS, in g, r, and z filters, with a nominal DES  exposure time of 90 seconds, and non-DES exposure time ranging from 60 to 200 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' In addition, the Legacy  Surveys contain deep field DECam observations (from surveys such as COSMOS, XMM-LSS, and SN-X3), with 800+  exposures for any given target in these fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Our pipeline has been applied to DECaLS and these deep observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  As we take the approach of a targeted search, we compile a database of 5807 strong lenses and candidates found  within DECaLS, with the majority from Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2020, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2021, and Storfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2022, and the rest  from Moustakas 2012, Carrasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2017, Diehl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2017, Jacobs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2017, Pourrahmani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2018, Sonnenfeld  & Leauthaud 2018, Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2018, and Jacobs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Note that Storfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2022 only included C-grade or  above candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' However on the project website1, they also included D-grade candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' These receive numerical  scores of 1 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Here we include those with the higher numerical score of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5, which in this paper we will call D+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' PIPELINE  The general framework of the pipeline consists of image reprojection and reference image generation (§ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1), image  subtraction (§ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2), and source detection and grouping (§ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3) for each of 5807 lensing systems and candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Image Reprojection and Reference Image Generation  The pipeline first collects all relevant exposures from DR9 for all targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The images are then reprojected onto the  same World Coordinate System (WCS) orientation, with the system centered in each 801×801 pixel (216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='27′′×216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='27′′)  cutout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For each filter, we use the median coadd as the reference image, in order to reduce the influence of a potential  transient (and other time-dependent systematics such as cosmic rays).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' These steps are done using the Montage software  package (Jacob et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For exposures of the same band within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5 days of each other, we combine them as a  mean coadd, as we do not expect significant change in flux for an astrophysical transient in that short of a time frame,  while increasing detection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Image Subtraction  To find transients, we perform image subtraction between each exposure and reference image for the same filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  This pipeline uses two different image subtraction algorithms: that of Bramich (2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' henceforth B08), and Saccadic  Fast Fourier Transform (SFFT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  The B08 algorithm fits for a spatially varying kernel that attempts to convolve the reference image to appear  comparable to the image of each exposure (“science” image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' B08 uses delta functions as its basis functions, and thus  fits for every pixel in the kernel to minimize the χ2 of the difference image between the science image and the convolved  reference image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Because it fits for every pixel in the kernel, it makes no assumption on the functional form of the  fitted kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  SFFT is a fully Fourier implementation of image subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' SFFT performs Fourier transforms on both reference  and science images, and fits for a convolutional, also spatially varying kernel for the reference image (thus similar to  B08, but in the Fourier space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  We have experimented with other well-known image subtraction algorithms (Alard 2000, Zackay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2016), but  found that B08 and SFFT best suit this pipeline’s application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Alard (2000, henceforth A00), implemented in the  HOTPANTS package (Becker 2015), is widely used for large surveys and is very similar to B08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The main difference  is that A00 uses a set of Gaussian priors, and thus assumes a functional form of the convolutional kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Rather  than fitting for each pixel of the kernel (as with B08), A00 only has to fit for the Gaussian parameters, which leads  to significant speed-up for large scale surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' But since we are conducting a targeted search, our pipeline can afford  to use the slower, more flexible B08 algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Zackay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2016 (also known as ZOGY) takes a different approach  to the image subtraction problem, by utilizing the concept of cross-filtering (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', two separate convolutional kernels)  for the difference image generation and solving for both kernels in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We opt to use the SFFT algorithm,  as their results indicate an improvement to addressing photometric mismatch within image subtraction over ZOGY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  1 https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='com/usfca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='edu/neuralens  2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Storfer, private communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  4  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Despite appearing as a publication only recently, SFFT has been extensively applied to time-domain observations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  For almost all cases, we find that the SFFT algorithm produces cleaner and more accurate image subtraction compared  to B08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Thus, though both algorithms are used for transient detection, we use the SFFT difference images for all  subsequent photometry and detection presentation in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Source Detection and Grouping  After generating difference images with both image subtraction algorithms, we use a Python implementation (SEP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Barbary 2018) of the source extraction algorithm from Bertin & Arnouts (1996) to detect any potential sources in all  difference images, with thresholds ranging from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5σ in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='25 increments as determined by SEP (with detections  > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5σ treated as the same detection level as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Henceforth, we denote a detection in a single difference algorithm  as a “sub-detection” (a given transient event in a single exposure can generate two sub-detections, by being detected  in difference images produced by both the B08 and SFFT algorithms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' All sub-detections (from both subtraction  algorithms, across all bands and across all exposures) are then grouped together spatially and temporally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' These  groupings contain all sub-detections that are within three pixels (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8′′) of one another, and are within 50 days of other  sub-detections in the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' If a given group has less than three sub-detections, the pipeline disregards them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' If a  group has three or more difference image sub-detections, it is labelled as a possible transient detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' That is, an  event must be observed by DECaLS at least twice, in at least two separate exposures, to be labelled as a possible  transient detection4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This is to reduce the number of false detections (from noise, cosmic rays, CCD artifacts, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=') and  inconclusive events, and improve the overall quality of detections that will be visually examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Had we conducted  this search live, O(100) 2+ sub-detection groups would have been identified, most of which would warrant follow-up  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  The entire process typically takes about two hours to run per system, with approximately one hour for data collection  and reprojection and one hour for image subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' To run this pipeline on 5807 systems (approximately 120,000  individual exposure cutouts), parallelization is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The full deployment is performed on the National Energy  Research Scientific Computing Center (NERSC) Cori supercomputer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Using 20 nodes, 32 CPUs per node, and 1 thread  per CPU, this requires that each thread run 9 to 10 systems, taking a total of 18 to 20 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' SFFT is capable of  being run on GPUs with significant speed gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We will take advantage of this capability in our pipeline in the near  future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Twenty-five of the 5807 candidate systems lies within a deep field survey footprint, and thus each has 800+ individual  exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For these systems, the amount of memory and time required makes naively running the pipeline infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Instead, we opt to split up the exposures from these systems into smaller groups of temporally-similar exposures, and  running the pipeline on each possible pair of groups, for all permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The groups were created such that every  exposure appears in at least three groups, as to ensure that the pipeline does not miss a possible lensed transient in  any exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We do not find any lensed transients within the 25 deep field systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Selection Criteria  Human inspection of the pipeline results is necessary to validate the pipeline detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For first round visual  inspection, for each system, two initial grades are assigned regarding its most convincing group of detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Firstly,  we assign a location grade to assess how close the detection location lies relative to any putative lensed features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We  use this grade to filter out transient candidates that are clearly not lensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Secondly, a transient grade is given by how  likely the detection is a transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Most systems were not given any grades;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', there were no convincing detections for these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' As our pipeline’s  focus is to detect transients that are lensed, very obvious transients far from any lensing features are given high transient  grades, but low location grades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' By using these metrics, we identified the most promising lensed transient candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  We apply our pipeline to these select candidates a second time, with the change of removing all exposures that  contain the suspected transient detection from the median coadd, in order to generate a reference free of possible  transient light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Preliminary Legacy Surveys Data Release 10 (DR10) data are included in this second run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The new  data also includes observations from the DECam i and Y bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Using Point Spread Functions (PSFs) modelled from  isolated stars within the entire CCD exposure brick5, PSF photometry is then applied to all SFFT difference images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  3 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Wang, private communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  4 For example, at the threshold (inclusive), a group with three sub-detections can correspond to two or three detections (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', in two or three  exposures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' In the case of two detections, one of them is detected by both subtraction algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' In the case of three detections, each are  detected by a single subtraction algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  5 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='legacysurvey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='org/dr10/description/  Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  5  For exposures that we are not able to fit with a PSF at the detection location (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', non-detections that take place  well before or after the peak of the suspected transient event), aperture photometry is applied to the location of the  possible transient in order to establish a baseline flux for light curve fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For promising candidates, we apply our  own final set of criteria to select the candidates with the highest potential of being a L-SN:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Strong Lensing Plausibility - Because the lens candidates included in the search are from multiple publications  and different search efforts, we determine how likely a candidate is a strong lensing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The criteria we use  are similar to those in Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Asteroid Filtering - If there are only a few detections minutes apart in a given night, and no detections after that  night, this is an indication that transient is possibly an asteroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' To confirm this, we can approximate the speed  of the asteroid between detections (using PSF fitting to precisely locate the punitive asteroid), and compare it  to the speed of a typical main-belt asteroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Location Consideration - In combination with the location grade above, we also take into consideration sur-  rounding objects (in some cases, modeling their light profiles) and assess the overall probability of the transient  candidate being lensed based on the detection location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Light Curve Fit Quality - To narrow the possible identities of the detection, we fit a SALT3 (Kenworthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  2021) SN Ia light curve model, and 161 different core-collapse (CC) SN models to the observed photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' As  the survey data is sparse and the search is retrospective, we use a photometric redshift prior or a spectroscopic  redshift in the fitting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' A fit with low χ2/DOF is an indication of a possible identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' When a light curve  model fits the photometry well, we also assess whether the best-fit light curve model parameters are reasonable  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', the SALT2 x1 parameter is generally between −3 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Amplification/Hubble Diagram Residual - From the best-fit light curve models, we can further deduce whether  the transient is amplified, and if the amplification is reasonable given the system configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For the case of  a SN Ia, we can find its Hubble residual and determine the amplification (if any).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' EXPECTATIONS  To assess the feasibility of finding L-SNe Ia in the DESI Legacy Imaging Surveys, we can simulate SNe Ia light curves  at various redshifts and lensing amplifications of a lensed SN, and calculate the amount of time a given simulated SNe  will be detectable in the DECam g, r, and z bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We will be using the following values: 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='47, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='43, and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='63  for g, r, and z bands respectively, based on the 5σ PSF detection thresholds from Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2019), adjusted to the  nominal time of 90 seconds per exposure in DR9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We use SNCosmo (Barbary 2014) to simulate SNe Ia light curves,  based on the SALT3 model (Guy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Kenworthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We assume x1 = 0 and c = 0 for the SALT3  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Figure 2 illustrates the length of time that a L-SN Ia is detectable in the DECam g, r, and z filters across  a range of reasonable redshifts and amplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This initial investigation indicated that L-SN Ia are discoverable  in the Legacy Surveys and motivated this search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  As a careful forecast of lensed SN rates is beyond the scope of this paper, we use the formulation of Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2018,  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' henceforth S18) to provide a first-order estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Following S18, we simulate the star formation rate for a  source galaxy at a given redshift in order to sample SN rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  We can now estimate the number of lensed Type Ia and CC SNe we expect to find in our retrospective search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We  start by simulating the source redshift of a given lensing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' As most of our lens candidates are from Huang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2020), Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2021), and Storfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2022), we generalize the lens galaxy spectroscopic or photometric  redshifts from those candidates as the lens galaxy redshift6 distribution in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We then multiply this with  a truncated normal distribution N(2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5), with a lower bound at 1, to represent the source galaxy redshift distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 3 shows the source galaxy redshift distribution from which we sample for our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Using SFR estimations of varying redshifts (from z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2 to z = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8) from Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2007), Smit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2012), and  Sobral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2012), we fit a polynomial function (degree of three, using Numpy’s polyfit algorithm) to log10(SFR),  with the uncertainties given by the polynomial fit covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We sample SFRs at a given source redshift from  this polynomial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  6 https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='com/usfca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='edu/neuralens  6  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Number of days when a L-SN Ia is detectable as a function of redshift and lensing amplification in DECam g, r,  and z bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Every pixel in each plot represents a simulated SALT3 light curve model, color-coded by the number of days it  exceeds the corresponding filter’s 5σ PSF detection limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  From S18, we can convert SFRs to CC SNe rates:  RCC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0068M −1  ⊙  SFR  1 + zS  [yr−1]  (1)  The CC SNe rates are the broken down into sub-rates for the different types, based on the percentages in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  We can estimate the SFH from the functional form (Madau & Dickinson 2014), normalized by the recent SFR (S18):  SFR(t(z)) = SFR ×  �1 + z(t)  1 + zS  �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='7  �  �  �  1 +  � 1+zS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='9  �5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6  1 +  �  1+z(t)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='9  �5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6  �  �  �  (2)  From S18, by assuming a delay time (tD) distribution, fD:  fD(tD) ∝ t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='07  D  ,  (3)  we can estimate SN Ia rates:  RIa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='00084M −1  ⊙  � t(zS)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1  SFR(t(zS) − tD)fD(tD) dtD  (1 + zS)  � t(z=0)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1  fD(tD) dtD  [yr−1],  (4)  Each system is assumed to have two or four lensed images (with probabilities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='7 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3 respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Oguri &  Marshall 2010), and each image has a magnification sampled from a lognormal distribution (mean = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5 and standard  deviation = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='35), with an expected magnification of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='765.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' As context, Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2021) used a constant magnification  of 5 for their targeted rates estimations, whereas Craig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2021) performs a targeted estimate on a set of 40 strong  lenses (from Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2017) with magnifications ranging [2, 105], with a median of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We sample time delays  between each lensed image from N(36, 4) days (Craig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  We do not simulate the times of exposures, but rather use the true exposure times of our 5807 targets, observed in  the DECam g, r, and z bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Conservatively, we assume 90 seconds for each exposure (since while occasionally an  exposure can be as low as 60 seconds, the vast majority of the exposures are 90 seconds or longer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Assuming all 5807 systems in our catalog are real lensing systems, we simulate the expected results for each system,  using their time of exposures in the DECam g, r, and z bands for our search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We sample their source redshifts, number  of lensed images, lensing amplifications, lensing time delays, and star formation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Using these sampled values, we  calculate the Ia and CC SN rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Based on these rates, we simulate SNe across the duration of the DR9 observation  range (from the date of the first exposure −100 days to the date of the last exposure +100 days;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the time frame being  significantly larger than the typical width of Ia and CC light curve widths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' In Table 1 are the SNCosmo light curve  g band  r band  z band  10  70  9  Number of days SNla is detectable  60  50  7  Amplification  6  40  5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  4  20  E  10  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6  80  10  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6  80  10  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8  10  12  RedshiftRetrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  7  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Distributions used to simulate necessary parameters for calculating the expected results of our pipeline’s deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Top Left: star formation rate versus redshift, with the polynomial fit (degree of 3) to the data (Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2007, Sobral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2012,  Smit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The uncertainty bounds (shown in grey) is generated using the the covariance matrix of the resulting fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Top  Right: the assumed source redshift distribution (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Bottom Left: CC SNe rates verses redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The same polynomial  fit and uncertainty bounds are used to calculate these rates, according to equation 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Bottom Right: SNe Ia rates versus  redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The same polynomial fit and uncertainty bounds are used to calculate these rates, according to equations 2, 3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  models used during simulations, as well as the absolute B band magnitude distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For simulated SNe Ia, we  sample the following parameters as: c ∼ N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1) and x1 ∼ N(0, 1) (Guy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Scolnic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For CC  SNe, we subdivide and simulate them as Ib, Ic, IIn, IIp, IIb, or IIL, with their MB sampled from Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For all  simulated SNe, we assume a small amount of host galaxy dust (E(B − V ) ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Finally, we check if a lensed SN  image for a given system in a given band is above the detection limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  The full simulation of our pipeline (on the 5807 target systems) was parallelized and run 1000 times on NERSC  to estimate the expected SNe rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The final results are shown in Figure 4 and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' In order to reliably find  transients, we require a threshold of three sub-detections or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' As mentioned earlier (§ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3), depending on the  system, this means at least two or three detections, corresponding to the highlighted rows in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Polynomial Fit  80  Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2007  Sobral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8  Smit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2018  60  Probability Density  SFR [Meyr]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  0  5  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  i  2  E  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content="0  50  10  1'5  2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  Redshift  Redshift  12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='01 yr]  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6  TO\'0]"  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  0  1  2  E  4  5  6  0  i  2  m  5  6  Redshift  Redshift8  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Supernova simulation percentages and brightness parameters  SNe Type  MB  σ  % Rate of  CC Occurrence  SNCosmo Template  Template  (1)  (2)  (3)  (4)  (5)  (6)  IIp  −16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='98  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='83  nugent-sn2p  Gilliland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (1999)  Ic  −17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='18  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='00  nugent-sn1bc  Levan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2005)  IIb  −16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='92  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='43  v19-2006t-corr  Vincenzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2019)  Ib  −17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='33  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='12  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='00  nugent-sn1bc  Levan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2005)  IIL  −17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='86  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='34  nugent-sn2l  Gilliland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (1999)  IIn  −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='32  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='36  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='40  nugent-sn2n  Gilliland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (1999)  Ia  −19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='50 salt3  Kenworthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2021)  Note—This table shows the supernova-related parameters used in the pipeline simulated results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' As with  Craig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2021), the rate for 87A-like SNe (1%) is uniformly distributed across IIp, IIb, and IIL SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  All MB and σ values are reported in Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' CC SNe percentages are reported in  Eldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Expected numbers of lensed SN detections  Number of detections  L-SNe Ia  L-CC SNe  (1)  (2)  (3)  1 or more  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='49 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='14  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='74 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='85  2 or more  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='58 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='57  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='35 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='08  3 or more  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='90 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='68  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='53 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='12  4 or more  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='31 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='16  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='43 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='55  5 or more  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='80  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='42 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='16  6 or more  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='87  7 or more  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='68  Note—This table shows the final results of the 1000  simulated for finding lensed supernovae in the 5807  lenses and candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' To compare this forecast with  our targeted search, we highlight the rows for two  and three detections or more (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' TESTING DETECTION AND PHOTOMETRY PIPELINES ON KNOWN SNE IA  To test the performance of our pipeline, we apply it to photometric data from known SNe Ia discovered in DES  (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  From the DES SNe Ia, we select well observed and modelled SNe that  had at least two DR10 exposures within −15 to +30 days of the time of peak brightness in B band (or t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This  results in a set of 32 SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The SNe were previously modelled using SALT2 (extended by Hounsell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2018)  parameterization7, with a host galaxy dust extinction model (Fitzpatrick 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Modelled SALT2 parameters include  redshift z, t0, the normalization factor x0 (normalized so that the peak B band apparent magnitude is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5 when  x0 = 1, per SNCosmo), the “stretch” factor x1, and the color parameter c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  We plot our photometry together  7 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='com/sam-dixon/sncosmo lc fits  Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  9  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Results of the 1000 simulated runs for finding lensed SNe in the 5807 targeted systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Left: A histogram illustrating  the average number of systems (out of 5807) with three or more detections verses redshift, as well as the scaled sampled source  redshift distribution outlined in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The redshift distribution of detectable systems clearly is lower than the distribution we  sample from, due to detection limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The integrals of the distributions are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Center: A histogram illustrating  the number of systems with three or more detections verses magnification, as well as the magnification distribution we sample  from outlined in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' There is a slight bias towards higher magnifications, as would be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Right: A bar diagram  illustrating the representation of each type of simulated SNe in the systems with three or more detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  with previously observed photometry and the SALT2 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' All coadded RGB images are made with the Legacy  Surveys’ RGB image generation scheme8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Below we present the results for four SNe Ia systems at different redshifts  (z = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='18, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='40, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='53, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='69]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Results for the full 32 DES SNe Ia test systems are shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' All light curves  presented are in the observer frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  8 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='com/legacysurvey/imagine/blob/main/map/views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='py  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='7  Sampling Distribution  Sampling Distribution  (arbitrary scale)  (arbitrary scale)  CC  CC  la  la  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content="0  0'0  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  cc  la  IIn  Ic  Ib  IIb  IIL  Redshift  Amplification  SN Type10  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Detection and photometry results of our pipeline, for a known SN Ia at (RA, dec) = (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6006, −42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2977), z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1838.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Top Left: The mean coadded RGB (generated from g, r, i, and z bands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' see text) image, generated using exposures that include  the SN event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Top Middle: The mean coadded RGB image, generated using exposures that exclude the supernova event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The  dotted red circle indicates the location of the SN Ia event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Top Right: Photometry for the detected SN from the SFFT difference  images (solid points), plotted with DES photometry (fainter points) for this SN and the best-fit DES light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We note the  good agreement between our results and DES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Below the dotted line: examples of single band images in chronological order,  alongside its corresponding SFFT difference image (bottom) and SN detection, labelled with band, date of exposure, the phase,  and the σ-level of detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  SALT2  g  20  21  Z  23  Z= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1838  X = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3 × 104 c = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1117  t。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' = 56613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='398  X, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1838  24  30  20  10  0  10  20  30  40  50  60  w/oDetections  Phase (days)  g  g  11/06/2013  12/07/2013  12/11/2013  12/24/2013  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='37  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='76  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='75  0+36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='67  Difference Image  Detection  = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  Detection  = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  Detectiong=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  11  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Detection and photometry results of our pipeline, for a known SN Ia at (RA, dec) = (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0802, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4498), z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  For the arrangement of panels, see the caption of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  21  SALT2  g  r  z  22  Magnitude  23  Z= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3970  X。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='9 x 10-5 c = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1126  t。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' = 56926.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='649  X, =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='222  24  30  20  i0  0  10  20  30  40  50  60  MeanCoaddwloDetections  Phase (days)  09/24/2014  09/30/2014  09/30/2014  09/30/2014,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='34  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='64  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='64  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='64  Difference Image  Detection 0 =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  Detection  -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  Detection=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  Detection o = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='512  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Detection and photometry results of our pipeline, for a known SN Ia at (RA, dec) = (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2919, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5649), z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  For the arrangement of panels, see the caption of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  SALT2  23  24  26  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5308  X = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1 × 10-6  C = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0403  t = 57350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='737  X, =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='9183  27  20  10  0  10  20  30  30  40  50  60  Phase (days)  12/05/2015  12/05/2015  12/05/201  12/09/2015  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='38Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  13  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Detection and photometry results of our pipeline, for a known SN Ia at (RA, dec) = (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3699, −28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4430), z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  For the arrangement of panels, see the caption of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Note that there is no detection for the last exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We therefore  performed aperture photometry at the detected location of the transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  From these results, we found that our detection pipeline can detect known SN events within the DR9/DR10 data,  with a 100% detection rate for the 32 test SNe Ia from DES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Furthermore, the new photometry points from our pipeline  are consistent with the DES photometry for these SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' RESULTS AND DISCUSSION  We have identified seven lensed SN candidates, one unlensed SN, and two asteroids detections with our pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This  section will focus on one Grade A and two Grade B lensed SN candidates, with the lower grade lensed SN candidates  and the unlensed SN in Appendix B, and the asteroid detections in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We would also note that all eight SN  detections (summarized in Table 3) would have warranted additional follow-up observations if found live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  22  SALT2  23  24  25  20  27  28  g  29  r  30 Z= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6894  X。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='9 x 10 c = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0754  z  t。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' = 56659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='936  X, = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='003  31  30  20  10  0  20  30  1o  40  Phase (days)  01/21/2014  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2114  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Lensed and Unlensed Supernova Candidates  System Name  Overall  Grade  Total Number  of Exposures  (g, r, i, z, Y)  Number of PSF  Photometry  Exposures  (g, r, i, z, Y)  Distance  Grade  Postulation Shown  Redshift  used in  LC Prior  Best-fit  zSN  χ2/DOF  Hubble  Residual  Required  Amplification  Model  Fitting  Grade  Reason  (1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10)  (11)  (12)  (13)  (14)  DESI-344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6252-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8977  A  (10, 8, 7, 7, 7)  (1, 2, 1, 0, 1)  A  uL-SN Ia  Y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='374 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='053  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='299 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='021  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='47  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='24 D  [1] [2]  L-SN Ia  Y  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='188 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='255  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='833 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='042  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='81  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='30  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='23+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='61  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='98  A  uL-CC SN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='374 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='053  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='393 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='026  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='93 D  [1]  L-CC SN  Y  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='188 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='255  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='731 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='049  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='75 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='28+52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='80  −22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='89  B  DESI-058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6486-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5959  B  (10, 11, 9, 9, 6)  (2, 2, 0, 0, 0)  B  uL-SN Ia  Y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1702  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1702  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='20 B−  L-SN Ia  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6735  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6735  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='61  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='04  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='82+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='54  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='52  C  [1]  uL-CC SN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1702  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1702  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='40 C  [1]  L-CC SN  Y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6735  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6735  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='43 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='91+82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='66  −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='92  B+  DESI-308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='7726-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2381  B  (12, 12, 10, 15, 8) (0, 1, 1, 1, 0)  A  uL-SN Ia  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='473 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='032  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='448 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='034  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='59  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='61 D  [1] [2]  L-SN Ia  Y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='869 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='021  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
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+page_content='391 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='011  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='41  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='20+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='12  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='11  D  [3]  uL-CC SN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='361 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='406 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='006  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='64 D  [1]  L-CC SN  Y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='593 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='146  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='819 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='064  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='02 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='05+153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='15  −55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='10  C  [1]  DESI-015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8465-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5450  N/A  (9, 7, 4, 9, 5)  (1, 1, 0, 0, 0)  N/A  uL-SN Ia  Y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='301 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='026  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='373 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='131  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='58  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='91 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='52 B  L-SN Ia uL-CC SN  Y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='301 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='026  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='289 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='036  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='30 B  L-CC SN Note—Summary table for our eight lensed and unlensed supernova candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Column 2: A wholistic grade on how likely a detection is a L-SN, based on  the criteria in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Column 5: The distance grade, based on the location of the transient (for first round visual inspection, § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Column 6: The four  postulations for each candidate, uL-SN Ia, L-SN Ia, uL-CC SN, L-CC SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Column 7: Only light curve models for postulations marked with “Y” in this column  are shown (in § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1 and Appendix B, above and below the horizontal line, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Column 8: For redshift prior in the fitting process, we use photometric  redshifts from Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2020, shown with three decimal places and uncertainties), or fix them to be the spectroscopic redshifts from Cikota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  (in  prep, shown with four decimal places and no uncertainties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Columns 14 and 15: Each postulation is given a grade based on the light curve fitting and/or  the Hubble residuals, as well as reasons for a C or D grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The reasons correspond to the following: [1] poor fit in comparision to the other postulations, [2]  large Hubble residual in the case of uL-SN Ia, and [3] best-fit zSN too close to the lens photo-z, and hence inconsistent with a lensing postulation (Note that,  to be complete, we nevertheless include the best-fit zSN and magnification information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  15  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Grade A & B Lensed Supernova Candidates  The first seven systems (of eight) in Table 3 are identified by the pipeline and determined by visual inspection to be  lensed SN candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For the eighth system, we believe that it is almost certainly not a strongly lensed system, and  have given it an overall grade of “N/A”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' In this section, we will present the best three lensed SN candidates, with the  remaining four (and one unlensed SN) candidates presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  For each system, we attempt to narrow the identity of the transients by fitting different light curve models to the  photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We test for four different postulations for each system: unlensed SN Ia (uL-SN Ia), lensed SN Ia (L-SN Ia),  unlensed CC SN (uL-CC SN), and unlensed CC SN (uL-CC SN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' In this paper, we present figures only for the most  probable scenarios (systems with “Y” in the “Shown” column 7 of Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  We account for Milky Way extinction according to Schlafly & Finkbeiner (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' When fitting a SN Ia light curve  model (for uL-SN Ia and L-SN Ia), we use the SALT3 model (Kenworthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2021), and fit for the parameters:  redshift z, time of B band peak t0, the normalization factor x0, the “stretch” factor x1, and the color parameter c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  We do not fit for host-galaxy dust, as the c parameter would be largely degenerate with small amounts of reddening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  We use the following priors in the fitting process: c ∼ N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1) and x1 ∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We then use the best-fit SALT  parameters for “stretch” and color corrections, and find the Hubble residual for the given model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This is plotted  together with SNe Ia from Suzuki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Though this data is modelled with SALT2, we expect negligible  differences in Hubble residuals of 7 ± 11 mmag (Kenworthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' To model CC SNe light curves, we fit for  161 separate CC SNe templates (as provided by SNCosmo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' All CC templates are parameterized by only t0, z, and  amplitude (a scaling term with arbitrary units), allowing for a small amount of host galaxy dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' In all the CC SN  light curves shown in this paper, the best-fit E(B − V ) values are very small (≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='01), and therefore we do not report  the reddening parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Except for one system, redshift priors used for both CC SNe and SNe Ia postulations  (see column 8 of Table 3) are photometric redshifts of objects identified by the forward modeling source extraction  algorithm, the Tractor (Lang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' All photometric redshifts in this paper are from Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For  DESI-058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6486-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5959, we fix the redshifts to be the spectroscopic redshifts (Cikota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  In the figures below, if a lensed scenario is postulated, we estimate the amplification, µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' If a CC SN scenario is  postulated, the expected peak B band magnitude for a given SN type (“X”) is represented with “E(MB | X),” where  the values from Table 1 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' DESI-344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6252-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8977  DESI-344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6252-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8977 is a strong lensing candidate discovered in Storfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2022) and assigned a C-grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' In  our analysis, we find this relatively-low grade was given because in the Legacy Surveys coadded image, the arc and  counterarc (objects 2 and 3 respectively, in Figure 9) appear to have somewhat different colors due to the transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  However, with an improved analysis of the color of the arc and counterarc by the criteria laid out in Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  (2021), taking into account of the presence of the transient and photo-z (see below), we now regrade this system as  an A-grade lensing candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  16  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Coadded RGB images (using g, r, i, and z bands) of L-SN Ia candidate DESI-344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6252-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8977 with and without  the transient detection (red dotted circle) exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The arrows are color-coded as the following postulated scenario: red for  the lens galaxy, green for the lensed source galaxy, and purple for a possible second lensed source or an interloper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Photometric  redshifts are displayed on the top right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Detection exposures for the transient in DESI-344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6252-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8977 in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Top row: single pass images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Bottom row: corresponding differencing images from SFFT, where the red circle indicates the location of the detected transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  DESI-344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6252-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8977  1: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='374 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='053  2: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='403 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='269  3: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='188 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='255  4: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='519± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='297  DetectionsCoadd  Mean  CoaddwoL08/17/2014  08/22/2014  09/26/2014Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  17  The lensed arc in DESI-344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6252-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8977 is located Southwest of the lens, with its counterimage appearing Northeast  of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The color and photo-z’s of the two lensed images agree with each other (within uncertainties), with both  photo-z’s being significantly higher than the lens photo-z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We note that for object 3, as the majority of the exposures  do not contain the transient, its light is unlikely to significantly affect the photo-z of object 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  As the lens and  source galaxies appear to be red elliptical galaxies, we consider their photo-z’s to be reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The image-counterimage  arrangement indicates a strong lensing configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The detected transient lies directly on the counterimage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' As with  all other light curves presented, the light curves below are constrained by both detection and non-detection exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Postulation 1: uL-SN Ia —Figure 11 shows the best-fit light curve model for the uL-SN Ia scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We see that it is  not a good fit (χ2/DOF = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='47), especially in the r band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Additionally, the resulting SALT3 model has a statistically  significant Hubble residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Thus, we believe this detection is unlikely to be an uL-SN Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Not shown is the uL-CC SN  scenario, which also (as with the uL-SN Ia scenario) has a large χ2/DOF (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Best-fit SALT3 model for DESI-344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6252-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8977 with a lens photo-z redshift prior of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='374 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For this and  Figures 12 and 13, solid photometry points correspond to the detection passes shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  HubbleDiagram  SALT3  21  46  Fitted Cosmology  g  SN la Detection  Union2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1 SNe la  44  Y  itude)  22  (Magni  42  40  23  Magnitude  24  36  34  Hubble residual=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='24  25  1  Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='299±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='021  X, = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='341 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='487  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='= 56902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='76  C=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='192±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='069  0  X = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='27) × 10-5  26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  20  10  0  10  20  30  40  50  60  Redshift  Phase (days)18  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Postulation 2: L-SN Ia —Figure 12 shows the best-fit light curve model for the L-SN Ia scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Due to the high  redshift, SALT3 cannot model the g band observation and it is ignored in the fitting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For the three redder  bands, the best-fit SALT3 curve model agrees well with the photometric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Based on the Hubble residual, the  implied amplification is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='23+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='61  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This is consistent with the expectation of a multiply imaged SN by a galaxy scale  lens (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Finally, the best-fit SN redshift, zSN = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='333 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='042, is consistent with the photo-z of the  source galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We note that this redshift value is in line with our preliminary investigation of the feasibility of our  search (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Best-fit SALT3 model for DESI-344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6252-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8977 with a source photo-z redshift prior of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='188 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  HubbleDiagram  SALT3  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  46  Fitted Cosmology  SN la Detection  Union2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1 SNe la  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  44  (Magnitude)  42  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  40  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  36  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  Hubbleresidual=-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='29±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='30  34  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  0  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='833 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='042  X,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='077±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='618  2  t。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='= 56911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='80 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='43  C=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='042±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='059  X= (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='74± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='53) × 10-5  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  20  20  40  60  Redshift  Phase (days)Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  19  Postulation 3: L-CC SN —Figure 13 shows the best-fit light curve model for the L-CC SN scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  As with the  previous postulation, this model cannot fit the g band datapoint due to the high redshift, and thus is ignored in the  fitting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The best-fit model (the “nugent-sn2l” SN IIL template) is in good agreement with the photometric  data from the redder bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' However, the implied magnification is quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Despite this, we note that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=') such  a magnification is not impossible (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', Quimby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2014), and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=') since CC SNe have a large range of MB, the  uncertainties of the amplification is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Therefore the L-CC SN scenario is still plausible for this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Best-fit core collapse template model for DESI-344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6252-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8977 with a lensed source photo-z redshift prior  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='188 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Conclusion —Any unlensed SN postulation seems unlikely, due to the poor agreement between the light curve model  and the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Additionally, for the uL-SN Ia scenario, the Hubble residual would be too high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' With all the evidence  considered, this is very likely a lensed SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' A L-SN CC is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' But, we believe it is most likely a L-SN Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This  conclusion is based on the following:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The red color and morphology seem to indicate that the putative lensed source is an elliptical galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The  foreground galaxy is clearly an elliptical galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Thus, the photo-z’s for the punitive lens and source are both  likely reliable, with the later being significantly higher than the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Based on this, combined with the classic  image-counterimage configuration, we regard this system as a grade-A lens candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The putative SN is situated directly on the counterimage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Given the source galaxy is likely an elliptical galaxy, the SN is more likely a Ia than CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The SN Ia light curve model is a good fit to the photometry and the Hubble residual is most consistent with it  being lensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For this scenario, the amplification is also consistent with galaxy-scale strong lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  If it is indeed a lensed SN, this would be the first galaxy-scale strongly lensed SN resolved by ground-based observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Furthermore, either for the case of L-CC SN or L-SN Ia, it is at a significantly higher redshift (zSN ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8) than the  nugent-sn2l : SN IIL  20  21  22  Magnitude  23  24  t。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' = 56881.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='72  Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='731 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='049  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='14  E(Mg / SN IIL) = -17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='90  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='89  25  20  0  20  40  60  80  100  Phase (days)20  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  other two resolved galaxy-scale strongly lensed SNe (Goobar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2017, Goobar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Given the Einstein  radius is ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5′′, the expected time delay would be on the order of weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' If caught live, it could have resulted in  a H0 measurement competitive with those from lensed quasars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Additionally, if it were a  L-SN Ia, the systematic effect of the mass sheet degeneracy could be significantly reduced due to the standardizability  of its brightness (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Therefore, if discovered live, this system would make a strong case for high  resolution imaging and spectroscopic follow-up observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='9  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' DESI-058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6486-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5959  DESI-058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6486-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5959 was discovered in Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2021), as a grade-A strong lensing candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Coadded RGB images (using g, r, i, and z bands) of DESI-058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6486-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5959 with and without the transient  detection (red dotted circle) exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Labelled objects are color-coded as the following postulated scenario: red as the lenses  and green as the source galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Photometric (shown with uncertainties) and spectroscopic (shown without uncertainties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Cikota  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' in prep) redshifts for the labelled objects are displayed on the top right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  9 At the present, it is still possible to obtain the source galaxy spectra to measure its star formation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This in turn would more precisely  quantify the SN Ia likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  DESI-058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6486-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5959  4: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6721  : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1768  2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1702  5: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='545 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='194  3: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6735  6: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='524 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='124  Possible  Detections Coadd  Mean  Coadd w/o DetectionsRetrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  21  Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Detection exposures for the transient in DESI-058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6486-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5959 in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' See caption of Figure 10  for the full description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Mean coadded images of DESI-058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6486-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5959 across different observed bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' These deeper images (especially  the r and i bands) seem to support the possibility of a faint lensed image between the detection (red dotted circle) and the arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  01/04/2016  01/04  201  01/16/2016  01/16/2016  mjd: 57403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='17DECamg  DECamr  Mean Coadd  MeanCoadd  DECami  DECamz  MeanCoadd  MeanCoadd22  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  DESI-058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6486-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5959 is a galaxy-group lensing system, with a prominent red arc Southeast of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Including  additional observations from DR10, there appears to be a faint, highly magnified blue arc Northwest of the lens as  well, somewhat further from the estimated center of mass for the foreground galaxy group than the red arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For this  system, we have obtained VLT MUSE spectroscopy with preliminary redshifts (Cikota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' in prep) for objects 1, 2,  3, and 4 in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The clearly detected transient lies ∼ 3′′ from the tip of the red arc (object 4, spectroscopically  confirmed to be in the background) in the deep i band image (Figure 16, lower left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The deep z band image  (Figure 16, lower right panel) seems to show that the arc curves towards the direction of the transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' There also  seems to be a very faint galaxy (object 6) between the red arc and the transient location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The source extraction  code for the Legacy Surveys, the Tractor, also identifies this object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' It has similar photo-z (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2020) as the  aforementioned blue arc (object 3, also spectroscopically confirmed to be in the background), and therefore could  possibly be its counter-image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Postulation 1: uL-SN Ia —Figure 17 shows the best-fit light curve model for the uL-SN Ia scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We see that the  second r band photometry point is not well accounted for in this SALT3 model, but not at an unreasonable level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Additionally, the Hubble residual indicates that this scenario is unusually faint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Best-fit SALT3 model for DESI-058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6486-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5959 with a lens photo-z redshift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For this and Figure 18,  solid photometry points correspond to the detection passes shown in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  HubbleDiagram  SALT3  20  46  Fitted Cosmology  g  SN la Detection  Union2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1 SNe la  44  (Magnitude)  21  42  40  22  Magnitude  23  36  34  Hubbleresidual=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='66±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1702  X, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='174 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='864  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='= 57405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='15± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='08  C= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='086±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='054  44  X = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='19) × 10-5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  30  20  10  0  10  20  30  40  50  60  Redshift  Phase (days)Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  23  Postulation 2: L-CC SN —Figure 18 shows the best-fit light curve model (with the best-fitting parameters of the best-  fitting CC SN templates) for the L-CC SN scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The best CC SN template to the data is the “nugent-sn2n”  SN IIn model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This model seems to provide the best overall fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The implied amplification would be 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='91+82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='66  −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Depending on the location of the SN relative to the lensing critical curve, a large amplification for a group-scale lens  is not impossible (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', SN Refsdal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Kelly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2016 and Rodney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We also note the large uncertainty, as  typical for CC SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Best-fit core collapse template model for DESI-058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6486-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5959 with a source photo-z redshift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Conclusion —The rise time for the transient in DESI-058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6486-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5959 is consistent with it being a SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' If so, there  are three possibilities for the host galaxy – objects 2, 4, or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Object 2 is clearly an elliptical galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Thus, if it hosts  a SN, it is more likely a Ia than CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Figure 17 shows that the SALT3 fit for a SN Ia at its spectroscopic redshift  cannot account well for the r band photometry near maximum light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Furthermore, the Hubble residual of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='66 mag,  is unusually large at > 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Object 4 appears to be at the greatest angular separation from the transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' However,  given the high degree of distortion due to lensing, without lens modeling, it is difficult to meaningfully assess how far  away the SN is from object 4 — e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', in terms of half-light radius or directional light radius (Sako et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2018) if the  delensed source is highly elliptical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Object 6 has the least angular separation from the transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' By color, location,  and photo-z, it appears to be the possible counterarc of the large arc (spectroscopically confirmed) to the Northwest  of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The L-CC SN postulation is also consistent with object 6 appearing to be a blue, and therefore likely  star-forming, galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Given the sparsity of the photometric data, it is difficult to be certain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' All factor considered, we  assigned a grade of B to this transient as a lensed SN (more likely a CC than Ia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Additional spectroscopic observation  of object 6 can test whether it is a lensed counterimage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' We also note that if this transient was detected live, real time  photometric and spectroscopic follow-up observation could be triggered to determine the nature of this transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' DESI-308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='7726-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2381  DESI-308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='7726-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2381 was discovered in Storfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2022), and is given a D+ grade strong lensing candidate (see  § 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' If it turns out to be a lensing system, the location of the detection would lie directly on the arc (Figure 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  nugent-sn2n : SN Iln  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  g  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  Magnitude  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  t。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='= 57384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='10  Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6735  M, = -22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='23  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  E(Mβ / SN Iln) = -18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='53 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='48  μ= 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='91 +82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='66  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='92  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  0  20  40  60  80  100  Phase (days)24  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Coadded RGB images (using g, r, i, and z bands) of DESI-308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='7726-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2381 with and without the transient  detection (red dotted circle) exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Labelled objects are color-coded as the following postulated scenario: red as the lens  galaxy and green as the source galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Photometric redshifts are displayed on the top right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Detection exposures for the transient in DESI-308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='7726-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2381 in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' See caption of Figure 10  for the full description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  DESI-308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='7726-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='238  : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='473 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='032  2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='948 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='375  3: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='427 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='340  4: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='748 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='408  5: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='652 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='363  DetectionsCoado  Mean Coadd w/oDete09/19/2014  09/23/2014  mid:56919.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='99  G:  DiffeRetrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  25  For DESI-308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='7726-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2381, there is a possible arc stretching from object 3 to 4 (in Figure 19), with object 2 as a  counterimage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' all three objects are identified by the Tractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The transient candidate is only detected twice: in the  r-band, and four days later in the z-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This system serves as an example of how our pipeline is able to detect  transients with only two detections, as the event was captured in at least three sub-detections (see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' These  detections are visually comparable to the difference and detection images of known high-redshift SNe Ia in the § 5  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', Figures 7 and 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' While there are only two detections, we are reasonably confident that this is an astrophysical  transient, as forced photometry in other bands at the detection location supports this postulation (Figures 21 to 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Postulation 1: L-SN Ia —Figure 21 shows the best-fit light curve model for the L-SN Ia scenario with no prior on the  redshift (given how broad the photometric redshifts of the punitive lensed images are).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The SALT3 parameters are all  reasonable with a best-fit redshift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The resulting SALT3 model seems to fit the photometric data reasonably  well, with an amplification of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='62+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='02  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Best-fit SALT3 model for DESI-308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='7726-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2381 with no redshift prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For this and Figures 22 and 23, solid  photometry points correspond to the detection passes shown in Figure 20, hollow points correspond to other exposures with  PSF photometry, and crosses correspond to measurements using aperture photometry (§ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  HubbleDiagram  SALT3  20  46  Fitted Cosmology  SN la Detection  Union2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1 SNe la  21  44  Y  (Magnitude)  42  22  40  23  36  25  Hubble residual =-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='91±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='32  34  26  0  27  Z= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='869± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='021  X, =-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='822±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='188  2  t= 56940.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='11  C = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='051± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='079  +  X。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='= (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='71) × 10-5  28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  20  0  20  40  60  Redshift  Phase (days)26  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Postulation 2: uL-CC SN —Figure 22 shows the best-fit light curve model for the uL-CC SN scenario, which appears to  be far worse compared with Postulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' To reiterate, this is the best-fitting CC SN template model of 161 templates  supplied by SNcosmo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Not shown is the uL-SN Ia fit, which results in a model too bright for what is expected of a SN Ia (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Best-fit core collapse template model for DESI-308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='7726-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2381 with a lens photo-z redshift prior of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='473 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  v19-2011hs-corr : SN lb  21  Z  22  23  Magnitude  24  25  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' = 56933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='56 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='61  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='465 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='026  26  M, = -18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='21 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='43  E(M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' / SN IIb) = -16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='93  20  0  20  40  60  Phase (days)Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  27  Postulation 3: L-CC SN —Figure 23 shows the best-fit light curve model for the L-CC SN scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The best-fit redshift  is at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='828, and the required amplification is 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='61, albeit with a large uncertainty, as is typical for CC SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The data  appear to rise more rapidly than the model, but this scenario may still be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Best-fit core collapse template model for DESI-308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='7726-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2381 with no redshift prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Conclusion —The transient found in DESI-308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='7726-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2381 is likely a lensed SN (more likely a Ia than CC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' As  with the previous two candidates, to be more confident of the lensing nature of this system, higher resolution and/or  spectroscopic observations are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' CONCLUSION  We have developed a pipeline for a targeted search for lensed transients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  For 5807 strong lensing systems and  candidates observed by the DESI Legacy Imaging Surveys, this pipeline first generates a median coadd for each  observed band as a reference image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' It then employs two image subtraction algorithms to identify transient detections  that are in close proximity both spatially and temporally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' By applying this pipeline to the DESI Legacy Imaging  Surveys DR9/10, we have found seven lensed SN candidates, one unlensed SN, and two asteroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  We have also  confirmed the variability of a large number of lensed quasars, which we will present in a subsequent paper (Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Of the seven lensed SN candidates, the one in DESI-344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6252-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8977 is very likely a galaxy-scale strongly  lensed SN, probably a Type Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Follow-up high resolution imaging and spectroscopy, as well as lens modeling, can help  reach a more definitive conclusion on whether some of these transient candidates are lensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Of our grade A and B candidates, the transients in DESI-344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6252-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8977 and DESI-308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='7726-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2381 are likely  L-SNe Ia, whereas the transient in DESI-058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6486-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5959 is likely a L-CC SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Preliminary results indicate that half  of the 5807 systems, for which we have conducted the search, are actually strong lenses (Tabares-Tarquinio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' in  prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Since the uncertainties of our forecast results in Table 2 are Poisson in nature, we adjust the number of L-SNe Ia  and L-CC SNe with two or more detections to be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='29 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='82 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='68 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='18, respectively, and the corresponding  v19-2010al-corr : SN llin  21  Z  22  Y  23  Magnitude  24  25  t。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' = 56941.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='86  26  Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='828 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='057  Mg = -21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='19  E(M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' / SN IIn) = -18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='53 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='48  μ= 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='61 +45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='94  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='66  27  20  0  20  40  60  Phase (days)28  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  numbers for three or more detections to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='45±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='19 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='27±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The results from our grade A and B candidates  are broadly consistent with these forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  We believe that these results demonstrate the very promising viability of our pipeline and its applicability to future  surveys such as the Vera C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Rubin Observatory Legacy Survey of Space and Time (LSST) and the Nancy Grace Roman  Space Telescope (RST) to find live lensed SNe and other types of transients, as well as lensed quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Assuming the  trend of three high grade lensed SN candidates for every 5807/2 ≈ 3000 systems found in our search, we can estimate  the lower-bound for the number of lensed SN candidates in these upcoming surveys: 17 for RST (∼ 17, 000 strong  lensed systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Weiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2020) and O(102) for LSST (∼ 105 strong lensed systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', Collett 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' These  bounds do not account for the obvious improvements to seeing and cadence these surveys have over the DESI Legacy  Imaging Surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Lensed transient searches in these future surveys will likely realize the potential to dramatically  improve lens modeling and possibly resolve the H0 tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This work was supported in part by the Director, Office of Science, Office of High Energy Physics of the US  Department of Energy under contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' DE-AC025CH11231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This research used resources of the National Energy  Research Scientific Computing Center (NERSC), a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Department of Energy Office of Science User Facility operated  under the same contract as above and the Computational HEP program in The Department of Energy’s Science Office  of High Energy Physics provided resources through the “Cosmology Data Repository” project (Grant #KA2401022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Huang acknowledges the University of San Francisco Faculty Development Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  This paper is based on observations at Cerro Tololo Inter-American Observatory, National Optical Astronomy  Observatory (NOAO Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' ID: 2014B-0404;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' co-PIs: D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Schlegel and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Dey), which is operated by the Association of  Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  This project used data obtained with the Dark Energy Camera, which was constructed by the Dark Energy Sur-  vey collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Funding for the DES Projects has been provided by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Department of Energy, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  National Science Foundation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the Ministry of Science and Education of Spain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the Science and Technology Facili-  ties Council of the United Kingdom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the Higher Education Funding Council for England,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the National Center for  Supercomputing Applications at the University of Illinois at Urbana-Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the Kavli Institute of Cosmolog-  ical Physics at the University of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the Center for Cosmology and Astro-Particle Physics at the Ohio State  University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Financiadora  de Estudos e Projetos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Funda¸c˜ao Carlos Chagas Filho de Amparo `a Pesquisa do Estado do Rio de Janeiro,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Con-  selho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico and the Minist´erio da Ciˆencia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Tecnologia e Inovac˜ao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the  Deutsche Forschungsgemeinschaft,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' and the Collaborating Institutions in the Dark Energy Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The Collaborating  Institutions are Argonne National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the University of California at Santa Cruz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Centro de Investigaciones En´ergeticas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Medioambientales y Tecnol´ogicas-Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the University of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' University  College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the DES-Brazil Consortium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the University of Edinburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the Eidgen¨ossische Technische Hochschule  (ETH) Z¨urich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Fermi National Accelerator Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the University of Illinois at Urbana-Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the Institut de  Ci`encies de l’Espai (IEEC/CSIC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the Institut de F´ısica d’Altes Energies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the  Ludwig-Maximilians Universit¨at M¨unchen and the associated Excellence Cluster Universe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  the National Optical Astronomy Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the University of Nottingham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the OzDES  Membership Consortium the University of Pennsylvania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the University of Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' SLAC National Accelerator  Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Stanford University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' the University of Sussex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' and Texas A&M University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  The work of Aleksandar Cikota is supported by NOIRLab, which is managed by the Association of Universities for  Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  We thank Alex Kim at the Lawrence Berkeley National Laboratory for insightful discussions on difference image  photometry, as well as Saul Perlmutter and Greg Aldering for general commentary on our paper results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Software: Astropy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2013, 2018), Montage (Jacob et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2010), SEP (Bertin & Arnouts  1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Barbary 2018), SNCosmo (Barbary 2014), NumPy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2020), Matplotlib (Hunter 2007)  Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  29  APPENDIX  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' PHOTOMETRY ON PREVIOUSLY DISCOVERED SNE IA  Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Our photometry (solid data-points) for new detections of 9 (of 32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' for the rest see Figures 25 and 26) known SNe Ia  from DES, shown along with DES photometry (faint points) and their best-fit SALT2 light curve11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For all panels, we follow  the color scheme of blue=g band, green=r band, yellow=i band, and red=z band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The redshift and the (RA, dec) is given on  the top right of every plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Note that our measurements match well with DES photometry, and provide additional photometry  points for these SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
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+page_content='689  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='274  24 -  (+53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5027, -28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6602)  (+53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3699,-28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4430)  (+41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1435, --1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2852)  23  26  21  24 -  28  25  26  22  32  27  te  28-  29  36  30  23 +  30  20  10  0  10  20  30  40  50  60  30  20  10  0  10  20  30  40  50  60  30-20  10  0  10  20  40  50  60  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='297  21 -  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='363  22  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='313  (+36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0028, -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4975)  (+42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8038, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6690)  (+36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1027, -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6752)  22  22 -  23  23  24  24  24 -  25  26  25  26  26 -  27  28  27  28  30 +  28  29-  30  20  10  0  10  20  30  40  50  60  30  20  10  0  10  20  30  40  50  60  30  20  10  0  10  20  30  40  50  60  22  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='425  22 -  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='612  (+35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4328, -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='9404)  (+36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2433, -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2508)  24 -  24  26  26 -  28  28  30  30  32  30  20  10  0  20  30  40  60  30  20  10  10  20  30  50  60  Phase (days)32  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' GRADE C & D LENSED SUPERNOVA AND UNLENSED SUPERNOVA CANDIDATES  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' DESI-034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3625-35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3563  DESI-034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3625-35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3563 is a C-grade strong lensing candidate, discovered in Storfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Coadded RGB images (using g, r, i, and z bands) of DESI-034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3625-35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3563 with and without the transient  detection (red dotted circle) exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Labelled objects are color-coded as the following postulated scenario: red as the lens  galaxy, green as the source galaxy, and cyan as an interloper or a second source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Photometric redshifts are displayed on the top  right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  DESI-034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3625-35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3563  1: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='240 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='008  2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='836 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='543  Probable aro  3:  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='657 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='556  4:  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='813 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='595  5: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='799 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='557  6:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='692 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='130  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='532 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='101  cl  Detections Coadd  Mean Coadd w/o DetectionsRetrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  33  Figure 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Detection exposures for the transient in DESI-034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3625-35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3563 in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' See caption of Figure 10  for the full description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Residual image after lens light subtraction for DESI-034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3625-35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3563, with the S´ersic light parameters (half-light  radius, axis ratio, semi-major axis orientation from the y axis) in the top left (modelled in DECam g filter), with the core of  the lens galaxy masked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  DESI-034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3625-35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3563 is a strong lensing candidate system with a single massive galaxy as the main lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' There  appear to be two faint arcs, located North (identified by Tractor as objects 4 and 5 in Figure 27) and South (objects  2 and 3) of the foreground galaxy, at approximately four arcseconds away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Given the morphology and similarity in  color, they quite possibly correspond to the same background source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The transient lies directly at the east end of the  first arc (object 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This transient is also only about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5 effective radii (Figure 29) away from the lens, and so it is  possible that the foreground galaxy is the host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  10/19/2017  10/20/2017  10/28/2017  58054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='33  DifferenceImage  ectiong== 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='331  q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='481  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='15  0=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='177  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0534  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Postulation 1: uL-SN Ia —Figure 30 shows the best-fit SALT3 light curve model for the uL-SN Ia scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This model  agrees well with the data, with reasonable light curve parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The inferred absolute magnitude is consistent with  the expectation for a SN Ia at the redshift of the foreground galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Therefore, this seems to be a likely identity the  transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Best-fit SALT3 model for DESI-034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3625-35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3563 with a lens photo-z redshift prior of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='240 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For this and  Figure 31, solid photometry points correspond to the detection passes shown in Figure 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  HubbleDiagram  SALT3  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  46  Fitted Cosmology  g  SN la Detection  Union2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1 SNe la  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  44  (Magnitude)  42  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  40  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  36  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  34  Hubble residual =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='25  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='238±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='011  X, =-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='192±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='861  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='= 58037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='17 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='69  C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='051±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='069  X。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='54) x 10-5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  30  20  i0  0  10  20  30  40  50  60  Redshift  Phase (days)Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  35  Postulation 2: L-CC SN —Figure 31 shows the best-fit light curve model for the L-CC SN scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This SN IIP  template fit has a slightly worse χ2/DOF compared to Postulation 1 (see Table 3), but this scenario is nevertheless  possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The model does require a fairly high amplification of 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='16+50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='83  −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='79, albeit with large uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Best-fit core collapse template model for DESI-034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3625-35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3563 with no redshift prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Conclusion —The photometry seems to suggest that this detection is an uL-SN Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' However, if additional high resolution  and/or spectroscopic observation can confirm the faint lensed arc North of the lens, the data would strongly support  the postulation of a L-CC SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This points to the importance of timely follow-up if this were a live detection, as both  the lensed and unlensed scenarios are possible, given the location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' DESI-035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1374+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4676  DESI-035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1374+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4676 was discovered in Storfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2022), as a C-grade strong lensing candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' However,  after viewing the Hyper Suprime-Cam (HSC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Aihara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2019) image (see Figure 32), we feel confident in moving  this into the A-grade lens candidate category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  snana-2007og : SN llP  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  9  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  Magnitude  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  t, = 58037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='14 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='25  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='530 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='218  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' = -20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='04  E(M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' / SN IIP) = -16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='97  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  20  10  0  10  20  30  40  50  Phase (days)36  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Coadded RGB images (using g, r, i, and z bands) of DESI-035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1374+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4676 with and without the transient  detection (red dotted circle) exposures, as well as the HSC DR2 image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Labelled objects are color-coded as the following  postulated scenario: red as the lens galaxy, green as the source galaxy, and cyan as an interloper or a member galaxy of the  foreground group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Photometric redshifts are displayed on the top right of the second image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Detection exposures for the transient in DESI-035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1374+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4676 in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' See caption of Figure 10  for the full description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  DESI  1: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='269 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='00  2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='776 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='118  3: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='486 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='068  HSC DR2 Image11/16/2017  12/07/2017  12/24/2017  mjd:58073.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='18  mjd:58094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='04  mjd:58111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='07  DifferenceImage  Detection0=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  37  Figure 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Residual image after lens light subtraction for DESI-035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1374+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4676, with the S´ersic light parameters (half-light  radius, axis ratio, semi-major axis orientation from the y axis) in the top left (modelled in DECam g filter), with the core of  the lens galaxy masked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  DESI-035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1374+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4676 appears to be a galaxy group-scale strongly lensing system, with the arc lying Northeast  of the main lensing galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The transient’s location is somewhat far from both the lens and lensed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' From the  best-fit foreground galaxy light parameters shown in Figure 34, the detection is approximately 4 half-light radii away  from the lensing galaxy, which does not exclude it from being the host galaxy of the transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' On the other hand,  if it is hosted by the lensed source galaxy, the distance between the transient and its center would be stretched along  the tangential direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Without lens modeling, which would provide the delensed source, it is difficult to estimate  how far the transient is from the source galaxy center in meaningful terms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=', half-light radius or directional light  radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Therefore neither is an impossible scenario based on the location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The possibility of a faint galaxy hosting  the transient seems remote, as such a galaxy does not appear even in the HSC image with superior seeing (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='58′′ in  the i band) and greater depth (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2 i band limiting magnitude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' see Figure 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  r。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='500°  q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='241  0=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='833  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  SersicFitResiduals  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='138  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Postulation 1: uL-SN Ia —Figure 35 shows the best-fit light curve model for the uL-SN Ia scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The SALT3 model  fits the four photometric data points well, and its Hubble residual is consistent with the Union 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1 best-fit cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  We consider this to be a possible identity of this transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Best-fit SALT3 model for DESI-035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1374+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4676 with a lens photo-z redshift prior of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='269 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For this and  Figure 36, solid photometry points correspond to the detection passes shown in Figure 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  HubbleDiagram  SALT3  21  46  Fitted Cosmology  g  SN la Detection  Union2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1 SNe la  44  z  (Magnitude)  22  42  40  23  Magnitude  24  36  34  Hubble residual =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='27±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  25  Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='269±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='008  X,= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='017± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='018  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  t。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='= 58077.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='53 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='56  C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='004±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='098  X。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='= (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='38) x 10-5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  20  0  20  40  60  Redshift  Phase (days)Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  39  Postulation 2: L-CC SN —Figure 36 shows the best-fit light curve model for the L-CC SN scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This model also fits  the available data well for a Type IIn SN template (“nugent-sn2n”), with an estimated amplification of 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='97+55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='29  −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  As this model is consistent with the data, we believe L-CC SN to be a possible identity of the transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Best-fit core collapse template model for DESI-035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1374+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4676 with a source photo-z redshift prior of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='776±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Conclusion —This transient in DESI-035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1374+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4676 appears to be consistent with an uL-SN Ia or a L-CC SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  With the data, it is difficult to discern which scenario is more likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  In the case of a uL-SN Ia, the supernova  would have occurred at approximately four effective radii away from the lensing galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' On the other hand, while  the detection is far from the center of the lensed galaxy, this separation may not rule out the background as the  host due to the tangential “stretching” from strong lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Lens modeling (using HSC DR2 data or follow-up higher  resolution observations) may shed more light on this possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' If detected live, this detection would warrant follow-up  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  nugent-sn2n : sN Iln  21  9  z  22  23  Magnitude  24  25  t。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='= 58017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='85  Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='795 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='050  E(M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='/SN IIn) =  :-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='53 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='48  μ= 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='97 +55:29  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='13  26  25  0  25  50  75  100  125  150  175  200  Phase (days)40  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' DESI-052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0083-37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2049  DESI-052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0083-37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2049 was discovered in Storfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2022) as a C-grade strong lensing candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Coadded RGB images (using g, r, i, and z bands) of DESI-052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0083-37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2049 with and without the transient  detection (red dotted circle) exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Labelled objects are color-coded as the following postulated scenario: red as the lens  galaxy and green as the source galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Photometric redshifts are displayed on the top right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Detection exposures for the transient in DESI-052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0083-37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2049 in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' See caption of Figure 10  for the full description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  DESI-052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0083-37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2049  1: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='292 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='029  2:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='782 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='610  3: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='272 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='412  Detections Coadd  Mean01/09/2015  1/10/2015  01/14/2015Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  41  DESI-052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0083-37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2049 is a possible strong lensing candidate, although a ring galaxy or a face-on spiral scenario is  not ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The transient, however, is unmistakably present, with multiple detections lying directly on the arc-like  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Postulation 1: uL-SN Ia —Figure 39 shows the best-fit light curve model for the uL-SN Ia scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Note that the  first r band point is near the peak, almost coincidental with a g band point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The SALT3 light curve model agrees  reasonably well with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The Hubble residual is somewhat large, but does not rule out this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Of note is  that the first z band point appears to be too bright for this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Best-fit SALT3 model for DESI-052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0083-37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2049 with a lens photo-z redshift prior of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='292 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For this and  Figure 40, solid photometry points correspond to the detection passes shown in Figure 38, hollow points correspond to other  exposures with PSF photometry, and crosses correspond to measurements using aperture photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  HubbleDiagram  SALT3  21  46  Fitted Cosmology  9  SN la Detection  Union2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1 SNe la  44  z  (Magnitude)  22  Y  42  40  23  Magnitude  24  36  34  Hubbleresidual=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='333± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='023  x,=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='74±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='585  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  C = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='188± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='047  X = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='11) x 10-5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  20  0  20  40  60  Redshift  Phase (days)42  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Postulation 2: L-CC SN —Figure 40 shows the best-fit light curve model for the L-CC SN scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  The SN IIP  template provides a reasonable fit for the photometry, with an amplification of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='60+18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='20  −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='45 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Similarly to Postulation  1, the first z band point is not well fit by this model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' nor is the second point of the r band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Best-fit core collapse template model for DESI-052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0083-37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2049 with no redshift prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Conclusion —The transient in DESI-052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0083-37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2049 appears to be either an uL-SN Ia or a L-CC SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' If high resolution  and/or spectroscopic observations reveal that the arc-like structure is part of a spiral or ring galaxy, that would  obviously rule out the L-CC SN scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Conversely, if it is strongly lensed arc/Einstein ring formation, then the  L-CC SN possibility becomes quite possible, considering the location of the detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' If found live, this detection  would warrant follow-up observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  snana-2007pg : SN lIP  21  g  r  i  z  22  Y  23  Magnitude  24  25  t = 57037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='36  Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='450 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='018  M,= -19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='03  E(M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' / SN IIP) = -16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='97  26  20  0  20  40  60  80  100  Phase (days)Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  43  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' DESI-084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8493-59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3586  DESI-084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8493-59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3586 was discovered in Storfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2022), labelled as a C-grade strong lensing candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Coadded RGB images (using g, r, i, and z bands) of DESI-084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8493-59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3586 with and without the transient  detection (red dotted circle) exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Labelled objects are color-coded as the following postulated scenario: red as the lens  galaxy, green as a source galaxy, and purple as a second source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Photometric redshifts are displayed on the top right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Detection exposures for the transient in DESI-084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8493-59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3586 in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' See caption of Figure 10  for the full description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  DESI-084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8493-59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3586  1: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='361± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='015  2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='593 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='146  3: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='980 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='578  4: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='049 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='478  Detections Coadd  Mean  CoaddwloDetections2/14/201  12/30/2013  Drefer44  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  DESI-084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8493-59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3586 is a single galaxy strongly lensed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' There is a red lensed arc to the East of the lens, and  the transient detection lies South of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Additionally, it is possible that objects 3 and 4 (Figure 41) correspond  to the same source galaxy, due to similarities in color and photo-z, with the possibility that object 3 is the host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Postulation 1: uL-SN Ia —Figure 43 shows the best-fit light curve model for the uL-SN Ia scenario, with the foreground  galaxy photo-z used as the redshift prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The SALT3 model agrees well with the data, with reasonable light curve  parameters and small Hubble residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Best-fit SALT3 model for DESI-084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8493-59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3586 with a lens photo-z redshift prior of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='361 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For this and  Figure 44, solid photometry points correspond to the detection passes shown in Figure 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  HubbleDiagram  SALT3  21  46  Fitted Cosmology  SN la Detection  Union2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1SNela  44  (Magnitude)  22  42  40  23  Magnitude  24  36  34  Hubbleresidual=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  25  Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='365±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='008  X,=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='806±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='310  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  t = 56645.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='31  C=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='046±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='028  X。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='= (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='06) x 10-5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  20  0  20  40  60  Redshift  Phase (days)Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  45  Postulation 2: L-CC SN —Figure 44 shows the best-fit light curve model for the L-CC SN scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The fit is significantly  inferior to the previous postulation in the z band, requiring a high magnification of 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='05+153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='15  −55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='10 , albeit with large  uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Best-fit core collapse template model for DESI-084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8493-59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3586 with a source photo-z redshift prior of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='593±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Conclusion —This transient in DESI-084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8493-59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='3586 is likely an unlensed SN Ia, though there is a small possibility of  it being a lensed CC SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Thus, we give this system an appropriately low lensed SN grade of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' If found live, follow-up  spectroscopic observations at the transient location could easily distinguish these two scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  v19-2014g-corr : SN II  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  g  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  Magnitude  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  0  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' = 56657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='78  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='819 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='064  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  M, = -21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='01  E(M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' / SN II) = -16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='98 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='11  μ = 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='05 +153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='15  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='10  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  20  15  i0  5  0  5  25  10  15  20  Phase (days)46  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' DESI-015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8465-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5450  DESI-015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8465-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5450 is a grade D+ strong lens candidate, discovered in Storfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Coadded RGB images (using g, r, i, and z bands) of DESI-015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8465-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5450 with and without the transient  detection (red dotted circle) exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Labelled objects are color-coded as the following postulated scenario: red as the main  galaxy and cyan as surrounding galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Photometric redshifts are displayed on the top right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Detection exposures for the transient in DESI-015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8465-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5450 in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' See caption of Figure 10  for the full description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  DESI-015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8465-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5450  1: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='301 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='026  2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='396 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='246  3: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='673 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='083  4: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='697 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='086  Detections Coadd  MeanCoadd wjoDetections12/11/2017  12/11/2017  mjd:58098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='09  mid:58098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='09  Difference  Image  Detection=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  47  Upon closer inspection of DESI-015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8465-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5450, we believe it is more likely a face-on spiral galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' In the detection  coadd image in Figure 45, the evidence for the spiral pattern (rather than lensed arcs) is especially strong in the g band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  There are only two detections of this transient, observed two minutes apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' However, we do not observe a shift in  detection location above the level of noise, and so we do not consider asteroid as a likely scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Postulation 1: uL-SN Ia —Figure 47 shows the best-fit light curve model for the uL-SN Ia scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Due to the sparsity  of the photometric data, the uncertainties of the SALT3 model parameters and Hubble residuals are large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' While there  are only two detection exposures, the pipeline identified this transient with four sub-detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' As with all other light  curves presented, the light curves below are constrained by both detection and non-detection exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Best-fit SALT3 model for DESI-015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8465-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5450 with a lens photo-z redshift prior of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='301 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' For this and  Figure 48, photometry points correlate to the detections shown in Figure 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  HubbleDiagram  SALT3  21  46  Fitted Cosmology  SN la Detection  Union2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1 SNe la  44  22  (Magnitude)  42  23  sninpo  40  36  25  34  Hubble residual = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='91 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='52  1  26  0  Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='373±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='131  X,=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='886±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='936  t,= 58109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='79  C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='247±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='376  1  X。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='01 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='18) x 10-5  2  27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4  20  10  0  10  20  30  40  50  60  Redshift  Phase (days)48  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Postulation 2: uL-CC SN —Figure 48 shows the best-fit light curve model for the uL-CC SN scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This SN Ic-LB  template (“v19-2002ap-coor”) is one of many templates that are consistent with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Figure 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Best-fit core collapse template model for DESI-015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8465-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5450 with a lens photo-z redshift prior of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='301 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Conclusion —The most likely scenario for the transient in DESI-015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='8465-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5450 is an unlensed supernova, as the  system is probably a spiral galaxy and not a strong lensing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' With the sparsity of photometric data, it is not  possible to determine the type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' It could also be the case that the host galaxy is object 4 (which could be lensed,  possibly with object 3 as its counterimage), as opposed to object 1 (see Figure 45), but it is infeasible to determine  with current data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' ASTEROIDS  We have found two asteroid candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The detections are observed on the same night (for each respective system),  separated by approximately one to two minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The locations of the two detections in each case are spatially close  enough for the pipeline to identify them as candidates (as a group of three to four sub-detections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' PSF fitting for the  transient detections shows that the movements between detections for both systems are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The approximate  speeds of transients are consistent with that of a main-belt asteroid (roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5′′ per minute near opposition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Cicco  2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  v19-2002ap-corr : SN Ic-Bl  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  g  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  Magnitude  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  t。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' = 58104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='45 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='42  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5  Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='289 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='036  M, = -17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='04  E(Mβ / SN Ic) = -17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='66 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='04  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='0  10  0  10  20  30  40  Phase (days)Retrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  49  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' DESI-008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6173+02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4228  Figure 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Above the dotted line are coadded images of DESI-008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6173+02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4228 with and without the transient detection  (red dotted circle) exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' From left to right, the images show 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=') The coadded image generated from mean coadding from  only the exposures with the transient detection, within each band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=') The coadded image, generated from mean coadding all  exposures excluding the detection exposures within each band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=') The HST image of the system (HST Proposal ID: 12884;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Ebeling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The RGB image incorporates g, r, i, and z bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Below the dotted line are detection exposures for the transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Each column is a single detection at the labelled date and band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The top row is the single exposure image, whereas the bottom is  the SFFT difference image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The detection location is marked with a red dotted circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The first and second columns show r and  g band detections on 01/18/2016 minutes apart, whereas the third column shows a set on nondetections six days afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  Note the slight but significant shift in the transient location between the two detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  For DESI-008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6173+02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4228, we find that the transient has moved 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='475′′±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='101 between the r and g band detections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  a movement of > 4σ significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' As the exposures were taken 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='98 minutes apart, the estimated speed of this asteroid  is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='240′′ ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='055 per minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' This is slower than the typical main-belt asteroid (near opposition) speed of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='5′′ per  DESI-008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='6173+02-4228  HST Image  Mean Coadd w/o Detection  11/18/2016  11/18/2016  11/24/2016  mjd: 57710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='08  mjd: 57710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='08  : 57716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='09  Difference lma50  Sheu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  minute, but not unreasonably so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The coordinates and time observed does not correlate with any known asteroid in  the IAU’s Minor Planet Center database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' DESI-150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4863+15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4209  Figure 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Above the dotted line are coadded images of DESI-150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4863+15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4209 with and without the transient detection (red  dotted circle) exposures (see Figure 49 caption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Below the dotted line are detection exposures for the transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The second  and third columns show detections on 01/17/2016, while the first column shows a nondetection on 01/12/2016 (5 days prior to  detection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The detection location is marked with a red dotted circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' As with the previous system, there is a significant shift  in detection location between the exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='  In DESI-150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4863+15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4209, we find that the transient has moved 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='155′′±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='372 between the r and g band detections,  with a significance of > 5σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The exposures were taken 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='23 minutes apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' Thus, the estimated speed of this asteroid  DESI-150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4863+15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='4209  DetectionCoaddImage  Mean Coadd w/o Detection  01/12/2016  01/17/2016  01/17/201  mjd:57399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='31  mjd: 57404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='23  midRetrospective Search for Strongly Lensed Supernovae in the DESI Legacy Imaging Surveys  51  is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='475′′ ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content='082 per minute, which is consistent with a main-belt asteroid near opposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
+page_content=' The coordinates and time  observed does not correlate with any known asteroid in the IAU’s Minor Planet Center database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQf_QZd/content/2301.03578v1.pdf'}
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+Emotion-Cause Pair Extraction as Question Answering
+Huu-Hiep Nguyen
+Cinnamon AI
+hubert@cinnamon.is
+Minh-Tien Nguyen
+Hung Yen University of Technology and Education
+tiennm@utehy.edu.vn
+Abstract
+The task of Emotion-Cause Pair Extraction
+(ECPE) aims to extract all potential emotion-
+cause pairs of a document without any anno-
+tation of emotion or cause clauses. Previous
+approaches on ECPE have tried to improve
+conventional two-step processing schemes by
+using complex architectures for modeling
+emotion-cause interaction. In this paper, we
+cast the ECPE task to the question answering
+(QA) problem and propose simple yet effec-
+tive BERT-based solutions to tackle it. Given a
+document, our Guided-QA model first predicts
+the best emotion clause using a fixed question.
+Then the predicted emotion is used as a ques-
+tion to predict the most potential cause for the
+emotion.
+We evaluate our model on a stan-
+dard ECPE corpus. The experimental results
+show that despite its simplicity, our Guided-
+QA achieves promising results and is easy to
+reproduce. The code of Guided-QA is also pro-
+vided.
+1
+Introduction
+Emotion Cause Extraction (ECE) is the task of
+detecting the cause behind an emotion given the
+emotion annotation (Lee et al., 2010; Gui et al.,
+2016), see Figure 1 (Top). The text was divided
+into clauses and the task was to detect the clause
+containing the cause, given the clause containing
+the emotion. However, the applicability of ECE
+is limited due to the fact that emotion annotations
+are required at test time. Recently, (Xia and Ding,
+2019) introduced the more challenging Emotion-
+Cause Pair Extraction (ECPE) task: extracting all
+possible emotion-cause clause pairs in a document
+without annotations. Figure 1 (Bottom) shows an
+example of the ECPE task. The input is a document
+of six clauses. Clauses c4 and c5 contain emotion
+with the emotion expressions “happy” and "wor-
+ried". The emotion c4 has two causes c3 and c2,
+the emotion c5 has one cause c6, so the expected
+output is {(c4,c2), (c4,c3), (c5,c6)}.
+c1: Yesterday morning
+c4: The old man was very happy (Emotion)
+c6: as he doesn’t know how to keep so much money. (Cause)
+c2: a policeman visited the old man with the lost money (Cause)
+c3: and told him that the thief was caught. (Cause)
+Emotion-Cause Pair Extraction (ECPE)
+Input: a document
+Output: emotion-cause pairs
+{(c4,c2), (c4, c3),(c5,c6)}
+c5: but he still feels worried, (Emotion)
+c1: Yesterday morning
+c4: The old man was very happy
+c6: as he doesn’t know how to keep so much money. (Cause)
+c2: a policeman visited the old man with the lost money
+c3: and told him that the thief was caught.
+Emotion-Cause Extraction (ECE)
+Input: a document and annotation of emotions c5
+Output: cause clause(s)
+{c6}
+c5: but he still feels worried, (Emotion)
+Figure 1: Illustration of ECE and ECPE tasks.
+Why cause-effect pair extraction? We argue that
+independent extraction of cause and emotion may
+be ineffective. For a given document, ECPE models
+may predict correct cause but incorrect emotion.
+This makes the output incomplete, and subsequent
+processing steps less reliable (Ding et al., 2020;
+Wei et al., 2020; Chen et al., 2020; Yan et al., 2021).
+We make a toy example of two models using the
+document in Figure 1. Model-1 predicts (c4,c1)
+and (c6,c3) as emotion-cause pairs. Its emotion,
+cause and pair accuracy scores are 0.5, 0.33 and 0.0.
+Model-2 predicts (c4, c2) and (c6, c1) as emotion-
+cause pairs. Its emotion, cause and pair accuracy
+scores are 0.5, 0.33 and 0.33. From the perspective
+of the pair extraction task, Model-2 is better.
+Previous studies addressed the ECPE task by us-
+ing sequence labeling (Lee et al., 2010; Cheng et al.,
+arXiv:2301.01982v1  [cs.CL]  5 Jan 2023
+
+2021), clause-level classification (Gui et al., 2016;
+Ding et al., 2020; Chen et al., 2020), ranking (Wei
+et al., 2020), or recurrent synchronization (Chen
+et al., 2022). The methods achieved promising re-
+sults, yet the use of interaction between emotion
+and cause clauses is still an open question. For
+example, c4 and c2 share "the old man" tokens,
+which refer to "him" in c3; and c5 and c6 share
+"he", which mentions "the old man" in c2 and c4.
+Based on this observation, we introduce a
+paradigm shift (Sun et al., 2022) for ECPE by us-
+ing span extraction. As far as we know, (Gui et al.,
+2017) is the first work that uses question answering
+for emotion-cause detection. However, their work
+addresses the ECE task only, which requires the
+annotation of emotion for cause prediction. In con-
+trast, our paradigm shift is applied to the ECPE task,
+which is more challenging and does not require the
+annotation of emotion for cause prediction. The
+paradigm bases on two hypotheses. First, infor-
+mation from emotion clauses can be used to infer
+cause clauses. Second, emotion and cause clauses
+share implicit interaction. The design of our model
+is based on these two hypotheses. For the first hy-
+pothesis, we form questions based on emotional in-
+formation which is used to predict emotion clauses.
+For the second hypothesis, we used predicted emo-
+tion as the guided question for cause prediction.
+The model is trained by using the BERT-QA archi-
+tecture (Devlin et al., 2018) in form of SQuAD task
+(Rajpurkar et al., 2016).
+Our paper makes three main contributions.
+• We formulate the ECPE task as a QA problem
+and propose a Guided-QA model to implicitly
+capture the relationship between emotion and
+cause clauses, in which the predicted emotion
+is used as a guided question for cause pre-
+diction. The model can capture the implicit
+interaction between emotions and causes with
+a simple but effective architecture. To the best
+of our knowledge, we are the first to address
+the ECPE task by using QA formulation.
+• We evaluate our model on the standard ECPE
+corpus (Xia and Ding, 2019; Fan et al., 2020).
+Experimental results show that our approach
+achieves promising results compared to previ-
+ous methods.
+• We promote the reproducibility (Houghton
+et al., 2020) by providing the source code
+of our methods as well as rerunning publicly
+available source codes of the compared meth-
+ods.
+2
+Related Work
+ECE and ECPE tasks
+The ECE task was for-
+mulated as sequence-labeling by (Lee et al., 2010)
+and refined as clause-level by (Gui et al., 2016).
+Recently, the more challenging ECPE task (Xia
+and Ding, 2019) has attracted a lot of contributions
+with several strong methods (Ding et al., 2020;
+Wei et al., 2020; Chen et al., 2020; Cheng et al.,
+2021; Chen et al., 2022). For example, (Ding et al.,
+2020) introduced ECPE-MLL, which uses a sliding
+window for a multi-label learning scheme. ECPE-
+MLL extracts the emotion and cause by using the
+iterative synchronized multitask learning. (Chen
+et al., 2022) proposed a similar approach, recur-
+rent synchronization network (RSN), that explic-
+itly models the interaction among different tasks.
+(Wei et al., 2020) presented RankCP, a transition-
+based framework, by transforming the ECPE prob-
+lem into directed graph construction, from which
+emotions and the corresponding causes can be ex-
+tracted simultaneously based on labeled edges. The
+PairGCN model (Chen et al., 2020) used Graph
+Convolutional Networks to model three types of
+dependency relations among local neighborhood
+candidate pairs and facilitate the extraction of pair-
+level contextual information.
+We share the purpose of addressing the ECE and
+ECPE tasks with prior studies, however, instead
+of using classification or sequence labeling, we ad-
+dress the tasks with a new paradigm shift by using
+span extraction. It allows us to take into account
+the implicit interaction between emotion and cause
+clauses and to design a simple but effective BERT-
+based model for ECE and ECPE.
+(Bi and Liu, 2020) derived a span-based dataset
+and formulated a new ECSP (Emotion Cause Span
+Prediction) task from (Xia and Ding, 2019) but it
+has not attracted much attention. The accessibility
+of the dataset and source code may be the reason.
+We leave span-based ECSP evaluation as future
+work.
+Paradigm shift in natural language processing
+A paradigm is a general modeling framework or
+a family of methods to solve a class of tasks.
+For instance, sequence labeling is a mainstream
+paradigm for Part-of-speech (POS) tagging and
+Named entity recognition (NER). The sequence-to-
+sequence (Seq2Seq) paradigm is a popular tool for
+
+summarization and machine translation. Different
+paradigms usually require different formats of in-
+put and output, and therefore highly depend on the
+annotation of the tasks.
+Paradigm shift indicates the job of solving one
+NLP task in a new paradigm by reformulating the
+task along with changing the input-output formats.
+Paradigm shift in NLP has been explored scatter-
+ringly in recent years and with the advent of pre-
+trained language models, it became a rising trend
+(Li et al., 2019; Khashabi et al., 2020). An excel-
+lent survey of paradigm shifts in NLP has been
+done by (Sun et al., 2022). In this work, we realize
+such a paradigm shift for the ECPE task, i.e., we
+reformulate the clause-based text classification task
+as span extraction.
+Span-based
+extractive
+question
+answering
+Our formulation for the tasks of ECE and ECPE
+relates to span-based extractive QA, which has
+been widely investigated (Khashabi et al., 2020).
+More precisely, we design our model based on the
+pretrained language models (PLMs) such as BERT
+(Devlin et al., 2018) or RoBERTa (Liu et al., 2019).
+This is because applying PLMs as the backbone of
+QA systems has become a standard procedure. For
+detailed information, please refer to (Devlin et al.,
+2018).
+Figure 2 reproduced from (Devlin et al., 2018)
+shows how BERT is applied to the extractive QA
+task. Tokens of question q = q1, .., qn and context
+C = c1, .., cm are concatenated before being en-
+coded by BERT. The contextual representations of
+tokens Ti are put into a feed-forward layer followed
+by a softmax. Each candidate span for the answer
+is scored as the product of start/end probabilities.
+The maximum scoring span is used as the predic-
+tion. The training objective is the loglikelihood of
+the correct start and end positions.
+By casting the ECPE to QA problem, our work
+leverages the powerful models of the BERT family
+(Devlin et al., 2018) to detect clause-level emotions
+and causes as well as emotion-cause pairs.
+3
+Method
+3.1
+Problem Statement
+Given a document of n clauses d = (c1, c2, .., cn),
+the goal of ECPE is to detect all potential emotion-
+cause pairs P = {..(ce, cc), ..} where ce is an
+emotion clause, and cc is the corresponding cause
+clause (Xia and Ding, 2019). We formulated the
+BERT
+CLS
+c_1
+q_1
+SEP
+q_n
+c_m
+...
+...
+Start/End Span
+T_CLS
+T_c_1
+T_q_1
+T_SEP
+T_q_n
+T_c_m
+...
+...
+E_CLS
+E_c_1
+E_q_1
+E_SEP
+E_q_n
+E_c_m
+...
+...
+Figure 2: BERT-based extractive Question Answering
+ECPE task as a QA problem.
+Given a set of
+questions {qe, qc} (qe is for emotion and qc is for
+cause) and a context document d with n clauses,
+the model learns to predict start and end posi-
+tions of each ce and cc: sce, ece = f(d, qe|Θ) and
+scc, ecc = f(d, qc|Θ) to form P. Θ can be learnt
+by using independent or guided extraction.
+3.2
+Independent Emotion, Cause Extraction
+We first introduce a simple version of our model,
+Indep-QA in Figure 3. Indep-QA receives a fixed
+question (for emotion or cause) and then pulls out
+corresponding emotion or cause clauses indepen-
+dently.
+Question
+formulation
+Because
+no
+emo-
+tion/cause information is provided beforehand, we
+have to detect them first with generic questions.
+It is possible to use pre-defined questions for ex-
+traction (Mengge et al., 2020), however, we argue
+that the definition of questions is time-consuming,
+needs domain knowledge, and does not guarantee
+the semantic relationship between the questions
+and context documents. Instead, we use two short
+questions "emotion" and "cause" as an implicit
+indicator that provides additional information for
+the model. We leave the analysis of using generic
+questions such as "What is the emotion?" and
+"What is the cause?" as future work.
+Learning and prediction
+Given a document d
+and a question ("emotion" or "cause"), we con-
+catenated all clauses of d and the question to form a
+single sequence C. The sequence was fed to a pre-
+trained language model (PLM) to obtain its hidden
+representations of tokens which were subsequently
+fed into a feed-forward layer followed by a soft-
+max layer. Each candidate span was scored as the
+
+answer start
+answer end
+c1
+c2
+c3
+c4
+question = "emotion"
+context =
+predicted Emotion clause = c2
+answer start
+answer end
+c1
+c2
+c3
+c4
+question = "cause"
+context =
+predicted Cause clause = c4
+predicted EC-pair = (c2,c4)
+Figure 3: Independent extraction Indep-QA.
+c1
+c2
+c3
+c4
+question =
+context =
+predicted Cause clause = c3
+c1
+c2
+c3
+c4
+context =
+predicted Emotion clause = c4
+predicted EC-pair = (c4,c3)
+question =
+c3
+answer start
+answer end
+answer start
+answer end
+c1
+c2
+c3
+c4
+question = 
+context =
+predicted Emotion clause = c2
+c1
+c2
+c3
+c4
+context =
+predicted Cause clause = c3
+predicted EC-pair = (c2,c3)
+question =
+c2
+answer start
+answer end
+answer start
+answer end
+"emotion"
+"cause"
+Figure 4: Guided pair extraction Guided-QA: Emotion is detected first (Left), Cause is detected first (Right).
+product of start/end probabilities. The maximum
+scoring span was used as the prediction.
+Mapping predicted answer span to clauses
+The predicted answer span may overlap with one
+or several clauses. We applied a span-to-clause
+mapping rule to determine which clauses are pre-
+dicted results: the clause that overlaps most with
+the predicted span is returned. The tie is broken
+arbitrarily. For instance, In Figure 3, the predicted
+span for "emotion" overlaps with clauses c2 and c3
+in which c2 is more overlapped. As a result, c2 is
+the predicted emotion.
+EC
+pair
+prediction
+Given
+predicted
+emo-
+tion/cause clauses ce and cc, Indep-QA simply pre-
+dicts (ce, cc) as an emotion-cause pair. As illus-
+trated in Figure 3, (c2, c4) is the predicted emotion-
+cause pair.
+3.3
+Guided Emotion-Cause Pair Extraction
+The Indep-QA model extracts emotion/clause
+clauses independently but does not exploit the rela-
+tionship between emotion and cause clauses, which
+plays an important role in the extraction of emotion-
+cause pairs (Ding et al., 2020; Wei et al., 2020;
+Chen et al., 2020; Cheng et al., 2021; Chen et al.,
+2022).
+To better model this relationship, we introduce
+Guided-QA in Figure 4. The model receives an
+emotion question and predicts the corresponding
+emotion clause. Then the predicted emotion clause
+is used as a question for cause extraction. Com-
+pared to Indep-QA, the Guided-QA takes into ac-
+count an implicit relationship from emotion for
+cause prediction.
+The Guided-QA model shares the question for-
+mulation, hidden representation learning, and the
+mapping process of the Indep-QA model.
+EC pair extraction
+We used the predicted
+(noisy) emotion clause as the question for cause
+extraction. The interaction between emotion and
+cause happens here. The predicted emotion clause
+may or may not be the true one but on average, it
+contains much more information for the QA model
+than the generic question (i.e., "emotion"). Note
+that the predicted (noisy) emotion as the question
+was used for the test set only. For the training set,
+as the model already knows which clauses are emo-
+tion or cause, it uses the true emotion clause as the
+question.
+By swapping the role, the model can detect cause
+clauses first and use the noisy causes as questions
+to predict the emotions. In Section 5 we compare
+Emotion-first and Cause-first, the two variants of
+Guided-QA and show that the gaps are tiny. In
+other word, the two variants are almost equivalent
+
+on the tested datasets.
+As our QA models use the best answer span for
+each question, only one emotion, one cause, and
+one EC pair are predicted for each document which
+are appropriate for the ECPE dataset. We also
+aware that the prediction of spans should be multi-
+ple and we aim to address this limitation in future
+work by using multiple span extraction methods
+(Nguyen et al., 2021; Fu et al., 2021).
+3.4
+Discussion
+Given a document of n clauses, existing schemes
+such as ECPE-MLL (Ding et al., 2020), RankCP
+(Wei et al., 2020) and PairGCN (Chen et al.,
+2020) attempt to reduce the O(n2) complexity of
+emotion-cause pair classification by using sliding
+window, transition graph techniques. However,
+these techniques may miss certain interaction be-
+tween the emotion-cause pair and the full context
+in the document. BERT-based QA models with
+full attention between the question and the con-
+text mitigate this issue. Through QA models, the
+emotion-cause relationship between all clauses is
+implicitly learned and we can leverage the power
+of existing QA methods.
+4
+Experimental Settings
+Datasets
+We followed the 10-split ECPE dataset
+provided by (Xia and Ding, 2019) and the 20-split
+TransECPE variant (Fan et al., 2020) to evaluate
+our methods. Each split is a random partition of
+the 1945 documents to train/dev/test sets with ratio
+8:1:1, i.e., the train set, dev set and test set contain
+approximately 1556, 194 and 195 documents. On
+average, each document contains 14.8 clauses.
+Table 1 shows the distribution of documents with
+different number of emotion-cause pairs. Most of
+the documents have only one emotion-cause pairs.
+This fact makes the detection of emotion/cause
+clauses as well as emotion-cause pairs challenging.
+Evaluation metrics
+We used the precision, re-
+call, and F1 score (Xia and Ding, 2019) as evalua-
+tion metrics for all three tasks of ECPE: emotion
+extraction, cause extraction and emotion-cause pair
+extraction. Let Te and Pe be the number of ground-
+truth and predicted emotion clauses respectively,
+the precision, recall and F1 score for emotion are
+as defined as follows.
+Pe = |Te ∩ Pe|
+|Pe|
+Re = |Te ∩ Pe|
+|Te|
+F1e = 2 ∗ Pe ∗ Re
+Pe + Re
+Metrics for cause clauses and emotion-cause
+pairs are defined similarly.
+Implementation details
+Our model was imple-
+mented using BERT classes provided by Hugging-
+face (Wolf et al., 2020). The model was trained
+in 5 epochs, with the learning rate of 5e − 5, and
+the batch size of 16. We used BERT (Devlin et al.,
+2018)1 and RoBERTa (Liu et al., 2019)2 for Chi-
+nese. All models were trained on a Tesla P100
+GPU.
+5
+Results and Discussion
+Guided-QA: Emotion-first vs. Cause-first
+We
+first compare the two variants Emotion-first and
+Cause-first of the Guided-QA method. Table 2
+shows that the two variants have almost equiva-
+lent performance on the tested datasets except the
+BERT-based results on 10-split ECPE. Also, the
+RoBERTa-based results are consistently better than
+the BERT-based, 1.1 to 2.0 points. In the next
+section, we pick the Emotion-first scores for com-
+paring Guided-QA with other methods.
+Guided-QA vs. Indep-QA
+We now compare
+Guided-QA and Indep-QA. For 10-split ECPE in
+the upper part of Table 3, the Guided-QA model
+is consistently better than Indep-QA for pair ex-
+traction. This is because Guided-QA takes into
+account the implicit interaction between emotion
+and cause clauses. For emotion or cause extraction,
+Indep-QA is competitive with Guided-QA. This
+is because they share the same formulation. The
+results in Table 4 also show similar observation.
+We also confirm the performance of our model
+by using RoBERTa to have better analysis. The
+results are consistent with the model using BERT,
+in which Guided-QA outputs better F-scores than
+the Indep-QA model. It also shows that our model
+can be improved further by using stronger PLMs.
+Guided-QA vs. strong baselines
+We compare
+our model with five strong methods for ECPE:
+ECPE-MLL3 (Ding et al., 2020), RankCP4 (Wei
+1https://huggingface.co/bert-base-chinese
+2https://huggingface.co/hfl/chinese-roberta-wwm-ext
+3https://github.com/NUSTM/ECPE-MLL
+4https://github.com/Determined22/Rank-Emotion-Cause
+
+Table 1: Histogram of the number of emotion-cause pairs per document.
+Number
+Percentage
+Documents with one emotion-cause pair
+1746
+89.77%
+Documents with two emotion-cause pairs
+177
+9.10%
+Documents with more than two emotion-cause pairs
+22
+1.13%
+All
+1945
+100%
+Table 2: Guided-QA Emotion-first vs. Cause-first on 10-split ECPE dataset and 20-split TransECPE dataset
+Model
+Emotion Extraction
+Cause Extraction
+EC Pair Extraction
+P
+R
+F1
+P
+R
+F1
+P
+R
+F1
+10-split ECPE
+Emotion-first (BERT)
+0.847
+0.908
+0.876
+0.719
+0.792
+0.754
+0.771
+0.692
+0.729
+Cause-first (BERT)
+0.831
+0.891
+0.860
+0.714
+0.787
+0.749
+0.763
+0.685
+0.722
+Emotion-first (RoBERTa)
+0.854
+0.916
+0.884
+0.732
+0.806
+0.767
+0.786
+0.706
+0.744
+Cause-first (RoBERTa)
+0.843
+0.904
+0.873
+0.733
+0.807
+0.768
+0.784
+0.704
+0.742
+20-split TransECPE
+Emotion-first (BERT)
+0.842
+0.906
+0.873
+0.710
+0.782
+0.744
+0.760
+0.689
+0.723
+Cause-first (BERT)
+0.833
+0.897
+0.864
+0.713
+0.785
+0.747
+0.761
+0.690
+0.724
+Emotion-first (RoBERTa)
+0.844
+0.909
+0.875
+0.723
+0.796
+0.757
+0.772
+0.700
+0.734
+Cause-first (RoBERTa)
+0.838
+0.902
+0.869
+0.724
+0.797
+0.758
+0.773
+0.701
+0.735
+et al., 2020), PairGCN5 (Chen et al., 2020), UTOS
+(Cheng et al., 2021), and RSN (Chen et al., 2022).
+For 10-split, our model using BERT follows ECPE-
+MLL, RankCP, and RSN. It shows that with a sim-
+ple architecture, our model can output competitive
+results compared to complicated methods. For 20-
+split TransECPE in Table 4, the trend is consistent
+with Table 3, in which the Guided-QA model is
+competitive for both ECE and ECPE tasks.
+Moreover, as we observe from all the compared
+methods, the gaps between the reported pair-f1
+scores for 10-split ECPE and 20-split TransECPE
+are 0.023 (=0.745-0.722) for ECPE-MLL, 0.042 for
+RankCP, 0.029 for UTOS, 0.003 for Indep-QA and
+0.006 for Guided-QA, i.e., largest gap in RankCP
+and smallest gaps in our models. Across the two
+settings, our models seem more robust than the
+compared methods.
+Reproducibility
+For fair comparison (Houghton
+et al., 2020), we also rerun publicly available
+source codes in the original setting. The repro-
+duced results confirm the gaps between reproduc-
+tion and original results. Compared to the repro-
+duced results, Guided-QA using BERT is the best
+for EC pair extraction.
+Compared to the results of reproduced methods,
+the Guided-QA is still better for both ECE and
+5https://github.com/chenying3176/PairGCN_ECPE
+ECPE tasks. This confirms our hypotheses stated in
+Section 1. Compared to the results of strong base-
+lines reported in papers, the F-scores of Guided-
+QA are still competitive. It shows that our simple
+model can output promising results compared to
+complicated ECPE methods (Ding et al., 2020; Wei
+et al., 2020; Chen et al., 2020; Cheng et al., 2021;
+Chen et al., 2022). The results from the original
+papers are just for reference because it seems there
+are gaps between the reproduced results and origi-
+nal results.6 . This is because several scholars tried
+to reproduce the results, but it seems there are gaps
+between the reproduced results and original results.
+For 20-split TransECPE in Table 4, the trend is
+consistent with Table 3. The Guided-QA is com-
+petitive for both ECE and ECPE tasks. The model
+using RoBERTa is still the best. After rerunning
+the source codes of the baselines, we found that
+PairGCN has the best reproducibility.
+By adopting the standardized pipeline of BERT-
+based question answering, our models inherit its
+simplicity and reproducibility which may become
+an issue in more complex methods like RankCP.
+Runtime comparison
+We also measured the run-
+ning time of our model and the baselines. In Table
+5, PairGCN which only uses BERT embeddings
+6https://github.com/Determined22/Rank-Emotion-
+Cause/issues/3
+
+Table 3: Experimental results of different models on 10-split ECPE dataset. * indicates reproduced results.
+Model
+Emotion Extraction
+Cause Extraction
+EC Pair Extraction
+P
+R
+F1
+P
+R
+F1
+P
+R
+F1
+Indep-QA (BERT)
+0.847
+0.908
+0.876
+0.714
+0.787
+0.749
+0.736
+0.661
+0.697
+Guided-QA (BERT)
+0.847
+0.908
+0.876
+0.719
+0.792
+0.754
+0.771
+0.692
+0.729
+Indep-QA (RoBERTa)
+0.854
+0.916
+0.884
+0.733
+0.807
+0.768
+0.761
+0.683
+0.720
+Guided-QA (RoBERTa)
+0.854
+0.916
+0.884
+0.732
+0.806
+0.767
+0.786
+0.706
+0.744
+ECPE-MLL (BERT)
+0.861
+0.919
+0.889
+0.738
+0.791
+0.763
+0.770
+0.724
+0.745
+RankCP (BERT)
+0.912
+0.900
+0.906
+0.746
+0.779
+0.762
+0.712
+0.763
+0.736
+PairGCN (BERT)
+0.886
+0.796
+0.838
+0.791
+0.693
+0.738
+0.769
+0.679
+0.720
+UTOS (BERT)
+0.882
+0.832
+0.856
+0.767
+0.732
+0.747
+0.739
+0.706
+0.720
+RSN (BERT)
+0.861
+0.892
+0.876
+0.773
+0.740
+0.755
+0.760
+0.722
+0.739
+ECPE-MLL (BERT)*
+—
+—
+—
+—
+—
+—
+0.688
+0.752
+0.718
+RankCP (BERT)*
+0.741
+0.744
+0.742
+0.614
+0.647
+0.627
+0.573
+0.625
+0.597
+PairGCN (BERT)*
+0.784
+0.883
+0.829
+0.686
+0.795
+0.735
+0.675
+0.772
+0.718
+Table 4: Experimental results of different models on 20-split TransECPE dataset. * indicates reproduced results.
+The authors of PairGCN and RSN did not tested their models on TransECPE.
+Model
+Emotion Extraction
+Cause Extraction
+EC Pair Extraction
+P
+R
+F1
+P
+R
+F1
+P
+R
+F1
+Indep-QA (BERT)
+0.842
+0.906
+0.873
+0.713
+0.785
+0.747
+0.730
+0.662
+0.694
+Guided-QA (BERT)
+0.842
+0.906
+0.873
+0.710
+0.782
+0.744
+0.760
+0.689
+0.723
+Indep-QA (RoBERTa)
+0.844
+0.909
+0.875
+0.724
+0.797
+0.758
+0.739
+0.670
+0.703
+Guided-QA (RoBERTa)
+0.844
+0.909
+0.875
+0.723
+0.796
+0.757
+0.772
+0.700
+0.734
+ECPE-MLL (BERT)
+0.847
+0.899
+0.872
+0.705
+0.770
+0.736
+0.749
+0.698
+0.722
+RankCP (BERT)
+0.894
+0.895
+0.894
+0.694
+0.747
+0.719
+0.658
+0.731
+0.692
+UTOS (BERT)
+0.865
+0.829
+0.849
+0.742
+0.708
+0.728
+0.710
+0.681
+0.691
+ECPE-MLL (BERT)*
+—
+—
+—
+—
+—
+—
+0.659
+0.714
+0.684
+RankCP (BERT)*
+0.896
+0.897
+0.896
+0.694
+0.749
+0.720
+0.657
+0.731
+0.691
+PairGCN (BERT)*
+0.804
+0.878
+0.839
+0.689
+0.770
+0.727
+0.677
+0.746
+0.709
+Table 5: Running time (train and test) on Tesla P100.
+ECPE
+TransECPE
+ECPE-MLL
+8.5h
+17h
+RankCP
+3h
+6h
+PairGCN
+42min
+85 min
+Indep-QA
+2h30
+5h
+Guided-QA
+2h30
+5h
+has the best running time. The other models take
+longer to run due to the fine-tuning of BERT mod-
+els. Our model is the second best, which is much
+faster than ECPE-MLL. It shows that our model
+can balance between competitive accuracy and high
+speed.
+6
+Conclusion
+This paper introduces a paradigm shift for the
+ECPE task. Instead of treating the task as the con-
+ventional formulation, we formulate the extraction
+as a QA problem. Based on that, we design a model
+which takes into account the implicit interaction be-
+tween emotion and cause clauses. Experimental re-
+sults on a benchmark Chinese dataset show that us-
+ing implicit interaction of emotions and causes can
+achieve competitive accuracy compared to strong
+baselines. Future work will consider explicit inter-
+action between emotion and cause clauses.
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diff --git a/HdA0T4oBgHgl3EQfB_8L/content/tmp_files/load_file.txt b/HdA0T4oBgHgl3EQfB_8L/content/tmp_files/load_file.txt
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@@ -0,0 +1,706 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf,len=705
+page_content='Emotion-Cause Pair Extraction as Question Answering  Huu-Hiep Nguyen  Cinnamon AI  hubert@cinnamon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='is  Minh-Tien Nguyen  Hung Yen University of Technology and Education  tiennm@utehy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='vn  Abstract  The task of Emotion-Cause Pair Extraction  (ECPE) aims to extract all potential emotion-  cause pairs of a document without any anno-  tation of emotion or cause clauses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Previous  approaches on ECPE have tried to improve  conventional two-step processing schemes by  using complex architectures for modeling  emotion-cause interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' In this paper, we  cast the ECPE task to the question answering  (QA) problem and propose simple yet effec-  tive BERT-based solutions to tackle it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Given a  document, our Guided-QA model first predicts  the best emotion clause using a fixed question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Then the predicted emotion is used as a ques-  tion to predict the most potential cause for the  emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  We evaluate our model on a stan-  dard ECPE corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The experimental results  show that despite its simplicity, our Guided-  QA achieves promising results and is easy to  reproduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The code of Guided-QA is also pro-  vided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  1  Introduction  Emotion Cause Extraction (ECE) is the task of  detecting the cause behind an emotion given the  emotion annotation (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Gui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=',  2016), see Figure 1 (Top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The text was divided  into clauses and the task was to detect the clause  containing the cause, given the clause containing  the emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' However, the applicability of ECE  is limited due to the fact that emotion annotations  are required at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Recently, (Xia and Ding,  2019) introduced the more challenging Emotion-  Cause Pair Extraction (ECPE) task: extracting all  possible emotion-cause clause pairs in a document  without annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Figure 1 (Bottom) shows an  example of the ECPE task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The input is a document  of six clauses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Clauses c4 and c5 contain emotion  with the emotion expressions “happy” and "wor-  ried".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The emotion c4 has two causes c3 and c2,  the emotion c5 has one cause c6, so the expected  output is {(c4,c2), (c4,c3), (c5,c6)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  c1: Yesterday morning  c4: The old man was very happy (Emotion)  c6: as he doesn’t know how to keep so much money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' (Cause)  c2: a policeman visited the old man with the lost money (Cause)  c3: and told him that the thief was caught.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' (Cause)  Emotion-Cause Pair Extraction (ECPE)  Input: a document  Output: emotion-cause pairs  {(c4,c2), (c4, c3),(c5,c6)}  c5: but he still feels worried, (Emotion)  c1: Yesterday morning  c4: The old man was very happy  c6: as he doesn’t know how to keep so much money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' (Cause)  c2: a policeman visited the old man with the lost money  c3: and told him that the thief was caught.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Emotion-Cause Extraction (ECE)  Input: a document and annotation of emotions c5  Output: cause clause(s)  {c6}  c5: but he still feels worried, (Emotion)  Figure 1: Illustration of ECE and ECPE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Why cause-effect pair extraction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' We argue that  independent extraction of cause and emotion may  be ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' For a given document, ECPE models  may predict correct cause but incorrect emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  This makes the output incomplete, and subsequent  processing steps less reliable (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  We make a toy example of two models using the  document in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Model-1 predicts (c4,c1)  and (c6,c3) as emotion-cause pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Its emotion,  cause and pair accuracy scores are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='33 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Model-2 predicts (c4, c2) and (c6, c1) as emotion-  cause pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Its emotion, cause and pair accuracy  scores are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='33 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' From the perspective  of the pair extraction task, Model-2 is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Previous studies addressed the ECPE task by us-  ing sequence labeling (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=',  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='01982v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='CL]  5 Jan 2023  2021), clause-level classification (Gui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020), ranking (Wei  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020), or recurrent synchronization (Chen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The methods achieved promising re-  sults, yet the use of interaction between emotion  and cause clauses is still an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' For  example, c4 and c2 share "the old man" tokens,  which refer to "him" in c3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' and c5 and c6 share  "he", which mentions "the old man" in c2 and c4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Based on this observation, we introduce a  paradigm shift (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2022) for ECPE by us-  ing span extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' As far as we know, (Gui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=',  2017) is the first work that uses question answering  for emotion-cause detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' However, their work  addresses the ECE task only, which requires the  annotation of emotion for cause prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' In con-  trast, our paradigm shift is applied to the ECPE task,  which is more challenging and does not require the  annotation of emotion for cause prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The  paradigm bases on two hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' First, infor-  mation from emotion clauses can be used to infer  cause clauses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Second, emotion and cause clauses  share implicit interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The design of our model  is based on these two hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' For the first hy-  pothesis, we form questions based on emotional in-  formation which is used to predict emotion clauses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  For the second hypothesis, we used predicted emo-  tion as the guided question for cause prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  The model is trained by using the BERT-QA archi-  tecture (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2018) in form of SQuAD task  (Rajpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Our paper makes three main contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  We formulate the ECPE task as a QA problem  and propose a Guided-QA model to implicitly  capture the relationship between emotion and  cause clauses, in which the predicted emotion  is used as a guided question for cause pre-  diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The model can capture the implicit  interaction between emotions and causes with  a simple but effective architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' To the best  of our knowledge, we are the first to address  the ECPE task by using QA formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  We evaluate our model on the standard ECPE  corpus (Xia and Ding, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Experimental results show that our approach  achieves promising results compared to previ-  ous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  We promote the reproducibility (Houghton  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020) by providing the source code  of our methods as well as rerunning publicly  available source codes of the compared meth-  ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  2  Related Work  ECE and ECPE tasks  The ECE task was for-  mulated as sequence-labeling by (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2010)  and refined as clause-level by (Gui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Recently, the more challenging ECPE task (Xia  and Ding, 2019) has attracted a lot of contributions  with several strong methods (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=',  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' For example, (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=',  2020) introduced ECPE-MLL, which uses a sliding  window for a multi-label learning scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' ECPE-  MLL extracts the emotion and cause by using the  iterative synchronized multitask learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' (Chen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2022) proposed a similar approach, recur-  rent synchronization network (RSN), that explic-  itly models the interaction among different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020) presented RankCP, a transition-  based framework, by transforming the ECPE prob-  lem into directed graph construction, from which  emotions and the corresponding causes can be ex-  tracted simultaneously based on labeled edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The  PairGCN model (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020) used Graph  Convolutional Networks to model three types of  dependency relations among local neighborhood  candidate pairs and facilitate the extraction of pair-  level contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  We share the purpose of addressing the ECE and  ECPE tasks with prior studies, however, instead  of using classification or sequence labeling, we ad-  dress the tasks with a new paradigm shift by using  span extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' It allows us to take into account  the implicit interaction between emotion and cause  clauses and to design a simple but effective BERT-  based model for ECE and ECPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  (Bi and Liu, 2020) derived a span-based dataset  and formulated a new ECSP (Emotion Cause Span  Prediction) task from (Xia and Ding, 2019) but it  has not attracted much attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The accessibility  of the dataset and source code may be the reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  We leave span-based ECSP evaluation as future  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Paradigm shift in natural language processing  A paradigm is a general modeling framework or  a family of methods to solve a class of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  For instance, sequence labeling is a mainstream  paradigm for Part-of-speech (POS) tagging and  Named entity recognition (NER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The sequence-to-  sequence (Seq2Seq) paradigm is a popular tool for  summarization and machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Different  paradigms usually require different formats of in-  put and output, and therefore highly depend on the  annotation of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Paradigm shift indicates the job of solving one  NLP task in a new paradigm by reformulating the  task along with changing the input-output formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Paradigm shift in NLP has been explored scatter-  ringly in recent years and with the advent of pre-  trained language models, it became a rising trend  (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Khashabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' An excel-  lent survey of paradigm shifts in NLP has been  done by (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' In this work, we realize  such a paradigm shift for the ECPE task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', we  reformulate the clause-based text classification task  as span extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Span-based  extractive  question  answering  Our formulation for the tasks of ECE and ECPE  relates to span-based extractive QA, which has  been widely investigated (Khashabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  More precisely, we design our model based on the  pretrained language models (PLMs) such as BERT  (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2018) or RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  This is because applying PLMs as the backbone of  QA systems has become a standard procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' For  detailed information, please refer to (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=',  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Figure 2 reproduced from (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2018)  shows how BERT is applied to the extractive QA  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Tokens of question q = q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='., qn and context  C = c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='., cm are concatenated before being en-  coded by BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The contextual representations of  tokens Ti are put into a feed-forward layer followed  by a softmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Each candidate span for the answer  is scored as the product of start/end probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  The maximum scoring span is used as the predic-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The training objective is the loglikelihood of  the correct start and end positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  By casting the ECPE to QA problem, our work  leverages the powerful models of the BERT family  (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2018) to detect clause-level emotions  and causes as well as emotion-cause pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  3  Method  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='1  Problem Statement  Given a document of n clauses d = (c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='., cn),  the goal of ECPE is to detect all potential emotion-  cause pairs P = {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='.(ce, cc), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='.} where ce is an  emotion clause, and cc is the corresponding cause  clause (Xia and Ding, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' We formulated the  BERT  CLS  c_1  q_1  SEP  q_n  c_m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Start/End Span  T_CLS  T_c_1  T_q_1  T_SEP  T_q_n  T_c_m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  E_CLS  E_c_1  E_q_1  E_SEP  E_q_n  E_c_m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Figure 2: BERT-based extractive Question Answering  ECPE task as a QA problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Given a set of  questions {qe, qc} (qe is for emotion and qc is for  cause) and a context document d with n clauses,  the model learns to predict start and end posi-  tions of each ce and cc: sce, ece = f(d, qe|Θ) and  scc, ecc = f(d, qc|Θ) to form P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Θ can be learnt  by using independent or guided extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='2  Independent Emotion, Cause Extraction  We first introduce a simple version of our model,  Indep-QA in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Indep-QA receives a fixed  question (for emotion or cause) and then pulls out  corresponding emotion or cause clauses indepen-  dently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Question  formulation  Because  no  emo-  tion/cause information is provided beforehand, we  have to detect them first with generic questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  It is possible to use pre-defined questions for ex-  traction (Mengge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020), however, we argue  that the definition of questions is time-consuming,  needs domain knowledge, and does not guarantee  the semantic relationship between the questions  and context documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Instead, we use two short  questions "emotion" and "cause" as an implicit  indicator that provides additional information for  the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' We leave the analysis of using generic  questions such as "What is the emotion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='" and  "What is the cause?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='" as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Learning and prediction  Given a document d  and a question ("emotion" or "cause"), we con-  catenated all clauses of d and the question to form a  single sequence C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The sequence was fed to a pre-  trained language model (PLM) to obtain its hidden  representations of tokens which were subsequently  fed into a feed-forward layer followed by a soft-  max layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Each candidate span was scored as the  answer start  answer end  c1  c2  c3  c4  question = "emotion"  context =  predicted Emotion clause = c2  answer start  answer end  c1  c2  c3  c4  question = "cause"  context =  predicted Cause clause = c4  predicted EC-pair = (c2,c4)  Figure 3: Independent extraction Indep-QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  c1  c2  c3  c4  question =  context =  predicted Cause clause = c3  c1  c2  c3  c4  context =  predicted Emotion clause = c4  predicted EC-pair = (c4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='c3)  question =  c3  answer start  answer end  answer start  answer end  c1  c2  c3  c4  question =  context =  predicted Emotion clause = c2  c1  c2  c3  c4  context =  predicted Cause clause = c3  predicted EC-pair = (c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='c3)  question =  c2  answer start  answer end  answer start  answer end  "emotion"  "cause"  Figure 4: Guided pair extraction Guided-QA: Emotion is detected first (Left),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Cause is detected first (Right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  product of start/end probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The maximum  scoring span was used as the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Mapping predicted answer span to clauses  The predicted answer span may overlap with one  or several clauses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' We applied a span-to-clause  mapping rule to determine which clauses are pre-  dicted results: the clause that overlaps most with  the predicted span is returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The tie is broken  arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' For instance, In Figure 3, the predicted  span for "emotion" overlaps with clauses c2 and c3  in which c2 is more overlapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' As a result, c2 is  the predicted emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  EC  pair  prediction  Given  predicted  emo-  tion/cause clauses ce and cc, Indep-QA simply pre-  dicts (ce, cc) as an emotion-cause pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' As illus-  trated in Figure 3, (c2, c4) is the predicted emotion-  cause pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='3  Guided Emotion-Cause Pair Extraction  The Indep-QA model extracts emotion/clause  clauses independently but does not exploit the rela-  tionship between emotion and cause clauses, which  plays an important role in the extraction of emotion-  cause pairs (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=',  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  To better model this relationship, we introduce  Guided-QA in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The model receives an  emotion question and predicts the corresponding  emotion clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Then the predicted emotion clause  is used as a question for cause extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Com-  pared to Indep-QA, the Guided-QA takes into ac-  count an implicit relationship from emotion for  cause prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  The Guided-QA model shares the question for-  mulation, hidden representation learning, and the  mapping process of the Indep-QA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  EC pair extraction  We used the predicted  (noisy) emotion clause as the question for cause  extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The interaction between emotion and  cause happens here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The predicted emotion clause  may or may not be the true one but on average, it  contains much more information for the QA model  than the generic question (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', "emotion").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Note  that the predicted (noisy) emotion as the question  was used for the test set only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' For the training set,  as the model already knows which clauses are emo-  tion or cause, it uses the true emotion clause as the  question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  By swapping the role, the model can detect cause  clauses first and use the noisy causes as questions  to predict the emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' In Section 5 we compare  Emotion-first and Cause-first, the two variants of  Guided-QA and show that the gaps are tiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' In  other word, the two variants are almost equivalent  on the tested datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  As our QA models use the best answer span for  each question, only one emotion, one cause, and  one EC pair are predicted for each document which  are appropriate for the ECPE dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' We also  aware that the prediction of spans should be multi-  ple and we aim to address this limitation in future  work by using multiple span extraction methods  (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='4  Discussion  Given a document of n clauses, existing schemes  such as ECPE-MLL (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020), RankCP  (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020) and PairGCN (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=',  2020) attempt to reduce the O(n2) complexity of  emotion-cause pair classification by using sliding  window, transition graph techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' However,  these techniques may miss certain interaction be-  tween the emotion-cause pair and the full context  in the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' BERT-based QA models with  full attention between the question and the con-  text mitigate this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Through QA models, the  emotion-cause relationship between all clauses is  implicitly learned and we can leverage the power  of existing QA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  4  Experimental Settings  Datasets  We followed the 10-split ECPE dataset  provided by (Xia and Ding, 2019) and the 20-split  TransECPE variant (Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020) to evaluate  our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Each split is a random partition of  the 1945 documents to train/dev/test sets with ratio  8:1:1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', the train set, dev set and test set contain  approximately 1556, 194 and 195 documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' On  average, each document contains 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='8 clauses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Table 1 shows the distribution of documents with  different number of emotion-cause pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Most of  the documents have only one emotion-cause pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  This fact makes the detection of emotion/cause  clauses as well as emotion-cause pairs challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Evaluation metrics  We used the precision, re-  call, and F1 score (Xia and Ding, 2019) as evalua-  tion metrics for all three tasks of ECPE: emotion  extraction, cause extraction and emotion-cause pair  extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Let Te and Pe be the number of ground-  truth and predicted emotion clauses respectively,  the precision, recall and F1 score for emotion are  as defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Pe = |Te ∩ Pe|  |Pe|  Re = |Te ∩ Pe|  |Te|  F1e = 2 ∗ Pe ∗ Re  Pe + Re  Metrics for cause clauses and emotion-cause  pairs are defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Implementation details  Our model was imple-  mented using BERT classes provided by Hugging-  face (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The model was trained  in 5 epochs, with the learning rate of 5e − 5, and  the batch size of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' We used BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=',  2018)1 and RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2019)2 for Chi-  nese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' All models were trained on a Tesla P100  GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  5  Results and Discussion  Guided-QA: Emotion-first vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Cause-first  We  first compare the two variants Emotion-first and  Cause-first of the Guided-QA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Table 2  shows that the two variants have almost equiva-  lent performance on the tested datasets except the  BERT-based results on 10-split ECPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Also, the  RoBERTa-based results are consistently better than  the BERT-based, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='1 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='0 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' In the next  section, we pick the Emotion-first scores for com-  paring Guided-QA with other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Guided-QA vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Indep-QA  We now compare  Guided-QA and Indep-QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' For 10-split ECPE in  the upper part of Table 3, the Guided-QA model  is consistently better than Indep-QA for pair ex-  traction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' This is because Guided-QA takes into  account the implicit interaction between emotion  and cause clauses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' For emotion or cause extraction,  Indep-QA is competitive with Guided-QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' This  is because they share the same formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The  results in Table 4 also show similar observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  We also confirm the performance of our model  by using RoBERTa to have better analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The  results are consistent with the model using BERT,  in which Guided-QA outputs better F-scores than  the Indep-QA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' It also shows that our model  can be improved further by using stronger PLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Guided-QA vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' strong baselines  We compare  our model with five strong methods for ECPE:  ECPE-MLL3 (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020), RankCP4 (Wei  1https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='co/bert-base-chinese  2https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
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+page_content='735  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020), PairGCN5 (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020), UTOS  (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2021), and RSN (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  For 10-split, our model using BERT follows ECPE-  MLL, RankCP, and RSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' It shows that with a sim-  ple architecture, our model can output competitive  results compared to complicated methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' For 20-  split TransECPE in Table 4, the trend is consistent  with Table 3, in which the Guided-QA model is  competitive for both ECE and ECPE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Moreover, as we observe from all the compared  methods, the gaps between the reported pair-f1  scores for 10-split ECPE and 20-split TransECPE  are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
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+page_content='722) for ECPE-MLL, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='042 for  RankCP, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='029 for UTOS, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='003 for Indep-QA and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='006 for Guided-QA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', largest gap in RankCP  and smallest gaps in our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Across the two  settings, our models seem more robust than the  compared methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Reproducibility  For fair comparison (Houghton  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020), we also rerun publicly available  source codes in the original setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The repro-  duced results confirm the gaps between reproduc-  tion and original results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Compared to the repro-  duced results, Guided-QA using BERT is the best  for EC pair extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Compared to the results of reproduced methods,  the Guided-QA is still better for both ECE and  5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='com/chenying3176/PairGCN_ECPE  ECPE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' This confirms our hypotheses stated in  Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Compared to the results of strong base-  lines reported in papers, the F-scores of Guided-  QA are still competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' It shows that our simple  model can output promising results compared to  complicated ECPE methods (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Wei  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
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+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The results from the original  papers are just for reference because it seems there  are gaps between the reproduced results and origi-  nal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
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+page_content=' This is because several scholars tried  to reproduce the results, but it seems there are gaps  between the reproduced results and original results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  For 20-split TransECPE in Table 4, the trend is  consistent with Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
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+page_content='  By adopting the standardized pipeline of BERT-  based question answering, our models inherit its  simplicity and reproducibility which may become  an issue in more complex methods like RankCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
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+page_content='  ECPE  TransECPE  ECPE-MLL  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='5h  17h  RankCP  3h  6h  PairGCN  42min  85 min  Indep-QA  2h30  5h  Guided-QA  2h30  5h  has the best running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' The other models take  longer to run due to the fine-tuning of BERT mod-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Our model is the second best, which is much  faster than ECPE-MLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' It shows that our model  can balance between competitive accuracy and high  speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  6  Conclusion  This paper introduces a paradigm shift for the  ECPE task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Instead of treating the task as the con-  ventional formulation, we formulate the extraction  as a QA problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Based on that, we design a model  which takes into account the implicit interaction be-  tween emotion and cause clauses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Experimental re-  sults on a benchmark Chinese dataset show that us-  ing implicit interaction of emotions and causes can  achieve competitive accuracy compared to strong  baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Future work will consider explicit inter-  action between emotion and cause clauses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  References  Hongliang Bi and Pengyuan Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' Ecsp: A new  task for emotion-cause span-pair extraction and clas-  sification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' arXiv preprint arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='03507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Fang Chen, Ziwei Shi, Zhongliang Yang, and Yongfeng  Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Recurrent synchronization network  for emotion-cause pair extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Knowledge-  Based Systems, 238:107965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content='  Ying Chen, Wenjun Hou, Shoushan Li, Caicong Wu,  and Xiaoqiang Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' End-to-end emotion-  cause pair extraction with graph convolutional net-  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
+page_content=' In Proceedings of the 28th International Con-  ference on Computational Linguistics, pages 198–  207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdA0T4oBgHgl3EQfB_8L/content/2301.01982v1.pdf'}
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+arXiv:2301.11820v1  [math-ph]  27 Jan 2023
+HYDRODYNAMIC AND CONSERVATIVE MODELS OF
+HYPERCOMPUTATION
+ROBERT CARDONA
+Abstract. Dynamical systems and physical models defined on idealized con-
+tinuous phase spaces are known to exhibit non-computable phenomena, ex-
+amples include the wave equation, recurrent neural networks, or Julia sets in
+holomorphic dynamics. Inspired by the works of Moore and Siegelmann, in
+this article we introduce new dynamical models of hypercomputation. First,
+we show that ideal fluids, modeled by the Euler equations, are capable of sim-
+ulating poly-time Turing machines with polynomial advice on compact three-
+dimensional domains. The complexity class that is shown to be computable
+by stationary ideal fluids is precisely the one considered by Siegelmann in her
+study of analog recurrent neural networks: the class P/poly. Then, we introduce
+a class of symbolic systems that can be embedded in conservative homeomor-
+phisms of the disk. These systems are shown to be capable of simulating Turing
+machines with advice in real-time, contrary to previously known models.
+1. Introduction
+Computational aspects of dynamical and physical systems can be studied from
+a variety of intertwined perspectives such as numerical simulation, computability
+theory, computational complexity, or analog computation. The last one under-
+stands a dynamical system as a computing device that takes an input (the initial
+condition) and reaches some region of the phase space encoding the output of the
+process. Combining symbolic dynamics and the Turing machine model, Moore
+showed in his seminal work [23, 22] that even low-dimensional dynamical systems
+are capable of universal computation, thus unveiling the undecidability of some
+of their properties. Since then, several dynamical systems coming from physical
+models have been shown capable of simulating universal Turing machines. Exam-
+ples include 3D optical systems [26], analog recurrent neural networks [30], high
+dimensional potential wells [31] and more recently incompressible fluids in various
+contexts [12, 14, 15].
+Robert Cardona acknowledges financial support from the Margarita Salas postdoctoral con-
+tract financed by the European Union-NextGenerationEU, as well as from the LabEx IRMIA, the
+Universit´e de Strasbourg and Instituto de Ciencias Matem´aticas. This work was partially sup-
+ported by the AEI grant PID2019-103849GB-I00 / AEI / 10.13039/501100011033, AGAUR grant
+2017SGR932 and the project Computational, dynamical and geometrical complexity in fluid dy-
+namics - AYUDAS FUNDACI ´ON BBVA A PROYECTOS INVESTIGACI ´ON CIENT´IFICA
+2021.
+1
+
+2
+ROBERT CARDONA
+Beyond Turing-computability arises hypercomputation: computational models
+that can compute more than the classical Turing machines, such as Turing’s oracle
+machines [34]. Dynamical systems modeled in continuous phase spaces allow the
+presence of real numbers and infinite precision, which can lead to non-computable
+phenomena and presence of hypercomputing capacities. Even if those dynamical
+systems can represent physical models that are highly idealized and hence physi-
+cally non-realizable, it is interesting from a theoretical of point of view to under-
+stand which models do admit these hypercomputing capacities. For example, the
+wave equation admits non-computable solutions even if one chooses computable
+initial data [25], see also [35, 19]. Some examples in purely dynamical contexts
+include the existence of non-computable Julia sets [9, 10], and polynomial planar
+flows with non-computable number of periodic orbits [20] . A complexity class that
+contains non-computable languages is P/poly, which is the set of languages rec-
+ognized by polynomial-time Turing machines with polynomial advice (see Section
+2.1 for details). In her influential work [27, 28, 29], Siegelmann showed that neural
+networks with real weights can simulate those machines in polynomial time, and
+hence that the model is capable of computing beyond Turing machines. Other dis-
+crete dynamical systems were shown to be computationally equivalent to P/poly
+by Bournez and Cosnard [4, 3].
+The first contribution of this paper is to establish that ideal fluids on three-
+dimensional geometric domains are also capable of simulating polynomial-time
+Turing machines with polynomial advice. Recall that given a three-dimensional
+manifold, with or without boundary, the motion of an ideal fluid (i.e. incompress-
+ible and without viscosity) is modeled by the Euler equations
+�
+∂
+∂tu + ∇uu
+= −∇p ,
+div u = 0 ,
+where p stands for the hydrodynamic pressure and u is the velocity field of the
+fluid, which is a non-autonomous vector field on M tangent to its boundary. Here
+∇uu denotes the covariant derivative of u along itself, and div is the divergence
+associated with the Riemannian metric g. A stationary solution to the Euler
+equations is an autonomous vector field on M whose integral curves represent the
+particle-paths of the fluid.
+Theorem 1. Given a polynomial-time Turing machine with polynomial advice
+(T, a), there exists a three-dimensional toroidal domain U equipped with some
+Riemannian metric g, and a stationary solution to the Euler equations in (U, g)
+that simulates T in polynomial time.
+There are several ways in which a continuous system can simulate a Turing
+machine, but generally it roughly means that each step-by-step computational
+process of the machine is encoded in the evolution of some orbit of the system.
+We introduce in this work a natural notion of simulation (see Definition 7) that
+
+HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION
+3
+is inspired by the classical ones in the literature [8, 23, 17, 18, 6], but that further
+provides a well-defined notion of time complexity in the context of conservative
+ODEs. In the statement, we have chosen for simplicity a domain diffeomorphic
+to a solid torus, however, one can choose as well any closed three-manifold or R3
+for example. To establish Theorem 1, we partially embed the dynamics of the
+symbolic systems introduced by Siegelmann [27] in the evolution of a stationary
+Euler flow, by following the strategy of [15]. We construct diffeomorphisms of the
+disk that simulate Turing machines with advice and embed them as a first-return
+map on a cross section of a suitably parametrized flow.
+In the second part of this paper, we first introduce a new family of symbolic
+systems that generalize those introduced by Moore [23] in a different way than the
+analog shift map [27]. We study some dynamical and computational properties
+of these systems, which we call countable generalized shifts, showing that they
+can compute P/poly. Besides the fact that it provides an alternative symbolic
+model of hypercomputation, it has some advantages with respect to previously
+known dynamical models such as real-time simulation of reversible polynomial-
+time Turing machines with polynomial advice. We show that these systems can
+be partially embedded in the evolution of conservative homeomorphisms of the
+disk, hence losing regularity when compared to the analogous result for classical
+generalized shifts [23, 15].
+Theorem 2. Given a (reversible) polynomial-time Turing machine with polyno-
+mial advice (T, a), there exists an area-preserving homeomorphism of the disk that
+simulates T in real-time.
+In this setting the simulation is given, as in Moore’s works [23], by the exis-
+tence of a computable semiconjugacy between the global transition function of
+the Turing machine and the homeomorphism of the disk. This provides another
+conservative model of hypercomputation. From a purely dynamical perspective, it
+would be interesting to understand if the symbolic systems that we introduce turn
+out to be a special feature of low-regularity area-preserving homeomorphisms of
+the disk. We point out that the area-preserving diffeomorphisms of the disk con-
+structed in the proof of Theorem 1 are already capable of hypercomputations (see
+Corollary 6). However, those simulations are not done in real-time, hence the dif-
+ference with the model introduced in Sections 5 and 6.
+The paper is organized as follows. In Section 2, we review the definitions of
+generalized shift and analog shift map. In Section 3, we prove that there ex-
+ists area-preserving diffeomorphisms of the disk that simulate in polynomial time
+Turing machines with advice. Section 4 starts by introducing time complexity in
+conservative ODEs by defining an appropriate notion of simulation. It is shown
+that there are Euler flows simulating any polynomial-time Turing machine with
+polynomial advice according to such definition. In Section 5 we introduce count-
+able generalized shifts, study some of their dynamical properties and analyze their
+
+4
+ROBERT CARDONA
+computational power. Finally, in Section 6 we prove Theorem 2 by showing how
+some countable generalized shifts can be partially embedded in area-preserving
+homeomorphisms of the disk.
+Acknowledgements: The author is grateful to Cristopher Moore, whose use-
+ful correspondence about transformations of the disk preserving the square Cantor
+set inspired this work. Thanks to Daniel S. Gra¸ca for helpful comments concern-
+ing time complexity of continuous systems, and to Daniel Peralta-Salas for useful
+discussions.
+2. Symbolic dynamics
+In this section, we recall several definitions, such as Turing machine with advice,
+Moore’s generalized shifts [23] and the analog shift map, a generalization proposed
+by Siegelmann [27].
+2.1. Turing machines with polynomial advice. We define a Turing machine
+T = (Q, q0, qhalt, Σ, δ) by the following data:
+- A finite set Q of “states” containing two particular (distinct) states: the
+initial state q0 ∈ Q and the halting state qhalt ∈ Q.
+- A finite set Σ which is the “alphabet” and that has cardinality at least
+two. It contains a specific symbol (denoted by 0) that is also called the
+“blank symbol”.
+- A transition function δ : Q \ {qhalt} × Σ −→ Q × Σ × {−1, 0, 1}.
+A pair (q, t) ∈ Q × ΣZ is a configuration of the machine if the symbols of
+t are all zero except for finitely many of them. We say in this case that the
+configuration, or the tape, is compactly supported. This can be assumed without
+loss of computational power: Turing machines with this condition are equivalent
+(as a computational model) to those which can have tapes with infinitely many
+symbols different from zero. Hence if A ⊂ ΣZ denotes the set of sequences that
+have all but finitely many symbols equal to zero, the space P = Q×A is the space
+of configurations of the machine. When writing a tape t = (ti) ∈ ΣZ, we will use
+a dot to specify that the position zero lies at the right of the dot:
+...t−1.t0t1...
+The evolution of a Turing machine is described as follows. At any given step of
+the algorithm, we will denote by q ∈ Q the current state, and by t = (tn)n∈Z ∈ ΣZ
+the current tape. Given an input tape s = (sn)n∈Z ∈ ΣZ the machine runs by
+applying the following algorithm:
+(1) We initialize the machine by setting the current state q to be q0 and the
+current tape t to be the input tape s.
+(2) If the current state is qhalt then halt the algorithm and return t as output.
+Otherwise compute δ(q, t0) = (q′, t′
+0, ε), with ε ∈ {−1, 0, 1}.
+(3) Change the symbol t0 by t′
+0, obtaining the tape ˜t = ...t−1.t′
+0t1....
+
+HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION
+5
+(4) Shift ˜t by ε obtaining a new tape t′. The new configuration is (q′, t′). Re-
+turn to step (2). Our convention is that ε = 1 (resp. ε = −1) corresponds
+to the left shift (resp. the right shift).
+A step of the algorithm determines a global transition function
+∆ : (Q \ {qhalt}) × A −→ P
+that sends a configuration to the configuration obtained after applying a step of
+the algorithm. The global transition function can also be trivially extended to
+(Q \ {qhalt}) × ΣZ, since a step of the algorithm is well-defined even for non-
+compactly-supported tapes.
+A polynomial-time Turing machine T is a machine that halts for any given
+input tin of size n in at most P(n) steps, where P(n) is some polynomially-
+bounded function. For our purposes, we shall follow the convention in [27, 29] and
+say that an input tin ∈ ΣZ is of size n if it is of the form
+tin = ...0.t0...tn0...,
+meaning by this that the tape only has zeroes away of the positions 0, ..., n + 1.
+A polynomial-time machine with polynomial-sized advice (T, a) comes equipped
+with an infinite collection of strings a = {an}n∈N such that an ∈ Σp(n) for some
+increasing polynomially-bounded function p(n). Given an input tin of size n, the
+machine has access to an in one computational step. To make a very concrete
+dynamical interpretation of this, given T and {an}n∈N, let us add a new state ˜q0
+in Q which will be an auxiliary initial state, and the machine with input (˜q0, tin)
+applies a preliminary step of the algorithm that sends (˜q0, tin) to the configuration
+(q0, ta
+in) where if tin = ...0.t0...tn0..., the tape ta
+in is
+ta
+in = ...0.t0...tnan
+1...an
+p(n)0...
+where we wrote the advice string as an = an
+1...an
+p(n). For such Turing machine, the
+global transition function ∆ can be extended to the set {˜q0}×ΣZ, by declaring that
+on elements of the form (q0, tin) we have ∆(q0, tin) = (q0, ta
+in), and any arbitrary
+extension for the other elements in Q × Σ. Throughout the paper, whenever we
+say Turing machine with advice, we refer to this polynomially-bounded version of
+it.
+2.2. Generalized shifts and analog shifts. Let us now review the two classes
+of symbolic systems that have been related respectively to Turing machines and
+Turing machines with advice.
+Generalized shifts. We recall here the definition of generalized shifts, follow-
+ing the original definition of Moore. Let A be a finite alphabet and s = (si)i∈Z ∈
+AZ an infinite sequence. A generalized shift is given by two maps. First, a map
+G : AZ −→ AZ,
+
+6
+ROBERT CARDONA
+together with a finite domain of dependence DG = {i, ..., i + l − 1} and a finite
+domain of effect De = {n, ..., n + m − 1}. These domains indicate that G(s) is
+equal to s except maybe along the positions in De, which are modified according
+to the symbols in positions DG of s.
+Secondly, a map
+F : AZ −→ Z.
+with a finite domain of dependence DF = {j, ..., j +r−1}, i.e. the image of F only
+depends on the symbols of the sequence in position DF . These functions take a
+finite number of different values since they depend on a finite number of positions.
+The image of s by the generalized shift φ : AZ → AZ is then defined as follows:
+- change s by G(s), this modifies potentially the symbols in positions De of
+s,
+- shift G(s) by F(s), and declare it by definition as φ(s).
+As we did for Turing machines, we choose by convention that a positive (negative)
+value of F(s) means that we shift |F(s)| times to the left (right).
+The previous definition is appropriate from a notational point of view to easily
+define the generalization of analog shift maps. However, generalized shifts admit
+an alternative simpler notation (as used in [15]) that we will use as well.
+Alternative notation. The map G, once we have fixed the domains of dependence
+and effect, is completely determined by assigning to each word in Al a word in
+Am. Hence, abusing notation, we will understand G as a map
+G : Al → Am,
+and write G(t1...tl) = t′
+1...t′
+m. Similarly, the image by F of s is completely deter-
+mined by the symbols in positions DF , so abusing notation we can think of F as
+a map
+F : Am → Z.
+Then, given a sequence s the sequence φ(s) is obtained by replacing the symbols in
+positions n, ..., n + m − 1 of s by G(si...si+l−1) and shifting the resulting sequence
+by F(sn...sn+m−1). Unless otherwise stated, and without loss of generality, we will
+assume that a generalized shift is such that De = DG.
+Let us state an obvious property of generalized shifts that we will use later.
+Lemma 3. Let φ be a generalized shift. Then for any sequence s ∈ AZ, s and
+its image φ(s) coincide after a shift −F(s), except along the symbols at positions
+DG.
+Proof. The sequence φ(s) is obtained by changing some of the symbols at posi-
+tions DG of s and shifting by F(si...si+r−1). Hence shifting φ(s) by −F(si...si+r−1)
+yields a sequence that coincides with s except maybe along the symbols in posi-
+tions in DG.
+□
+
+HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION
+7
+The analog shift map. The analog shift maps [27] are defined similarly to
+generalized shifts, except that the domain of effect of G can be infinite (in one or
+both directions). Hence in this case an analog shift map φ : AZ → AZ is specified
+by
+F : AZ −→ Z, DF = {i, ..., i + r − 1}
+with a finite domain of dependence DF , and
+G : AZ → AZ,
+with a finite domain of dependence DG = {j, ..., j+l−1}. However, the domain of
+effect of G depends on the symbols in position DG of the sequence s and can be a
+finite number of consecutive integers, a one-sided infinite sequence of consecutive
+integers or all Z.
+3. Simulation with area-preserving diffeomorphisms
+In this section, we show how to simulate Turing machines with advice via com-
+pactly supported area-preserving diffeomorphisms of the disk. To prove this fact,
+we will first show how to simulate such machines via a globally injective piece-
+wise linear area-preserving map defined on a finite number of disjoint rectangular
+domains on a disk. These domains will mostly be Cantor blocks, whose definition
+we recall below.
+Definition 4. The square Cantor set is the product set C2 := C ×C ⊂ I2, where
+C is the (standard) Cantor ternary set in the unit interval I = [0, 1]. A Cantor
+block is a block of the form B = [ a
+3i , a+1
+3i ] × [ b
+3j , b+1
+3j ] ⊂ I2, where i, j are two
+non-negative integers and a < 3i, b < 3j are two non-negative integers such that
+there are points of C2 in the interior of B.
+We identify sequences s = (...s−1.s0s1...) in {0, 1}Z with points C2 via the
+following bijection.
+e : {0, 1}Z −→ C2
+(si)i∈Z �−→ (
+∞
+�
+i=1
+s−i
+2
+3i ,
+∞
+�
+i=0
+si
+2
+3i+1 )
+(1)
+If we choose an alphabet with k symbols instead of only two, we can encode all
+the sequences of the alphabet in a square Cantor set Ck2 where Ck is a Cantor
+set of the interval obtained by iteratively removing k − 1 open subintervals. Then
+Cantor blocks are defined analogously. In the statement below D denotes some
+disk of big enough radius and that contains the unit square I2.
+Theorem 5. Given a Turing machine with advice (T, a), there exist two families
+B = B1, ..., BN−1, BN and B′ = B′
+1, ..., B′
+N−1 of pairwise disjoint Cantor blocks,
+a rectangular domain B′
+N ⊂ D disjoint from B′, and a piecewise linear area-
+preserving map F : ⊔Bi → B′
+i such that F|C2∩B simulates (T, a) in polynomial
+time.
+
+8
+ROBERT CARDONA
+A concrete property that formalizes simulation in polynomial time in this con-
+text is that there exists a computable map ϕ that maps the configurations of
+(T, a) to points on a square Cantor set ˜C2 ∩ B such that if (T, a) halts with input
+(q0, tin) and output (qhalt, tout) in k steps, then the orbit of F through ϕ(q0, tin)
+reaches ϕ(qhalt, tout) in q(k) iterations of F, where q is some polynomially-bounded
+function.
+Proof. As explained by Siegelmann, the analog shift map is capable of simulating,
+in polynomial time, a given polynomial-time Turing machine with polynomial
+advice. Let us first recall how this is done to make suitable changes and obtain
+globally injective dynamics.
+Given a Turing machine with advice (T, a), consider T as a classical Turing
+machine. Let us denote by a∞ = ...a2a1a0 the one-sided infinite word obtained
+by concatenation of all the advice of T. Following [29, Chapter 12], there exists
+a Turing machine ˜T whose set of states and symbols contains those of T but
+have as well an extra marker symbol d in its alphabet and an extra set of states
+˜q1, ..., ˜qr. The initial state of ˜T is ˜q1 (instead of q0 ∈ Q) and it simulates T on
+inputs of the form (...0.t0...tn0...) in polynomial time as follows. If we consider the
+configuration (˜q1, ˜t) with ˜t = (...a2a1a0.t0t1...tn0...), then in g(n) steps (where g
+is some polynomially-bounded function) it reaches the configuration (˜qr, t′) with
+t′ = (...an+1d.t0...tnan0...),
+and from there simulates T in polynomial time. This is done step by step as in T,
+while moving the marker d to the left when necessary to keep track of the relevant
+part of the left side of the tape. To keep the notation simple, let Q and Σ denote
+the set of states and the alphabet of ˜T.
+We now consider the alphabet ˜A = Q ⊔ Σ ⊔ {˜q0}, and we identify the configu-
+rations of ˜T to AZ by the map
+ϕ : Q × ΣZ ֒→ ˜AZ
+(q, t) �−→ (...t−1.qt1...).
+Using [23, Theorem 7], there exists a generalized shift φGS with DF = DG =
+De = {−1, 0, 1} defined on the alphabet A =
+˜A \ {˜q0} such that the global
+transition function of ˜T is semi-conjugate to φ by this identification. We extend
+this generalized shift to an analog shift map φ : ˜AZ → ˜AZ satisfying:
+�
+G(0.˜q0t)
+= (a∞.˜q1t) for any t ∈ Σ,
+F(0.˜q0t)
+= 0 for any t ∈ Σ.
+The dot in the words is used to specify that DG is still {−1, 0, 1}, but the domain
+of effect of G for the words of the form (0˜q0t) is now every integer smaller or
+equal to 1. For each (s−1s0s1) where φGS was already defined, the map G still
+only changes the symbols in position −1, 0, 1.
+
+HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION
+9
+We encode an input (...0.t0...tn0...) in ˜AZ as the sequence (...0.˜q0t0...tn0...).
+After one step of ˜φ, we reach (a∞.˜q1t0...tn0...), and then proceed simulating ˜T.
+Notice that we can assume that ˜T is reversible. Indeed, a classical method of
+Benett [2] shows how to simulate T by some reversible Turing machine with three
+tapes. The simulation is polynomial in time, in fact if ˜T halts in K steps with
+some input, then the reversible machine halts in O(K1+ε). A reversible three-tape
+Turing machine can be simulated by a reversible one-tape Turing machine, such
+simulation is also polynomial in time (in this case the simulation is quadratic see
+e.g. [1, Proposition 1]). We can also assume that the initial state of ˜T, which is
+˜q1, is not in the image of the transition function of ˜T as discussed in [24, Section
+6.1.2]. Let ˜∆ : (Q \ {qhalt} × ΣZ → Q × ΣZ be the global transition function of ˜T,
+and do not define it on halting configurations. It is injective because we assumed
+that ˜T is reversible.
+An application of [23, Lemma 0] implies that φGS (which restricted to AZ is
+equal to φ) is induced by a piecewise area-preserving linear map by blocks of some
+square Cantor set ˜C (associated to an alphabet of | ˜A| symbols). This set of blocks
+never lie in the block determined by fixing the zero symbol of a sequence to be ˜q0,
+since φGS is defined on AZ ⊂ ˜AZ. Each of these blocks is contained in the block
+determined by an element (s−1, s0, s1) ∈ A3 of the domain of dependence. We
+will ignore the block associated with halting configurations (i.e. those for which
+s0 = qhalt) and its image, since once we reach a sequence encoding a halting
+configuration, the computational process if finished. The fact that ˜T is reversible
+implies that the family of blocks B1, ..., BN−1 of ˜C is pairwise disjoint, and the
+family of image blocks B′
+1, ..., B′
+N−1 is also pairwise disjoint. Moore’s lemma also
+provides a linear area-preserving map F : �N−1
+i=1 Bi → �N−1
+i=1 B′
+i that induces
+on ˜C the generalized shift φGS, again ignoring those sequences whose symbol at
+position zero is qhalt. Here we are identifying sequences in ˜AZ with points in ˜C
+via an injective map, computable for compactly supported sequences, of the form
+of (1).
+Exactly as it happened with the blocks Bi, the image blocks B′
+i will not intersect
+either the block determined by those sequences that have at position zero the
+symbol ˜q0. This easily follows from the application that we did of [23, Theorem
+7] and [23, Lemma 0]: we only need to use the blocks determined by words of
+the form (t−1.qt0) ∈ Σ × Q × Σ, whose image by φGS never has a ˜q0 in its zero
+position. We will now extend F to another rectangular domain: we will first define
+this rectangle and its image. The first rectangle, which will be denoted by BN,
+is given by the Cantor block of ˜C corresponding to those sequences that have at
+position zero the symbol ˜q0 (this is a horizontal rectangle in I2). Let ˜B be the
+block given by those sequences that have at position zero the symbol ˜q1, this is
+another horizontal rectangle in I2. No block B′
+i intersects ˜B, since ˜q′
+1 is not in the
+image of the transition function of ˜T, and hence ˜q′
+1 cannot be in the zero position
+of a sequence in the image blocks. Now we define B′
+N as the block which is the
+
+10
+ROBERT CARDONA
+image of ˜B by the translation
+τ(x, y) := (x + α, y),
+where α is the number associate do the left-infinite sequence (a∞.0...) via a map
+like (1). That is, if the alphabet was of two symbols and a∞ = (...s−2s−1.0...),
+then α = �∞
+i=1 s−i 2
+3−i .
+It is clear that B′
+N remains disjoint from B′
+1, ..., B′
+N−1, since we just slightly
+translated the horizontal Cantor block ˜B horizontally, i.e. the part of B′
+N that
+is not in ˜B, lies outside the unit square I2 and hence cannot intersect any other
+Cantor block. We define F|BN to be a composition of the trivial translation of BN
+into B′
+N and τ. Observe that if we start with a sequence (...0.˜q0t0...tn0...) in ˜AZ,
+it is encoded to BN ⊂ ˜C, and applying F it is sent to the point whose associated
+sequence is (a∞.˜q1t0...tn0). In other words, given an input (...0.˜q0t0...tn0...), in
+one step all the advice of the machine is included in the sequence, and then we
+proceed by computing as specified by φGS. Hence F : �N
+i=1 Bi → �N
+i=1 B′
+i induces
+on ˜C a map that simulates T in polynomial time.
+□
+Observe that even if the initial machine T is already reversible, the proof of
+Theorem 5 shows that the simulation is only done in polynomial time, not in real
+time. This is because one needs to preprocess a∞ by introducing the auxiliary
+machine ˜T.
+The map F extends to a compactly supported area-preserving diffeomorphism
+of the disk arguing exactly as in [15, Proposition 5.1].
+Corollary 6. Given a polynomial-time Turing machine with polynomial advice
+(T, a), there exists a compactly supported area-preserving diffeomorphism of a disk
+D that simulates T in polynomial time.
+The simulation is concretely described as follows. The symbolic system φ :
+AZ → AZ (that simulates (T, a) in polynomial time) that is constructed in the
+proof of Theorem 5 is semiconjugate, along the sequences that encode configura-
+tions of (T, a), to the area-preserving diffeomorphism of the disk.
+Regarding computability, the encoding (1) is of course computable on com-
+pactly supported sequences, and so is the map by blocks except along the block
+that simulates the first step of the machine (block BN in the proof of Theorem
+5). This is necessary, since the dynamics applied to that block adds to the com-
+putational process the possibly non-computable advice string.
+4. Hypercomputation in ideal fluid motion
+In this section, we introduce a notion of simulation of Turing machines by
+volume-preserving autonomous flows that admits a well-defined notion of time
+complexity. We show that there exist stationary solutions to the Euler equations
+on compact three-dimensional geometric domains equipped with some Riemann-
+ian metric that are capable of simulating a given Turing machine with advice.
+
+HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION
+11
+4.1. Simulation and complexity with conservative ODEs. In Section 5.2,
+we introduced a computational model that computes beyond Turing machines,
+following [27, 28]. We showed that countable generalized shifts are capable of sim-
+ulating polynomial-time Turing machines with polynomial advice in real-time, i.e.
+a step of the Turing machine corresponds to an iteration of a countable general-
+ized shift. However, if we are interested in showing that a continuous dynamical
+system simulates a polynomial-time Turing machine, we need to specify how to
+measure “time” to compare it to a step of the algorithm of the Turing machine.
+Through this section, every object we consider, like a manifold or a vector field,
+will be assumed to be smooth. Let X be an autonomous (i.e. time-independent)
+vector field on a manifold M (for concreteness, one might think of an ordinary
+differential equation defined on Rn). Given a point p ∈ M the integral curve of X
+through p is the solution to the system
+�
+y′(t) = X(y(t)),
+y(0) = p.
+(2)
+There are several ways to define continuous-time simulation of a Turing machine T
+by a vector field X. One approach, introduced in [31] and followed in [15, 12, 14], is
+to define simulation by requiring that the halting (perhaps with a prescribed finite
+part of the output) of the machine for a given input is equivalent to the integral
+curve of X through a computable point p ∈ M intersecting a computable open
+set U ⊂ M. This definition does not impose a “step-by-step” simulation, as it is
+more customary in previous works [8, 17, 18, 21], although the constructions in
+[31, 15, 12, 14] do in fact provide step-by-step simulations. Inspired by [17, 18], we
+consider the following definition of step-by-step simulation that takes into account
+the behavior of the time parameter.
+Definition 7. Let X be a vector field on a manifold M. Let T be a Turing
+machine a denote by ∆ its global transition function, and by ϕ : P ֒→ M an
+injective encoding of the configurations P of T in M. We say that X simulates T
+if for each initial configuration (q0, s) ∈ P, there is a constant Ks such that the
+solution y(t) to y′(t) = X(y(t)) with initial condition y(0) = ϕ(q0, s) satisfies
+y(Ks.n) = ϕ(∆n(q0, s)),
+(3)
+for each n ≤ N, where N ∈ N ⊔ {+∞} is the halting time of T with input (q0, s).
+With this definition, the values of t that are integer multiples of Ks measure
+the steps of the algorithm. So given one computation (an initial configuration),
+each step of the algorithm is performed in the same amount of continuous time.
+Remark 8. Definition 7 defines real-time simulation, i.e. one step of the ma-
+chine corresponds to one step (measured by time multiples of Ks) of the system.
+Polynomial-time simulation can be defined analogously, by replacing y(Ks.n) by
+y(Ks.Q(n)) in Equation (3) for some polynomially-bounded function Q(n).
+
+12
+ROBERT CARDONA
+We will next see why this definition behaves well for conservative autonomous
+flows.
+Remark 9. We can further require that the positive trajectory y(t) as above
+intersects some open set Us (for example, a ε-neighborhood of the image by ϕ
+of the halting configurations) if and only if T halts with input (q0, s). This is
+interesting because the halting of T with a given input is equivalent to an integral
+curve reaching an explicit open set. In addition, we can require that the orbit y(t)
+either intersects Us or remains at a positive distance (bounded from below) from
+it.
+Remark 10. It is also possible to use an encoding ϕ that assigns an open set to
+each configuration (e.g. as in [14]), and then require that y(Kti.n) lies in the open
+set that encodes ∆n(q0, tin). Definition 7 readily generalizes to robust simulations
+as in [17].
+Given an autonomous vector field simulating a Turing machine as in Definition
+7, one is tempted to use the time-parameter t of an integral curve as a measure
+of time complexity. According to the definition of step-by-step simulation, the
+solution at time t = k corresponds to the kth step of the algorithm. However, as
+classically treated in the literature [6, 7], the parameter t is not a well-defined
+measure of time for a very simple reason. One can rescale the vector field X
+defined in M by considering ˜X = fX for some positive function f ∈ C∞(M), so
+that the orbit can compute faster or slower depending on the step of the algorithm
+that is being simulated. The same happens if we consider a vector field simulating
+a Turing machine as in Definition 7, taking as the time step the values of the
+continuous-time parameter given by integer multiples of Ks. However, if X is
+assumed to preserve some volume form µ ∈ Ωn(M) (where n = dim M), the
+following proposition shows that there is a well-defined notion of time complexity.
+Lemma 11. Let X be a vector field on M, preserving some volume form µ ∈
+Ωn(M), and simulating a Turing machine T as in Definition 7. Let �
+X = f.X, with
+f ∈ C∞(M) a positive function, be a reparametrization of X that also preserves
+µ. Then �
+X also satisfies Definition 7 with the same encoding and constants �Ks =
+Ks
+f(ϕ(q0,s)).
+Proof. Assume that X satisfies Definition 7 with encoding ϕ and constant Ks
+for a given initial configuration (q0, s). Let �
+X = f.X be a reparametrization of
+the vector field X that preserves µ, for some positive function f ∈ C∞(M). This
+implies that L ˜
+Xµ = 0, and is equivalent to
+dιfXµ = df ∧ ιXµ = 0.
+Contracting this equation with X again, we see that a necessary and sufficient
+condition is that ιXdf = 0, i.e. that f is a first integral of X. Let us show that
+˜X satisfies Definition 7 with the same encoding as X. Let (q0, s) be an initial
+
+HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION
+13
+configuration of the machine T. Let ˜y(t) be the integral curve of ˜X with initial
+condition ϕ(q0, s), hence the solution to the system
+�
+˜y′(t) = f.X(˜y(t)),
+˜y(0) = ϕ(q0, s).
+(4)
+Since f is a first integral of X, it is constant along any orbit of X, so we might
+replace f by the constant Cs = f(ϕ(q0, s)) in Equation (4). Let y(t) denote the
+solution to the system (2). It is standard that ˜y(t) satisfies
+˜y(t) = y(Cs.t).
+Consider the constant ˜Ks = Ks
+Cs . Using that X simulates T with constants Ks and
+encoding ϕ, we deduce that
+˜y( ˜Ksn) = y(Ks.n) = ϕ(∆n(q0, s)),
+for each n ≤ N, where N is the halting time of T with input (q0, s).
+□
+It is clear that if a vector field X satisfies the property described in Remark
+9, any reparametrization also satisfies it. The time complexity of a computation
+can be measured in a well-defined way by using the values of t that are integer
+multiples of Ks. From a physical point of view, the value of f along an orbit of
+X measures the norm of X along that orbit. It is reasonable that computations
+along an orbit where X has greater norm (though of as a measure of the “energy”
+of the system along that orbit) occur faster in terms of the continuous measure of
+time t. However, the time complexity measured discretely is invariant under these
+reparametrizations.
+4.2. Stationary Euler flows computing P/poly. Having introduced a well-
+defined notion of time complexity for conservative vector fields, we will prove in
+this subsection that given any polynomial-time Turing machine with polynomial
+advice, there exists a solution to the stationary Euler equations in some compact
+Riemannian three-manifold that simulates it (polynomially in time) according to
+Definition 7.
+The Euler equations model the dynamics of an ideal (incompressible and with-
+out viscosity) fluid on a Riemannian manifold (M, g) where they take the form
+�
+∂tu + ∇u.u = −∇p,
+div u = 0.
+Here u is the velocity field of the fluid, the scalar function p is the pressure
+function, and all the differential operators are defined with the ambient metric g. A
+stationary solution is a solution satisfying ∂tu = 0, it is hence a time-independent
+vector field whose integral curves define the particle paths of the fluid. The second
+equation ensures that u is always a volume-preserving vector field with respect
+to the Riemannian volume. In order to prove that there exist stationary solutions
+that simulate polynomial-time Turing machines with polynomial advice, our main
+
+14
+ROBERT CARDONA
+tool will be the connection between Euler flows and Reeb flows in contact geometry
+established by Etnyre and Ghrist [16]. This connection was used in [15] to prove
+that there exists Turing complete steady Euler flows in three-dimensional compact
+manifolds.
+Let us recall that on a three-dimensional manifold M, a (cooriented) contact
+structure is a plane distribution ξ defined as the kernel of a one-form α ∈ Ω1(M)
+that satisfies the non-integrability condition α∧dα ̸= 0. We call α a contact form,
+and any positive multiple γ = f.α with f ∈ C∞(M) is another contact form
+defining ξ. Each contact form γ uniquely defines a vector field R called the Reeb
+vector field, which is determined by the equations
+�
+γ(R) = 1,
+ιRdγ = 0.
+Reeb fields will play a role in the proof of the following theorem.
+Theorem 12. Let (T, a) be a polynomial-time Turing machine with polynomial
+advice. There exists a metric g in S3 and a stationary solution to the Euler equa-
+tions X in (S3, g) that simulates T polynomially in time.
+The simulation will be according to Definition 7 and Remark 8.
+Proof. Let (T, a) be a polynomial-time Turing machine with polynomial advice.
+By Corollary 6 there exists a compactly supported area-preserving diffeomorphism
+of a disk
+H : D −→ D
+that simulates (T, a) in polynomial-time. Concretely, as done in Theorem 5, there
+exists a symbolic system φ : AZ → AZ and a computable map E : P → AZ
+encoding the configurations of the machine such that φ simulates (T, a) in poly-
+nomial time. Then H satisfies H(e(s)) = e(φ(s)) for every s ∈ E(P) ⊂ AZ, where
+e denotes an encoding as in Equation (1) (perhaps using an expansion in base k
+instead of two) into some square Cantor set ˜C2.
+Fix the contact manifold (S3, ξstd), where S3 is the three-sphere and ξstd the
+standard tight contact structure. By [15, Theorem 3.1], there exists a contact
+form α whose Reeb field R exhibits a Poincar´e disk-like section DM ⊂ M whose
+first-return map is conjugate to H. This means that DM is an embedded disk
+transverse to the flow, and that there exists a smooth function τ : DM → R such
+that the flow of R, that we denote by ϕt : M −→ M, satisfies
+ϕτ(p)(p) = ψ ◦ H ◦ ψ−1(p), for p ∈ DM
+where ψ : D −→ DM is a chart identifying D with the disk-like section DM.
+We will now choose a suitable positive rescaling function h ∈ C∞(M) so that the
+first-return time of the flow of X = h.R at DM is constant and equal to one. First,
+up to multiplying R by a small enough constant, we can assume that τ(p) < 1 for
+all p ∈ DM. Choose flow-box coordinates (x, y, z) of U = {ϕz(D′) | z ∈ [−ε, ε]} ∼=
+D2 ×[−ε, ε], where D′ is a slightly bigger disk-like section containing DM. Denote
+
+HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION
+15
+by F the first-return map on D′, it satisfies F|DM = ϕτ(p)(p). In these coordinates
+R =
+∂
+∂z, and the integral curve with initial condition (x, y, −ε) takes exactly time
+ε to hit DM.
+Construct a smooth function g : D′ × [−ε, ε] → R constantly equal to 1 near
+(D′ × ({−ε, ε}) ∪ (D′ × [0, ε)) and such that
+� 0
+−ε
+1
+1 + g(x, y, z)dz = 1 − τ(F −1(x, y)) + ε, for (x, y) ∈ DM.
+It is clear that such a function exists: for a fixed point (x, y), we are choosing a
+function depending on z with a given integral value. Varying smoothly the value
+of the integral we can smoothly vary the function with respect to z, parametrically
+with respect to two parameters x and y.
+Consider the vector field X = h.R. We claim that the first-return time of X
+to DM is constant and equal to one. Consider p ∈ DM, then the solution to the
+ODE defined by X and initial condition p hits DM × {−ε} at a point (x, y, −ε)
+after time τ(p) − ε, since X = R along the piece of orbit outside of D′ × [−ε, 0).
+In particular, we have (x, y) = F(p). On the other hand, the solution u(t) to the
+ODE defined by X and initial condition (x, y, −ε) satisfies
+t =
+� z
+−ε
+1
+1 + g(x, y, z)dz.
+It follows that the solution intersects DM = {z = 0} when t = 1− τ(F −1(x, y)) =
+1− τ(p). Hence the time that the flow of X takes to send a point p back to DM is
+˜τ(p) = τ(p) − ε + 1 − τ(p) + ε = 1
+as claimed.
+We have thus constructed a reparametrized Reeb field X = h.R that has a
+disk-like Poincar´e section, with first-return time equal to one, and conjugated
+to H. It was proved in [16] that any such vector field is a stationary solution
+to the Euler equations for some Riemannian metric in the ambient manifold. It
+only remains to check that X does simulate the Turing machine T according to
+Definition 7. Given a configuration c ∈ P of the machine, it is mapped to an
+element of AZ by the map E : P −→ AZ. The set of sequences AZ in the image of
+E is injectively mapped to the square Cantor set ˜C2 by the map e. As encoding,
+we choose ˜e = ψ ◦ e ◦ E and as constants we take Ks = 1 for all s. The first-
+return map FX of X at any point p ∈ DM is given by the flow of X at time
+1. Hence, given an initial configuration (q0, s) of the machine T, we consider the
+solution y(t) to the ODE defined by X and initial condition ˜e(q0, s). Using that
+FX = ψ ◦ H ◦ ψ−1(p) and that y(k) = F k
+X(p), we deduce that
+y(Q(n)) = ˜e(∆n(q0, s)),
+
+16
+ROBERT CARDONA
+for each n smaller than the halting time of T with input (q0, s), where Q(n) is a
+polynomially-bounded function that comes from the polynomial-time simulation
+of (T, a) by φ. This concludes the proof that X simulates T according to Definition
+7, polynomially in time as described in Remark 8.
+□
+We point out that, as mentioned in the context of neural networks [27], a given
+computation requires only polynomial time with respect to the size of the input
+and hence is simulated by a finite portion of the associated integral curve. Hence,
+a given computation is robust to perturbations of some size that depends on the
+size of the input. Indeed, only finitely many positions of the sequence need to be
+read to simulate a finite number of iterations of the symbolic system in ˜C2.
+Remark 13. The reparametrization argument used in the proof of Theorem 12
+can be applied to the Turing complete Reeb flows constructed in [15]. This yields a
+stationary solution to the Euler equations in some Riemannian three-sphere that
+simulates a universal Turing machine according both to the definition used in [15]
+and to Definition 7 that takes into account time complexity.
+Theorem 3.1 in [15] can be applied to any fixed closed contact three-manifold
+(M, ξ), or open contact three-manifold such as (R3, ξstd). Furthermore, observe
+that if the machine is taken to be reversible, then no increase of time is required
+for the simulation and the Euler flow simulates in real-time. This corresponds to
+the statement of our main Theorem 1.
+In addition, the strong property described in Remark 9 is also true, namely
+that there exists an open set U such that for any initial configuration (q0, s), the
+orbit of X through the explicit point ps ∈ M associated with (q0, s) intersects U
+if and only if T halts with input (q0, s). The orbit either intersects U or stays at
+a positive distance from U uniformly bounded from below. Even if a polynomial-
+time Turing machine halts in every input, the last property is not meaningless.
+One could, for example, modify the Turing machine so that halting only occurs
+on accepted inputs. Then the previous property would mean that the trajectory
+associated to an input that is accepted intersects some domain U, but if the input
+is not accepted then the associated trajectory remains at a positive distance of
+U. This is seen as follows.
+Let y(t) be the trajectory associated with an initial configuration (q0, s). All
+the points representing a halting configuration are encoded in a finite collection of
+blocks of the square Cantor set contained in DM, and no non-halting configuration
+is encoded there. Let UD be a small enough neighborhood of those blocks not
+intersecting any other block, and U be defined as
+U = {φX
+t (p) | p ∈ UD, t ∈ (−ε, ε)},
+where ϕX
+t : M −→ M denotes the flow defined by X. It is clear that y(t) intersects
+U if and only if T halts with input (q0, s). In particular, since H(e(p)) = e(φ(p))
+is satisfied for all the points encoding configurations of the machine (T, a), the
+
+HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION
+17
+orbit of the initial configuration by the first-return map will always remain in the
+blocks with non-halting configurations. These blocks are distance greater than δ,
+for some δ > 0, of the halting configuration blocks. This shows that the orbit
+of a non-halting initial configuration stays at a positive distance (bounded from
+above) of U.
+4.3. Variations of the model. We end up this section by discussing other mod-
+els, either of complexity or of ideal fluids, that could be considered in this context.
+Time complexity via orbit lenght. In [6], another possible measure of the
+time complexity of simulations with ODEs was proposed. When there is a metric
+in the ambient space (for example, the Euclidean metric for an ODE defined in
+Rm), a measure of time that is invariant with respect to reparametrizations is the
+length of the orbit with respect to the Riemannian metric. This approach is also
+reasonable in the context of hydrodynamics, since the space is endowed with a
+natural Riemannian metric, the one for which the flow solves the Euler equations.
+This point of view can as well be taken in our construction in Theorem 12. Indeed,
+the vector field X has no zeroes, and the ambient manifold is compact, hence there
+are c, C ∈ R constants such that
+c < g(X, X) < C,
+(5)
+where g denotes the metric for which X is a stationary solution to the Euler
+equations. In particular, the length of an injective piece of an integral curve grows
+linearly with time. For the flow X, the computational steps are given by Ks = 1
+(as in Definition 7). By hypothesis, given an input of size n the machine halts
+after P(n) steps (where P is a polynomially-bounded function and n the size of
+the input), the integral curve simulates the process in time P(n). By Equation
+(5), the length of the curve up to time P(n) is polynomial as well, so polynomial
+complexity is well-defined using the approach proposed in [6]. Note that it is also
+possible to construct a metric ˜g for which X is a stationary solution to the Euler
+equations as well, and X has constant norm equal to one. In that case, the length
+coincides with the time. The construction of ˜g is done using the arguments ex-
+plained in [11, Section 1.3.4 page 85], by considering a one-form α (which is not
+anymore of contact type everywhere, but is instead closed in the solid torus) such
+that α(X) = 1 in the solid torus.
+Other hydrodynamical systems. Ideal fluid flows capable of universal com-
+putation have been constructed in other situations, besides stationary flows on
+geometric three-dimensional domains endowed with an adapted (not fixed a pri-
+ori) Riemannian metric. Indeed, a natural requirement is to impose that the metric
+is a fixed natural one, such as the flat metric on the three-torus or the Euclidean
+metric in R3. In [14], it was shown that at the high cost of losing compactness, one
+can construct stationary solutions to the Euler equation in R3 with the Euclidean
+
+18
+ROBERT CARDONA
+metric that can simulate a universal Turing machine. One can check that the sim-
+ulation is not as good as the one defined in Definition 7, because the simulation
+has an exponential slow-down in terms of the continuous-time parameter of the
+ODE. Even if we use the orbit-length approach to time complexity, one cannot
+simulate polynomial-time Turing machines in polynomial time using the construc-
+tion done in [14]. A natural question is then whether there is another construction
+of stationary solutions to the Euler equations in Euclidean space that simulate
+either in real-time or in polynomial time any Turing machine (with or without
+advice), according to some natural definition of simulation and time complexity.
+Similarly, it was proved in [12] that there are time-dependent solutions to the
+Euler equations in some high enough dimensional closed manifold that are capa-
+ble of simulating a universal Turing machine. Those solutions not only have an
+exponential slow-down as well but also rely on constructions of polynomial ODEs
+[18] that simulate any Turing machine which might not hold for Turing machines
+with advice. Hence another question is if there are time-dependent solutions to the
+Euler equations modeling hypercomputation. Perhaps this can be shown as well
+by doing a construction that can use the embedding results in [33, 32], as done
+in [12]. The question of whether viscous fluids, as modeled by the Navier-Stokes
+equation, can simulate a universal Turing remains open in any possible context
+[15]. Of course, the same question can be asked about the class P/poly.
+5. Countable generalized shifts and P/poly
+In this section, we introduce a class of symbolic dynamical systems that contains
+in particular generalized shifts. As we will see, our generalization is different from
+the analog shift map.
+5.1. Countable generalized shifts. The broader class of symbolic systems that
+we introduce in this section should be thought of as a countable version of gen-
+eralized shifts. Instead of changing the sequence according to a finite portion of
+it of fixed size (like DF and DG), we change it according to a finite portion of a
+variable but arbitrarily large size. This is a different generalization than analog
+shift maps, where only a finite portion of fixed size determines the image of the
+sequence, but infinitely many symbols can be changed in one step.
+A countable generalized shift φ : AZ → AZ is defined by the following informa-
+tion:
+(1) a set P of pairs {(nj, Ij) ∈ Z × Amj}, with j ∈ {0, ..., N} or j ∈ N, such
+that for each s ∈ AZ there is at most one j such that snj...snj+mj−1 = Ij,
+(2) a map J assigning to each element of p = (nj, Ij) ∈ P a word J(p) = I′
+j ∈
+Amj,
+(3) a map H : P → Z.
+
+HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION
+19
+We denote by SP ⊂ AZ the set of sequences s ∈ AZ such that there is some
+(nj, Ij) ∈ P such that snj...snj+mj−1 = Ij.
+The dynamical system is described as follows. Given some s ∈ AZ, if s ̸∈ SP
+then φ(s) := s. Otherwise, let p = (nj, Ij) be the only pair assigned to s. The
+sequence φ(s) is obtained by changing the symbols in positions nj, ..., nj + mj − 1
+by J(p), and then shifting by H(p).
+Notation. To simplify notation, given a pair (nj, Ij) we say that the symbols of
+the word Ij are at positions nj...nj + mj − 1 of Ij (instead of positions 1, .., mj).
+Furthermore, given a pair p ∈ P we say that a sequence s ∈ AZ coincides with
+p ∈ P (or with Ij) if
+snj...snj+mj−1 = Ij.
+(6)
+Similarly, if we denote a finite word with indices w = wnwn+1...wm−1wm with
+n, m ∈ Z, we say that s coincides with w if
+sn...sm = wn...wm,
+(7)
+where the left hand side denotes the symbols in position n, n + 1, ..., m of s.
+It is easy to see that a generalized shift is, in particular, a countable generalized
+shift.
+Lemma 14. Any generalized shift is a countable generalized shift with a finite set
+P.
+Proof. Let φ : AZ −→ AZ be a generalized shift. Without loss of generality, we can
+assume that DF = DG, simply by taking the union of both domains and redefining
+F and G appropriately. Hence φ is defined by F : Al → Z and G : Al → Al, where
+DF = DG = {i, ..., i + l − 1}. Let us define a countable generalized shift that is
+equal to φ. As space of pairs, we choose P = {(i, (t1...tl)) | (t1, ..., tl) ∈ Al}. Now
+we define J((i, (t1...tl))) = G(t1...tl) and H((i, (t1...tl)) = F(t1...tl). We obtain a
+countable generalized shift ψ such that ψ(s) = φ(s) for each s ∈ AZ.
+□
+An interesting property of the set SP ⊂ AZ is that it can never be equal to AZ.
+The diagonal argument used in the proof of this lemma will be useful throughout
+the paper.
+Lemma 15. There is no countable generalized shift satisfying that P is not a
+finite set and SP = AZ.
+Proof. Let φ be a countable generalized shift such that P is not a finite set. Then
+there is an infinite sequence pik = (nik, Iik) ∈ P, with Iik ∈ A⋗ℶℸ such that |mik|
+or |nik| go to infinity as k goes to infinity. To simplify, assume that we found a
+family Iik such that mik → ∞, an analogous argument works for the other cases.
+Choose a family of sequences sk coinciding with each pik. Endow AZ with the
+
+20
+ROBERT CARDONA
+metric
+d(t, t′) =
+k
+�
+i=0
+(2N)−k(|tk − t′
+k| + |t−k − t′
+−k|),
+where N is the cardinality of A. Then AZ is compact with this metric and the
+sequence sk admits a convergent subsequence skr such that skr → ˜s as kr → ∞.
+We will show that ˜s ̸∈ SP. Indeed, assume that there is some ˜p = (˜n, ˜I) ∈ P (with
+˜I of size ˜m) such that ˜s coincides with ˜p. Choose some M such that |˜n| < M
+and | ˜m| < M. Since skr → ˜s, there is some K0 such that |skr − ˜s| < N −M for
+every kr > K0. This implies that skr and ˜s are equal for symbols in the positions
+−M, ..., M, and hence skr coincides with ˜p for every kr > K0. We deduce that
+skr coincides both with ˜p and pkr. For a big enough kr the element pik is such
+that mik > M, and hence ˜p ̸= pkr. This is a contradiction with the definition of
+a countable generalized shift: every sequence coincides with at most one element
+in P.
+□
+It is possible to characterize, although with a property that is difficult to verify
+for a given example, those countable generalized shifts that are generalized shifts
+too.
+Lemma 16. Let φ be a countable generalized shift, it is a generalized shift if
+and only if there is some N ∈ N for which we can associate to each word w =
+w−N...wN ∈ A2N+1 an integer kw ∈ Z and a word w′ = w′
+−N...w′
+N satisfying
+the following conditions. Given s ∈ AZ, whose symbols at positions −N, .., N
+determines a unique word ws, we have:
+- If s ∈ SP coincides with some p = (nj, Ij), then H(p) = kws and J(p) is
+such that for every position r ∈ {−N, ..., N}, either r ∈ {nj, ..., nj+mj−1}
+and J(p)r = (ws)r or r ̸∈ {nj, ..., nj + mj − 1} and then (w′
+s)r = (ws)r.
+- If s ̸∈ SP , then changing the symbols of s at positions −N, ..., N by w′
+s
+and shifting by kws recovers s.
+Proof. Let φ : AZ −→ AZ be a countable generalized satisfying the property in
+the statement. Define a generalized shift ˆφ : AZ −→ AZ by taking DF = DG =
+{−N, ..., N} and functions
+F(w) = kw, for each w ∈ A2N+1
+and
+G(w) = w′, for each w ∈ A2N+1.
+It follows that if s ∈ SP then it follows from the first item above that ˆφ(s) = φ(s).
+Furthermore if s ̸∈ SP then φ(s) = s, but by the second item above ˆφ(s) = s too.
+Conversely, let φ : AZ −→ AZ be a countable generalized such that there is
+some generalized shift ˜φ : AZ −→ AZ, defined by functions
+F ′ : Ab−a → Z,
+
+HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION
+21
+where DF ′ = {a, ..., b} and
+G′ : Ad−c → Ad−c,
+where DG′ = {c, ..., d} and such that φ(s) = ˜φ(s) for every s ∈ AZ.
+First, to simplify, we can easily construct another generalized shift ˆφ : AZ −→
+AZ with associated functions F, G such that DF , DG = {−N, ..., N} for some N
+and ˆφ(s) = ˜φ(s) for every s ∈ AZ. To do so, let N denote the greatest absolute
+value of the elements of DF ′ and DG′. To define F, given a word w−N...wN ∈
+A2N+1, we define F(w) := F ′(wa...wb). To define G, given a word w−N...wN if
+G′(wc, ..., wd) = w′
+c...w′
+d then we define
+G(w−N...wN) := w−N...wc−1w′
+c...w′
+awa+1...wN.
+Let us now show that for this N the claimed property is satisfied. Fix any word
+w = w−N...wN, and denote F(w) and G(w) by kw ∈ Z and ω′ = w′
+−N...w′
+N
+respectively. Given any sequence s that coincides with w, since φ(s) = ˆφ(s), the
+image of s is obtained by replacing the symbols at positions −N, ..., N by w′ and
+shifting by kw. If s ̸∈ SP, then φ(s) = s and item two above is satisfied. If s ∈ SP ,
+then there is some p = (nj, Ij) such that s coincides with p. If Ij = s′
+nj...s′
+nj+mj−1,
+let s′ be the sequence
+s′ = ...0w−N...wnj−1s′
+nj...s′
+nj+mj−1wnj+mj...wN10...
+which also coincides with p. Now the image of s′ can be computed either by φ
+or by ˆφ. By looking at the one placed at the last position of s′ which is not
+zero, we deduce that necessarily H(p) = kw. This immediately implies that the
+sequence obtained from s′ either by changing symbols at position −N, ..., N by
+w′, or symbols at position nj, ..., nj +mj −1 by J(p) are the same. In other words,
+J(p) and w′ are the same in the positions they have in common, and w′ coincides
+with w at those positions which are not in {nj, ..., nj + mj − 1}. This finishes the
+proof of the lemma.
+□
+Let us give a sufficient criterion that is easy to check in practice to determine
+when a countable generalized shift is not a generalized shift.
+Definition 17. We asy that a countable generalized shift φ “modifies at infinity”
+if there is a sequence of numbers |rik| → ∞ with rik ∈ {nik, ..., nik + mik − 1}
+for some pk = (nik, Iik) ∈ P, such that the symbol at position rik of Iik does not
+coincide with the symbol at position rik of J(pk).
+Lemma 18. If a countable generalized shift φ modifies at infinity, then φ|SP is
+not induced by a generalized shift.
+Proof. Let φ be a countable GS defined by P, J and H, and assume that it modifies
+at infinity.
+There is a sequence of numbers |rik| → ∞ with rik ∈ {nik, ..., nik + mik − 1} for
+some pk = (nik, Iik) ∈ P satisfying Definition 17. By Lemma 3, there is always a
+shifted version of φ(s) that coincides with s except maybe along positions in DG.
+
+22
+ROBERT CARDONA
+We will prove that this is not the case under our hypotheses. Assume that φ is
+a generalized shift given by P, J and H. For a fixed k, let sk be a sequence that
+has zeroes everywhere except in positions nik, ..., nik + mik − 1 where it coincides
+with Iik = anik ...anik +mik−1, and also has a 1 at position nik + mik, i.e sk is of
+the form
+sk = ...0anik ...ak
+nik +mik −110...
+The image φ(sk) is obtained by changing the symbols in position nik, ..., nik +
+mik − 1 and shifting by H(pk).
+Let us show that for any integer r, the sequence sk and the sequence φ(sk)
+do not coincide in some element in position greater or equal to rik. To see this,
+given an arbitrary r let φ(sk)r be the r-shifted sequence of φ(sk). If r > 0 (i.e.
+left shift), then the symbol in position nik + mik > rik of sk (which is a one) does
+not coincide with that of φ(sk)r, which is a zero. If r < 0 (right shift) then the
+symbol in position nik +mik +r > rik (which is a zero) does not coincide with the
+symbol of φ(sk)r in that position (which is a one). Finally when r = 0, we have
+that the symbol in position rik does not coincide with that of φ(sk) by hypothesis.
+We conclude that some symbol at a position greater or equal than rik of sk and
+of any shifted version of φ(sk) are not equal. Since rik is arbitrarily large choosing
+an arbitrarily large k, this gives a contradiction Lemma 3. We conclude that φ is
+not a generalized shift.
+□
+Lemma 18 can be used to easily construct examples of countable generalized
+shifts that are not generalized shifts, even bijective ones. We shall call a countable
+generalized shift that is not a generalized shift an “infinite generalized shift”. We
+will refer to a countable generalized shift that is a generalized shift as a “finite”
+generalized shift.
+Remark 19. As for generalized shifts [23, Lemma 1], any countable generalized
+shift is conjugate (perhaps injectively semi-conjugate depending on the cardinality
+of the alphabet) to another one whose alphabet is Σ = {0, 1}. This is done by
+identifying the symbols of the alphabet with large enough blocks of zeroes and
+ones.
+5.2. Computational power of countable generalized shifts. In [23], Moore
+showed that generalized shifts are equivalent to Turing machines, both from a
+dynamical and computational point of view. In this section, we analyze the com-
+putational power of countable generalized shifts and show that they can simu-
+late Turing machines with advice. As done in previous sections, we restrict to
+polynomial-time Turing machines with polynomial advice, which define the com-
+plexity class P/poly.
+Technical assumptions on Turing machines with advice. Given a Turing
+machine with advice (T, a), we will assume that the first symbol of each advice
+string is always a zero, and we will only consider inputs tin of size n such that
+
+HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION
+23
+ti ̸= 0 for i = 0, .., n. It is clear that this does not restrict the computational power
+of the resulting polynomial-time Turing machines with polynomial advice. When
+the Turing machine T that we consider is assumed to be reversible, we will assume
+as well that q0 is not in the image of the transition function δ. This can easily be
+assumed, as discussed in previous sections, even if we restrict to reversible Turing
+machines [24, Section 6.1.2]. The latter assumption ensures as well that if we take
+any Turing machine that is reversible, then adding advice to it keeps the global
+transition function injective.
+Let us show how to simulate polynomial-time Turing machines with polynomial
+advice using countable generalized shifts.
+Theorem 20. Let (T, a) be a polynomial-time Turing machine with polynomial
+advice a = {an}n∈N. Then there is a countable generalized shift φ with some
+alphabet A and an injective map ϕ : P −→ SP ⊂ AZ such that ∆ = ϕ−1 ◦φ|SP ◦ϕ.
+If ∆ is injective, then we can assume that φ|SP is injective in all SP.
+Proof. Let (T, a) be a polynomial-time Turing machine with polynomial advice.
+Take the alphabet of the countable generalized shift to be A = Q∪Σ∪{d}, where
+d is a symbol disjoint from Q and Σ. Let ϕ be the encoding function, which maps
+injectively the configurations of T to sequences in AZ, defined as follows.
+ϕ : P −→ AZ
+(8)
+(q0, tin = (...0.t0...tn0...)) �−→ (...0d.q0t0...tnd0...), ti ̸= 0, n ≥ 0
+(9)
+(q, (ti)) �−→ (...t−1.qt0...) otherwise.
+(10)
+Let us define φ in terms of the space of pairs P and the maps J and H. The space
+of pairs P is defined by P1 ⊔ P2, where P1 is infinite and given by
+P1 = {(−1, (dq0t0...tnd 0...0
+����
+p(n)-1
+)) | n ∈ N and ti ̸= 0 for i = 0, ..., n},
+where p(n) is the polynomially-bounded function assigning to an input of size n
+its advice of size p(n). The second set of pairs P2 is given by
+P2 = {(−1, (t−1qt0) | (t−1qt0) ∈ Σ × Q × Σ}.
+We define then the map J on P1 as
+J((−1, (dq0t0...tnd 0...0
+����
+p(n)-1
+)) = (0q0t0...tnan
+1...an
+p(n)),
+where an
+1...an
+p(n) is the advice string assigned to inputs of size n. To define it on
+P2, for any (q, t0) let δ(q, t0) = (q′, t′, ε).
+J((−1, (t−1qt0)) =
+
+
+
+
+
+(t−1t′q′) if ε = +1
+(q′t−1t′) if ε = −1
+(t−1q′t′) if ε = 0
+
+24
+ROBERT CARDONA
+Finally, define H as H(x) = 0 for each x ∈ P1 and H((−1, (t−1qt0)) = ε. Observe
+that a given sequence s ∈ AZ coincides with at most one element in P = P1 ∪ P2,
+so P can be used to define a countable generalized shift. The countable general-
+ized shift φ is such that ∆ = ϕ−1 ◦ φSP ◦ ϕ.
+Assume that ∆ is injective, and by contradiction assume further that φ|SP
+is not injective. Then there are two sequences s, t ∈ SP such that s ̸= t and
+φ(s) = φ(t). The countable generalized shift φ changes at most a finite number
+of symbols of each sequence, which is then shifted by at most 1 position. Assume
+to simplify that both sequences are not shifted (an analogous argument works if
+they are shifted in any direction). Take the two unique pairs p1 = (−1, I1) and
+p2 = (−1, I2) such that s, t coincide respectively with p1 and p2. Then φ(s) is equal
+to s except maybe at those symbols in positions D1 = {−1, ..., m1 − 2} and φ(t)
+coincides with t except maybe at those symbols in position D2 = {−1, ..., m2 −2}.
+In particular, for each k ∈ D1 \ D2 we deduce that tk is equal to the symbol in
+position k of J(p1), and for each r ∈ D2 \ D1 we deduce that sr is equal to the
+symbol in position r of J(p2). For each position j ∈ D1 ∩ D2, we must have that
+the symbol at position j of J(p1) is equal to the symbol in position j of J(I2).
+Consider the sequence s′ defined by
+s′
+i =
+
+
+
+
+
+0 if i ̸∈ D1 ∪ D2,
+si if i ∈ D1
+J(p2)i if i ∈ D2 \ D1,
+(11)
+and the sequence t′ defined by
+t′
+i =
+
+
+
+
+
+0 if i ̸∈ D1 ∪ D2,
+ti if i ∈ D2
+J(p1)i if i ∈ D1 \ D2
+(12)
+Our previous discussion shows that φ(s′) = φ(t′). On the other hand, we know
+that
+- s and t are equal in any position away from D1 ∪ D2,
+- s and t are equal to s′ and t′ respectively in positions D1 ∪ D2,
+- s ̸= t,
+so we deduce that we must have s′ ̸= t′.
+Using the description of φ and the fact that ∆ is injective, we will reach a
+contradiction. Let us analyze case by case depending on p1 and P2. If p1, p2 ∈
+P2, then D1 = D2 = {−1, 0, 1} and we deduce that s′ = ...0s−1.q0s10... and
+t′ = ...0t−1.q0t10.... It follows that s′, t′ ∈ ϕ(P) as per equation (10), which is a
+contradiction with the fact that ∆ is injective. If p1 ∈ P1 and p2 ∈ P2, then
+s′ = ...0d.q0s0...sjd0...
+t′ = ...0t−1.qt1... with t−1, t1 ∈ Σ.
+
+HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION
+25
+The sequence φ(s′) has a q0 in the zero position, while φ(t′) has a q ̸= q0 in the
+zero position (since we assumed that q0 is not in the image of δ). We reached a
+contradiction.
+The last case is when p1, p2 ∈ P1, then p1 = (−1, I1) and p2 = (−1, I2) with
+I1 = (dq0s0...sm1d0...0) and I2 = (dq0t0...tm20...0). Assume that m1 ≥ m2 and
+then s′ and t′ are of the form
+s′ = ...0d.q0s0...sjd 0...0
+����
+p(j)-1
+0...,
+and t′ is of the form
+t′ = ...0d.q0t0...trd 0...0
+����
+p(r)-1
+sp(r)+r+2...ska1...ap(j)0...
+if r + 2 + p(r) ≤ j
+or
+t′ = ...0d.q0t0...trd 0...0
+����
+p(r)-1
+ar+2+p(r)...ap(j)0...
+if r + 2 + p(r) > j,
+where the input (...0.s0...sj0...) has an advice (a1...ap(j)) and the input (...0.t0...tr0...)
+has an advice (b1...bp(r)). Then φ(s′) = φ(t′) implies that (...0.q0s0...sja1...ap(j)0...)
+is equal to
+(...0.q0t0...trb1...bp(r)ap(r+1)...ap(j)0...),
+if p(r) ≥ j, or to
+(...0.q0t0...trb1...bp(r)sp(r)+1...sja0....ap(j)0...),
+if p(r) < j. If m1 = m2 then we deduce that t′ = s′ which is a contradiction. In
+general, we deduce that ti = si for each i = 1, ..., r, and that sr+1 = b1. However,
+by our technical assumptions, we know that b1 = 0 and that sr+1 ̸= 0, hence
+finishing the proof of the theorem.
+□
+The countable generalized shift constructed in the proof of Theorem 20 is an
+infinite generalized shift. Indeed, the description of J on P1 implies that the
+countable generalized shift modifies at infinity, so by Lemma 18 it is an infinite
+generalized shift. The simulation by a countable generalized shift of a polynomial-
+time Turing machine is done in real-time: a step of the countable generalized shift
+corresponds to a step of the machine. This is an advantage with respect to the
+simulation via analog shifts [27], where the first step is necessarily simulated in
+polynomial time with respect to the size of the input.
+6. Area-preserving homeomorphisms of the disk
+In this last section, we show that some countable generalized shifts can be
+embedded, at least partially, in the evolution of a compactly supported area-
+preserving homeomorphism of a disk. This will be used to prove Theorem 2.
+
+26
+ROBERT CARDONA
+6.1. Cantor set and map by blocks. From now on, we will make the sim-
+plifying assumption that a countable generalized shift is defined on the alphabet
+A = {0, 1}, see Remark 19. As done in previous sections we identify sequences
+s = (...s−1.s0s1...) in {0, 1}Z with points C2 via the bijection introduced in Equa-
+tion (1). The following lemma shows that countable generalized shifts are induced
+by countably piecewise linear maps of blocks of C2, just as for standard generalized
+shift [23, Lemma 0].
+Lemma 21. Given a countable generalized shift φ, there exists a piecewise linear
+and area-preserving map f defined over a countable set of blocks into another
+countable set of blocks of the square Cantor set such that φ|SP = e−1 ◦ f ◦ e. The
+following are equivalent:
+- φ|SP : SP → AZ is injective,
+- the image blocks are disjoint.
+Proof. Each element of pj = (j, Ij) ∈ P determines a block Bj of the square Cantor
+determined by all those sequences (si) such that snj...snj+mj−1 coincides with Ij.
+Each block Bj is first translated into the block determined by J(pj), then Baker’s
+map is applied H(pj) times. The block Bj might be cut into rj ≤ 2|H(pj)| connected
+pieces when applying the Baker’s map (or its inverse if H(pj) is negative) is applied
+|H(pj)| times. Let Bk
+j denote the preimages of those pieces. Then define the map
+f :
+�
+j∈N
+rj
+�
+k=0
+Bk
+j → I2
+which coincides in each block Bk
+j with the translation and iteration of Baker’s
+map associated to Bj. Clearly f corresponds to Φ when applied to a point of
+the Cantor set. Observe that two image blocks intersect if and only if there is a
+point of the Cantor set in both blocks, which happens if and only if f|e(SP ) is not
+injective, which happens if and only if φ|SP is not injective.
+□
+It follows from the construction that if there is a finite word w = (wn, ..., wn+m)
+such that any sequence s ∈ AZ that coincides with w is not in SP, then we can
+as well define f to be the identity in the block defined by w. Hence the piece-wise
+map f and the conjugacy with φ can be extended for other sequences that are not
+in SP. The only problem, if we want a piece-wise map defined on blocks, arises
+with those sequences that are obtained as the limit of a family of words obtained
+from pairs pj = (nj, Ij) ∈ P with a size that tends to infinity.
+6.2. Extension to area-preserving homeomorphism. In the case that φ|SP
+is injective, the image blocks are disjoint and we can extend the map to an
+area-preserving homeomorphism of a disk containing the unit square. This fact is
+proven by generalizing [15, Proposition 5.1] to the case of countably many pairwise
+disjoint blocks. Taking a non-finite family of disks will imply a loss of regularity
+of the homeomorphism, which is only continuous.
+
+HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION
+27
+Proposition 22. Let φ be a countable generalized shift such that φ|SP is in-
+jective. Then there exists a compactly supported area-preserving homeomorphism
+H : D −→ D of some disk D strictly containing the unit square such that
+H(e(s)) = e(φ(s)) for every s ∈ SP.
+Proof. Let φ be a countable generalized shift such that φ|SP is injective. By Lemma
+21, there exists two countable families of pairwise disjoint blocks �∞
+i=1 Bi and
+�∞
+i=1 B′
+i and an area-preserving piece-wise linear bijective map
+F :
+∞
+�
+i=1
+Bi −→
+∞
+�
+i=1
+B′
+i,
+such that F(Bi) = B′
+i and e ◦ φSp = F ◦ e. Let D be a big enough disk containing
+in its interior the unit square, for example a disk of radius 2 centered at (1/2, 0)
+is enough. We will construct an area-preserving continuous isotopy ϕt : D −→ D,
+compactly supported, and such that ϕ1|�∞
+i=1 Bi = F. Since the area of Bi and B′
+i
+are the same, we can choose two small closed smooth neighborhoods Di and D′
+i
+of Bi and B′
+i respectively, both diffeomorphic to a closed disk and with the same
+area. Here the area refers to the integral of the standard area-form ωstd = dx ∧ dy
+along that disk. Notice that the distance between any two disjoint blocks of the
+square Cantor set is always at positive, so we can assume all the Di are pairwise
+at a positive distance, and that all the D′
+i are pairwise at a positive distance. The
+linear area-preserving bijective map F|�∞
+i=1 Bi : Bi → �∞
+i=1 B′
+i can be naturally
+extended to an area-preserving diffeomorphism F : �∞
+i=1 Di → �∞
+i=1 D′
+i satisfying
+F|�∞
+i=1 Bi ≡ F, for example by taking the same linear map.
+The set A = D \
+�
+⊔∞
+i=1 Di ∪ D′
+i
+�
+can be assumed to have a non-empty interior,
+since the blocks are all contained in I2, and D contains I2 in its interior. Since D
+is big enough, we can find a countable family of pairwise disjoint disks �Di such
+that �Di ⊂ D \ A and area( ˜Di) = area(Di). Starting with D1 and ˜D1, choose an
+embedded arc τ1 such that
+- τ(0) ∈ D1,
+- τ(1) ∈ �D1,
+- τ1(I) ∩
+� �∞
+i=1 Di ∪ �Di
+�
+⊂ D1 ∪ �
+D1.
+This is possible because D \
+� �∞
+i=1 Di ∪ �Di
+�
+is path-connected. Let �U1 be a small
+enough connected set (diffeomorphic to a disk) containing D1, �D1 and τ1, and
+such that �U1 ∩
+� �∞
+i=1 Di ∪ �Di
+�
+⊂ D1 ∪ �D1. Since D \
+�� �∞
+i=1 Di ∪ �Di
+�
+∪ �U1
+�
+is
+still path-connected, we can find similarly a set �U2 containing D2, �D2 and a path
+τ2 connecting them and that only intersects those two disks in the family Di, �Di.
+Inductively, we find a family of embedded closed disks �Ui ⊂ D that are disjoint
+and contain Di and �Di. We easily find for each �Ui a compactly supported family
+of embeddings ei
+t : Di → �Ui such that ei
+0(Di) = Di and ei
+1(Di) = �Di, for example
+by compressing Di into a small enough disk, moving it to the center of �Di and
+
+28
+ROBERT CARDONA
+undoing the compression1. By the isotopy extension property, we find for each �Ui
+a compactly supported isotopy ξi
+t : �Ui → �Ui such that ξi
+1|Di is a translation with
+image �Di.
+Using the relative Moser’s path method, exactly as in [15, Proof of Proposition
+5.1], we obtain from ξi
+t an area-preserving isotopy ηi
+t such that ηi
+1|Di is a translation
+with image �Di.
+Define the isotopy ϕt : D → D for t ∈ [0, 1/2] as follows.
+ϕt(p) =
+�
+ηi
+2t(p) if p ∈ �Ui,
+p otherwise,
+t ∈ [0, 1/2].
+This is a well-defined continuous area-preserving isotopy of D, since the isotopies
+ηi
+t restrict as the identity near the boundary of �Ui and each of them preserves
+the measure induced by the standard area form. However, regularity is lost by
+the extension in the complement of the countably many domains �Ui. Indeed, the
+derivatives might not be continuous at any point where the family Bi accumulates.
+We argue now similarly with the family of disks { �Di | i = 1, ...} and {D′
+i | i =
+1, ...}, which are pairwise at positive distance. Let γ1 : I −→ D be an embedded
+arc such that
+- γ1(0) ∈ �D1,
+- γ1(1) ∈ D′
+1,
+- γ1(I) ∩
+� �∞
+i=1 �Di ∪ D′
+i
+�
+⊂ �D1 ∪ D′
+1.
+Choose an open set U1 containing D1, D′
+1 and a small enough neighborhood of γ1,
+such that D \
+�
+U1 ∪
+� �∞
+i=1 �Di ∪ D′
+i
+��
+is path-connected. We will now construct a
+compactly supported isotopy of U1
+G1
+t : U1 → U1, t ∈ [0, 1]
+such that G1
+1|D1 is area-preserving, G1
+0| �
+D1 = �D1 and G1
+1| �
+D1 = D′
+1. Actually we will
+impose as well that G1
+1 ◦ ϕ1/2|Di = F|Di, hence prescribing the homeomorphism
+between �D1 and D′
+1.
+To do so, choose small enough disks d1 ⊂ �D1 and d′
+1 ⊂ D′
+1 obtained by applying
+contracting homotheties hs, h′
+s centered respectively at the center of �D1 and D′
+1.
+Define a parametric family of embeddings et : �D1 → U1, t ∈ [0, 4] as follows
+(1) For t ∈ [0, 1/4], we shrink �D1 into d1 using hs.
+(2) For t ∈ [1/4, 1/2], we isotope d1 through the open neighborhood of γ1 up
+to a small disk ˜d1 centered at the center of D′
+1 obtained by a translation
+of d1. That is e1/2 corresponds to a homothety and a translation.
+(3) For t ∈ [1/2, 3/4] we apply an isotopy that first transforms ˜d1 to d′
+1 in a
+way that e3/4 = (h′
+1)−1F ◦ ϕ−1
+1/2,
+1To simplify, we could have chosen each �Di to be a translation of Di.
+
+HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION
+29
+(4) For t ∈ [3/4, 1] we expand d′
+1 to D′
+1 by using the inverse homothety (h′
+s)−1,
+hence e1 = F ◦ ϕ−1
+1/2.
+The family et extends, by the isotopy extension theorem, to a compactly supported
+isotopy G1
+t of U1 that satisfies the claimed properties.
+We can iterate this process for each pair of disks �Di, D′
+i. Indeed, D \
+�
+U1 ∪
+� �∞
+i=1 �Di ∪ D′
+i
+��
+is path-connected, so we can find an arc γ2 connecting �D2 and
+D′
+2 as before, and a connected open set U2 ⊂ D \ U1 containing �D2 and D′
+2 such
+that D \
+�
+U1 ∪ U2 ∪
+� �∞
+i=1 �Di ∪ D′
+i
+��
+is connected. We obtain a countable number
+of disjoint embedded disks Ui and compactly supported isotopies Gi
+t : Ui → Ui.
+Arguing as before, these can be assumed to be area-preserving using the relative
+Moser’s path method while still satisfying Gi
+1 ◦ ϕ1/2|Di = F Di. We extend ϕt for
+t ∈ [1/2, 1] as
+ϕt(x) =
+�
+G2(t−1/2)(p) if p ∈ Ui,
+p otherwise,
+The homeomorphism H = ϕ1 is a area-preserving and satisfies H|Bi = F|Bi for
+each i, since H|Di = ϕ1|Di = F|Di. Thus H(e(s)) = e(φ(s)) follows from the fact
+that e(SP ) ⊂ �∞
+i=1 Bi, which is satisfied by the construction of the map by blocks
+in Lemma 21.
+□
+Remark 23. From the previous proof, one extracts the following general state-
+ment. Let Di and D′
+i be two families of countably many embedded disks on a
+disk D (or any connected surface), such that in each family the elements are pair-
+wise at positive distance and D \ (�∞
+i=1 Di ∪ D′
+i) has non-empty interior. Assume
+that area(Di) = area(D′
+i) and that we are given an area-preserving diffeomor-
+phism Fi : Di → D′
+i. Then there exists a compactly supported area-preserving
+homeomorphism H of D such that H|Di ≡ Fi.
+Finally, Theorem 2 stated in the introduction is obtained by combining Theo-
+rem 20 and Proposition 22.
+References
+[1] H. B. Axelsen. Time complexity of tape reduction for reversible Turing machines. Interna-
+tional Workshop on Reversible Computation. Springer, Berlin, Heidelberg, 2011.
+[2] C. H. Bennett. Logical reversibility of computation. IBM Journal of Research and Develop-
+ment 17.6 (1973): 525-532.
+[3] O. Bournez. How much can analog and hybrid systems be proved (super-) Turing. Applied
+Mathematics and Computation 178.1 (2006): 58-71.
+[4] O. Bournez, M. Cosnard. On the computational power of dynamical systems and hybrid
+systems. Theor. Comput. Sci., 168-2 (1996), 417-459.
+[5] O. Bournez, D.S. Gra¸ca, E. Hainry. Computation with perturbed dynamical systems. J. Com-
+put. Syst. Sci. 79 (2013) 714–724.
+[6] O. Bournez, D. S. Gra¸ca, A. Pouly. Polynomial time corresponds to solutions of polyno-
+mial ordinary differential equations of polynomial length. Journal of the ACM (JAMC) 64.6
+(2017): 1-76.
+
+30
+ROBERT CARDONA
+[7] O. Bournez, A. Pouly. A survey on analog models of computation. Handbook of Computabil-
+ity and Complexity in Analysis. Springer, Cham, 2021. 173-226.
+[8] M. S. Branicky. Universal computation and other capabilities of hybrid and continuous dy-
+namical systems. Theor. Comput. Sci. 138.1 (1995): 67-100.
+[9] M. Braverman, M. Yampolsky. Non-computable Julia sets. J. Amer. Math. Soc. 19.3 (2006),
+551-578.
+[10] M. Braverman, M. Yampolsky. Constructing locally connected non-computable Julia sets.
+Comm. Math. Phys. 291.2 (2009), 513-532.
+[11] R. Cardona. The geometry and topology of steady Euler flows, integrability and singular
+geometric structures. PhD Thesis, Universitat Polit`ecnica de Catalunya, 2021.
+[12] R. Cardona, E. Miranda, D. Peralta-Salas. Turing universality of the incompressible Euler
+equations and a conjecture of Moore. Int. Math. Res. Not. 22 (2022), 18092-18109.
+[13] R. Cardona, E. Miranda, D. Peralta-Salas. Looking at Euler flows through a contact mirror:
+universality and undecidability. To appear in the Proceedings of the 8th European Congress
+of Mathematics (2022).
+[14] R. Cardona, E. Miranda, D. Peralta-Salas. Computability and Beltrami fields in Euclidean
+space. J. Math. Pures Appl. (9) 169 (2023), 50-81.
+[15] R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. Constructing Turing complete Euler
+flows in dimension 3. Proc. Natl. Acad. Sci. 118 (2021) e2026818118.
+[16] J. Etnyre, R. Ghrist. Contact topology and hydrodynamics I. Beltrami fields and the Seifert
+conjecture. Nonlinearity 13 (2000) 441–458.
+[17] D. S. Gra¸ca, M. L. Campagnolo, J. Buescu. Robust simulations of Turing machines with
+analytic maps and flows. In: S.B. Cooper, B. L¨owe, L. Torenvliet (Eds.), CiE 2005: New
+Computational Paradigms, in: Lecture Notes Comput. Sci., vol. 3526, Springer, 2005, pp.
+169-179.
+[18] D. S. Gra¸ca, M. L. Campagnolo, J. Buescu. Computability with polynomial differential equa-
+tions. Adv. Appl. Math. 40 (2008) 330–349.
+[19] D. S. Gra¸ca, N. Zhong. Robustness of non-computability. arXiv preprint arXiv:2109.15080
+(2021).
+[20] D. Gra¸ca, N. Zhong. Computing the exact number of periodic orbits for planar flows. Trans-
+actions of the American Mathematical Society, 375.8 (2022), 5491-5538.
+[21] P. Koiran, C. Moore. Closed-form analytic maps in one and two dimensions can simulate
+universal Turing machines. Theor. Comput. Sci. 210 (1999) 217–223.
+[22] C. Moore. Unpredictability and undecidability in dynamical systems. Phys. Rev. Lett. 64
+(1990) 2354–2357.
+[23] C. Moore. Generalized shifts: unpredictability and undecidability in dynamical systems. Non-
+linearity 4 (1991) 199–230.
+[24] K. Morita, Theory of reversible computing, Springer Japan, 2017.
+[25] M. B. Pour-El, I. Richards. The wave equation with computable initial data such that its
+unique solution is not computable. Adv. Math. 39.3 (1981): 215-239.
+[26] J.H. Reif, J.D. Tygar, A. Yoshida. Computability and complexity of ray tracing. Discrete
+Comput. Geom. 11 (1994) 265–288.
+[27] H. T. Siegelmann. Computation beyond the Turing limit. Science 268-5210 (1995), p. 545-548.
+[28] H. T. Siegelmann. The simple dynamics of super Turing theories. Theor. Comput. Sci. 168.2
+(1996): 461-472.
+[29] H. T. Siegelmann. Neural networks and analog computation: beyond the Turing limit.
+Springer Science & Business Media, 2012.
+[30] H.T. Siegelmann, E.D. Sontag. On the computational power of neural nets. J. Comput. Syst.
+Sci. 50 (1995) 132–150.
+[31] T. Tao. On the universality of potential well dynamics. Dyn. PDE 14 (2017) 219-238.
+
+HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION
+31
+[32] T. Tao. On the universality of the incompressible Euler equation on compact manifolds.
+Discrete Cont. Dyn. Sys. A 38 (2018) 1553–1565.
+[33] F. Torres de Lizaur. Chaos in the incompressible Euler equation on manifolds of high di-
+mension. Invent. Math. 228 (2022) 687–715.
+[34] A. M. Turing. Systems of logic based on ordinals. Proc. Lond. Math. Soc. 2.45 (1939), 161-
+228.
+[35] K. Weihrauch, N. Zhong. Is wave propagation computable or can wave computers beat the
+Turing machine? Proc. Lond. Math. Soc., 85.3 (2002), 312-332.
+Robert Cardona, Laboratory of Geometry and Dynamical Systems, Department
+of Mathematics, Universitat Polit`ecnica de Catalunya and BGSMath Barcelona
+Graduate School of Mathematics, Avinguda del Doctor Mara˜non 44-50, 08028 ,
+Barcelona
+Email address: robert.cardona@upc.edu
+
diff --git a/INFKT4oBgHgl3EQfdi4B/content/tmp_files/load_file.txt b/INFKT4oBgHgl3EQfdi4B/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1360 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf,len=1359
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='11820v1  [math-ph]  27 Jan 2023  HYDRODYNAMIC AND CONSERVATIVE MODELS OF  HYPERCOMPUTATION  ROBERT CARDONA  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Dynamical systems and physical models defined on idealized con-  tinuous phase spaces are known to exhibit non-computable phenomena, ex-  amples include the wave equation, recurrent neural networks, or Julia sets in  holomorphic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Inspired by the works of Moore and Siegelmann, in  this article we introduce new dynamical models of hypercomputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' First,  we show that ideal fluids, modeled by the Euler equations, are capable of sim-  ulating poly-time Turing machines with polynomial advice on compact three-  dimensional domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The complexity class that is shown to be computable  by stationary ideal fluids is precisely the one considered by Siegelmann in her  study of analog recurrent neural networks: the class P/poly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Then, we introduce  a class of symbolic systems that can be embedded in conservative homeomor-  phisms of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' These systems are shown to be capable of simulating Turing  machines with advice in real-time, contrary to previously known models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Introduction  Computational aspects of dynamical and physical systems can be studied from  a variety of intertwined perspectives such as numerical simulation, computability  theory, computational complexity, or analog computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The last one under-  stands a dynamical system as a computing device that takes an input (the initial  condition) and reaches some region of the phase space encoding the output of the  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Combining symbolic dynamics and the Turing machine model, Moore  showed in his seminal work [23, 22] that even low-dimensional dynamical systems  are capable of universal computation, thus unveiling the undecidability of some  of their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Since then, several dynamical systems coming from physical  models have been shown capable of simulating universal Turing machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Exam-  ples include 3D optical systems [26], analog recurrent neural networks [30], high  dimensional potential wells [31] and more recently incompressible fluids in various  contexts [12, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Robert Cardona acknowledges financial support from the Margarita Salas postdoctoral con-  tract financed by the European Union-NextGenerationEU, as well as from the LabEx IRMIA, the  Universit´e de Strasbourg and Instituto de Ciencias Matem´aticas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This work was partially sup-  ported by the AEI grant PID2019-103849GB-I00 / AEI / 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='13039/501100011033, AGAUR grant  2017SGR932 and the project Computational, dynamical and geometrical complexity in fluid dy-  namics - AYUDAS FUNDACI ´ON BBVA A PROYECTOS INVESTIGACI ´ON CIENT´IFICA  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  1  2  ROBERT CARDONA  Beyond Turing-computability arises hypercomputation: computational models  that can compute more than the classical Turing machines, such as Turing’s oracle  machines [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Dynamical systems modeled in continuous phase spaces allow the  presence of real numbers and infinite precision, which can lead to non-computable  phenomena and presence of hypercomputing capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Even if those dynamical  systems can represent physical models that are highly idealized and hence physi-  cally non-realizable, it is interesting from a theoretical of point of view to under-  stand which models do admit these hypercomputing capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' For example, the  wave equation admits non-computable solutions even if one chooses computable  initial data [25], see also [35, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Some examples in purely dynamical contexts  include the existence of non-computable Julia sets [9, 10], and polynomial planar  flows with non-computable number of periodic orbits [20] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' A complexity class that  contains non-computable languages is P/poly, which is the set of languages rec-  ognized by polynomial-time Turing machines with polynomial advice (see Section  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In her influential work [27, 28, 29], Siegelmann showed that neural  networks with real weights can simulate those machines in polynomial time, and  hence that the model is capable of computing beyond Turing machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Other dis-  crete dynamical systems were shown to be computationally equivalent to P/poly  by Bournez and Cosnard [4, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  The first contribution of this paper is to establish that ideal fluids on three-  dimensional geometric domains are also capable of simulating polynomial-time  Turing machines with polynomial advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Recall that given a three-dimensional  manifold, with or without boundary, the motion of an ideal fluid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' incompress-  ible and without viscosity) is modeled by the Euler equations  �  ∂  ∂tu + ∇uu  = −∇p ,  div u = 0 ,  where p stands for the hydrodynamic pressure and u is the velocity field of the  fluid, which is a non-autonomous vector field on M tangent to its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Here  ∇uu denotes the covariant derivative of u along itself, and div is the divergence  associated with the Riemannian metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' A stationary solution to the Euler  equations is an autonomous vector field on M whose integral curves represent the  particle-paths of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Given a polynomial-time Turing machine with polynomial advice  (T, a), there exists a three-dimensional toroidal domain U equipped with some  Riemannian metric g, and a stationary solution to the Euler equations in (U, g)  that simulates T in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  There are several ways in which a continuous system can simulate a Turing  machine, but generally it roughly means that each step-by-step computational  process of the machine is encoded in the evolution of some orbit of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  We introduce in this work a natural notion of simulation (see Definition 7) that  HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION  3  is inspired by the classical ones in the literature [8, 23, 17, 18, 6], but that further  provides a well-defined notion of time complexity in the context of conservative  ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In the statement, we have chosen for simplicity a domain diffeomorphic  to a solid torus, however, one can choose as well any closed three-manifold or R3  for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' To establish Theorem 1, we partially embed the dynamics of the  symbolic systems introduced by Siegelmann [27] in the evolution of a stationary  Euler flow, by following the strategy of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We construct diffeomorphisms of the  disk that simulate Turing machines with advice and embed them as a first-return  map on a cross section of a suitably parametrized flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  In the second part of this paper, we first introduce a new family of symbolic  systems that generalize those introduced by Moore [23] in a different way than the  analog shift map [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We study some dynamical and computational properties  of these systems, which we call countable generalized shifts, showing that they  can compute P/poly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Besides the fact that it provides an alternative symbolic  model of hypercomputation, it has some advantages with respect to previously  known dynamical models such as real-time simulation of reversible polynomial-  time Turing machines with polynomial advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We show that these systems can  be partially embedded in the evolution of conservative homeomorphisms of the  disk, hence losing regularity when compared to the analogous result for classical  generalized shifts [23, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Given a (reversible) polynomial-time Turing machine with polyno-  mial advice (T, a), there exists an area-preserving homeomorphism of the disk that  simulates T in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  In this setting the simulation is given, as in Moore’s works [23], by the exis-  tence of a computable semiconjugacy between the global transition function of  the Turing machine and the homeomorphism of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This provides another  conservative model of hypercomputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' From a purely dynamical perspective, it  would be interesting to understand if the symbolic systems that we introduce turn  out to be a special feature of low-regularity area-preserving homeomorphisms of  the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We point out that the area-preserving diffeomorphisms of the disk con-  structed in the proof of Theorem 1 are already capable of hypercomputations (see  Corollary 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' However, those simulations are not done in real-time, hence the dif-  ference with the model introduced in Sections 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In Section 2, we review the definitions of  generalized shift and analog shift map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In Section 3, we prove that there ex-  ists area-preserving diffeomorphisms of the disk that simulate in polynomial time  Turing machines with advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Section 4 starts by introducing time complexity in  conservative ODEs by defining an appropriate notion of simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' It is shown  that there are Euler flows simulating any polynomial-time Turing machine with  polynomial advice according to such definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In Section 5 we introduce count-  able generalized shifts, study some of their dynamical properties and analyze their  4  ROBERT CARDONA  computational power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Finally, in Section 6 we prove Theorem 2 by showing how  some countable generalized shifts can be partially embedded in area-preserving  homeomorphisms of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Acknowledgements: The author is grateful to Cristopher Moore, whose use-  ful correspondence about transformations of the disk preserving the square Cantor  set inspired this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Thanks to Daniel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Gra¸ca for helpful comments concern-  ing time complexity of continuous systems, and to Daniel Peralta-Salas for useful  discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Symbolic dynamics  In this section, we recall several definitions, such as Turing machine with advice,  Moore’s generalized shifts [23] and the analog shift map, a generalization proposed  by Siegelmann [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Turing machines with polynomial advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We define a Turing machine  T = (Q, q0, qhalt, Σ, δ) by the following data:  A finite set Q of “states” containing two particular (distinct) states: the  initial state q0 ∈ Q and the halting state qhalt ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  A finite set Σ which is the “alphabet” and that has cardinality at least  two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' It contains a specific symbol (denoted by 0) that is also called the  “blank symbol”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  A transition function δ : Q \\ {qhalt} × Σ −→ Q × Σ × {−1, 0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  A pair (q, t) ∈ Q × ΣZ is a configuration of the machine if the symbols of  t are all zero except for finitely many of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We say in this case that the  configuration, or the tape, is compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This can be assumed without  loss of computational power: Turing machines with this condition are equivalent  (as a computational model) to those which can have tapes with infinitely many  symbols different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Hence if A ⊂ ΣZ denotes the set of sequences that  have all but finitely many symbols equal to zero, the space P = Q×A is the space  of configurations of the machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' When writing a tape t = (ti) ∈ ΣZ, we will use  a dot to specify that the position zero lies at the right of the dot:  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='t0t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  The evolution of a Turing machine is described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' At any given step of  the algorithm, we will denote by q ∈ Q the current state, and by t = (tn)n∈Z ∈ ΣZ  the current tape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Given an input tape s = (sn)n∈Z ∈ ΣZ the machine runs by  applying the following algorithm:  (1) We initialize the machine by setting the current state q to be q0 and the  current tape t to be the input tape s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  (2) If the current state is qhalt then halt the algorithm and return t as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Otherwise compute δ(q, t0) = (q′, t′  0, ε), with ε ∈ {−1, 0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  (3) Change the symbol t0 by t′  0, obtaining the tape ˜t = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='t′  0t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='.  HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION  5  (4) Shift ˜t by ε obtaining a new tape t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The new configuration is (q′, t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Re-  turn to step (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Our convention is that ε = 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' ε = −1) corresponds  to the left shift (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' the right shift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  A step of the algorithm determines a global transition function  ∆ : (Q \\ {qhalt}) × A −→ P  that sends a configuration to the configuration obtained after applying a step of  the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The global transition function can also be trivially extended to  (Q \\ {qhalt}) × ΣZ, since a step of the algorithm is well-defined even for non-  compactly-supported tapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  A polynomial-time Turing machine T is a machine that halts for any given  input tin of size n in at most P(n) steps, where P(n) is some polynomially-  bounded function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' For our purposes, we shall follow the convention in [27, 29] and  say that an input tin ∈ ΣZ is of size n if it is of the form  tin = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=',  meaning by this that the tape only has zeroes away of the positions 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  A polynomial-time machine with polynomial-sized advice (T, a) comes equipped  with an infinite collection of strings a = {an}n∈N such that an ∈ Σp(n) for some  increasing polynomially-bounded function p(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Given an input tin of size n, the  machine has access to an in one computational step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' To make a very concrete  dynamical interpretation of this, given T and {an}n∈N, let us add a new state ˜q0  in Q which will be an auxiliary initial state, and the machine with input (˜q0, tin)  applies a preliminary step of the algorithm that sends (˜q0, tin) to the configuration  (q0, ta  in) where if tin = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', the tape ta  in is  ta  in = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tnan  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='an  p(n)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  where we wrote the advice string as an = an  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='an  p(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' For such Turing machine, the  global transition function ∆ can be extended to the set {˜q0}×ΣZ, by declaring that  on elements of the form (q0, tin) we have ∆(q0, tin) = (q0, ta  in), and any arbitrary  extension for the other elements in Q × Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Throughout the paper, whenever we  say Turing machine with advice, we refer to this polynomially-bounded version of  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Generalized shifts and analog shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let us now review the two classes  of symbolic systems that have been related respectively to Turing machines and  Turing machines with advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Generalized shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We recall here the definition of generalized shifts, follow-  ing the original definition of Moore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let A be a finite alphabet and s = (si)i∈Z ∈  AZ an infinite sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' A generalized shift is given by two maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' First, a map  G : AZ −→ AZ,  6  ROBERT CARDONA  together with a finite domain of dependence DG = {i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', i + l − 1} and a finite  domain of effect De = {n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', n + m − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' These domains indicate that G(s) is  equal to s except maybe along the positions in De, which are modified according  to the symbols in positions DG of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Secondly, a map  F : AZ −→ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  with a finite domain of dependence DF = {j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', j +r−1}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' the image of F only  depends on the symbols of the sequence in position DF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' These functions take a  finite number of different values since they depend on a finite number of positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  The image of s by the generalized shift φ : AZ → AZ is then defined as follows:  change s by G(s), this modifies potentially the symbols in positions De of  s,  shift G(s) by F(s), and declare it by definition as φ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  As we did for Turing machines, we choose by convention that a positive (negative)  value of F(s) means that we shift |F(s)| times to the left (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  The previous definition is appropriate from a notational point of view to easily  define the generalization of analog shift maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' However, generalized shifts admit  an alternative simpler notation (as used in [15]) that we will use as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Alternative notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The map G, once we have fixed the domains of dependence  and effect, is completely determined by assigning to each word in Al a word in  Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Hence, abusing notation, we will understand G as a map  G : Al → Am,  and write G(t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tl) = t′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='t′  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Similarly, the image by F of s is completely deter-  mined by the symbols in positions DF , so abusing notation we can think of F as  a map  F : Am → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Then, given a sequence s the sequence φ(s) is obtained by replacing the symbols in  positions n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', n + m − 1 of s by G(si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='si+l−1) and shifting the resulting sequence  by F(sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='sn+m−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Unless otherwise stated, and without loss of generality, we will  assume that a generalized shift is such that De = DG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Let us state an obvious property of generalized shifts that we will use later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let φ be a generalized shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Then for any sequence s ∈ AZ, s and  its image φ(s) coincide after a shift −F(s), except along the symbols at positions  DG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The sequence φ(s) is obtained by changing some of the symbols at posi-  tions DG of s and shifting by F(si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='si+r−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Hence shifting φ(s) by −F(si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='si+r−1)  yields a sequence that coincides with s except maybe along the symbols in posi-  tions in DG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  □  HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION  7  The analog shift map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The analog shift maps [27] are defined similarly to  generalized shifts, except that the domain of effect of G can be infinite (in one or  both directions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Hence in this case an analog shift map φ : AZ → AZ is specified  by  F : AZ −→ Z, DF = {i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', i + r − 1}  with a finite domain of dependence DF , and  G : AZ → AZ,  with a finite domain of dependence DG = {j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', j+l−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' However, the domain of  effect of G depends on the symbols in position DG of the sequence s and can be a  finite number of consecutive integers, a one-sided infinite sequence of consecutive  integers or all Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Simulation with area-preserving diffeomorphisms  In this section, we show how to simulate Turing machines with advice via com-  pactly supported area-preserving diffeomorphisms of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' To prove this fact,  we will first show how to simulate such machines via a globally injective piece-  wise linear area-preserving map defined on a finite number of disjoint rectangular  domains on a disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' These domains will mostly be Cantor blocks, whose definition  we recall below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The square Cantor set is the product set C2 := C ×C ⊂ I2, where  C is the (standard) Cantor ternary set in the unit interval I = [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' A Cantor  block is a block of the form B = [ a  3i , a+1  3i ] × [ b  3j , b+1  3j ] ⊂ I2, where i, j are two  non-negative integers and a < 3i, b < 3j are two non-negative integers such that  there are points of C2 in the interior of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  We identify sequences s = (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='s0s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=') in {0, 1}Z with points C2 via the  following bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  e : {0, 1}Z −→ C2  (si)i∈Z �−→ (  ∞  �  i=1  s−i  2  3i ,  ∞  �  i=0  si  2  3i+1 )  (1)  If we choose an alphabet with k symbols instead of only two, we can encode all  the sequences of the alphabet in a square Cantor set Ck2 where Ck is a Cantor  set of the interval obtained by iteratively removing k − 1 open subintervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Then  Cantor blocks are defined analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In the statement below D denotes some  disk of big enough radius and that contains the unit square I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Given a Turing machine with advice (T, a), there exist two families  B = B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', BN−1, BN and B′ = B′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', B′  N−1 of pairwise disjoint Cantor blocks,  a rectangular domain B′  N ⊂ D disjoint from B′, and a piecewise linear area-  preserving map F : ⊔Bi → B′  i such that F|C2∩B simulates (T, a) in polynomial  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  8  ROBERT CARDONA  A concrete property that formalizes simulation in polynomial time in this con-  text is that there exists a computable map ϕ that maps the configurations of  (T, a) to points on a square Cantor set ˜C2 ∩ B such that if (T, a) halts with input  (q0, tin) and output (qhalt, tout) in k steps, then the orbit of F through ϕ(q0, tin)  reaches ϕ(qhalt, tout) in q(k) iterations of F, where q is some polynomially-bounded  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' As explained by Siegelmann, the analog shift map is capable of simulating,  in polynomial time, a given polynomial-time Turing machine with polynomial  advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let us first recall how this is done to make suitable changes and obtain  globally injective dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Given a Turing machine with advice (T, a), consider T as a classical Turing  machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let us denote by a∞ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='a2a1a0 the one-sided infinite word obtained  by concatenation of all the advice of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Following [29, Chapter 12], there exists  a Turing machine ˜T whose set of states and symbols contains those of T but  have as well an extra marker symbol d in its alphabet and an extra set of states  ˜q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', ˜qr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The initial state of ˜T is ˜q1 (instead of q0 ∈ Q) and it simulates T on  inputs of the form (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=') in polynomial time as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' If we consider the  configuration (˜q1, ˜t) with ˜t = (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='a2a1a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='t0t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='), then in g(n) steps (where g  is some polynomially-bounded function) it reaches the configuration (˜qr, t′) with  t′ = (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='an+1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tnan0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='),  and from there simulates T in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This is done step by step as in T,  while moving the marker d to the left when necessary to keep track of the relevant  part of the left side of the tape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' To keep the notation simple, let Q and Σ denote  the set of states and the alphabet of ˜T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  We now consider the alphabet ˜A = Q ⊔ Σ ⊔ {˜q0}, and we identify the configu-  rations of ˜T to AZ by the map  ϕ : Q × ΣZ ֒→ ˜AZ  (q, t) �−→ (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='qt1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Using [23, Theorem 7], there exists a generalized shift φGS with DF = DG =  De = {−1, 0, 1} defined on the alphabet A =  ˜A \\ {˜q0} such that the global  transition function of ˜T is semi-conjugate to φ by this identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We extend  this generalized shift to an analog shift map φ : ˜AZ → ˜AZ satisfying:  �  G(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='˜q0t)  = (a∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='˜q1t) for any t ∈ Σ,  F(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='˜q0t)  = 0 for any t ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  The dot in the words is used to specify that DG is still {−1, 0, 1}, but the domain  of effect of G for the words of the form (0˜q0t) is now every integer smaller or  equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' For each (s−1s0s1) where φGS was already defined, the map G still  only changes the symbols in position −1, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION  9  We encode an input (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=') in ˜AZ as the sequence (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='˜q0t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  After one step of ˜φ, we reach (a∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='˜q1t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='), and then proceed simulating ˜T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Notice that we can assume that ˜T is reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Indeed, a classical method of  Benett [2] shows how to simulate T by some reversible Turing machine with three  tapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The simulation is polynomial in time, in fact if ˜T halts in K steps with  some input, then the reversible machine halts in O(K1+ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' A reversible three-tape  Turing machine can be simulated by a reversible one-tape Turing machine, such  simulation is also polynomial in time (in this case the simulation is quadratic see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' [1, Proposition 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We can also assume that the initial state of ˜T, which is  ˜q1, is not in the image of the transition function of ˜T as discussed in [24, Section  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let ˜∆ : (Q \\ {qhalt} × ΣZ → Q × ΣZ be the global transition function of ˜T,  and do not define it on halting configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' It is injective because we assumed  that ˜T is reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  An application of [23, Lemma 0] implies that φGS (which restricted to AZ is  equal to φ) is induced by a piecewise area-preserving linear map by blocks of some  square Cantor set ˜C (associated to an alphabet of | ˜A| symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This set of blocks  never lie in the block determined by fixing the zero symbol of a sequence to be ˜q0,  since φGS is defined on AZ ⊂ ˜AZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Each of these blocks is contained in the block  determined by an element (s−1, s0, s1) ∈ A3 of the domain of dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We  will ignore the block associated with halting configurations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' those for which  s0 = qhalt) and its image, since once we reach a sequence encoding a halting  configuration, the computational process if finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The fact that ˜T is reversible  implies that the family of blocks B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', BN−1 of ˜C is pairwise disjoint, and the  family of image blocks B′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', B′  N−1 is also pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Moore’s lemma also  provides a linear area-preserving map F : �N−1  i=1 Bi → �N−1  i=1 B′  i that induces  on ˜C the generalized shift φGS, again ignoring those sequences whose symbol at  position zero is qhalt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Here we are identifying sequences in ˜AZ with points in ˜C  via an injective map, computable for compactly supported sequences, of the form  of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Exactly as it happened with the blocks Bi, the image blocks B′  i will not intersect  either the block determined by those sequences that have at position zero the  symbol ˜q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This easily follows from the application that we did of [23, Theorem  7] and [23, Lemma 0]: we only need to use the blocks determined by words of  the form (t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='qt0) ∈ Σ × Q × Σ, whose image by φGS never has a ˜q0 in its zero  position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We will now extend F to another rectangular domain: we will first define  this rectangle and its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The first rectangle, which will be denoted by BN,  is given by the Cantor block of ˜C corresponding to those sequences that have at  position zero the symbol ˜q0 (this is a horizontal rectangle in I2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let ˜B be the  block given by those sequences that have at position zero the symbol ˜q1, this is  another horizontal rectangle in I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' No block B′  i intersects ˜B, since ˜q′  1 is not in the  image of the transition function of ˜T, and hence ˜q′  1 cannot be in the zero position  of a sequence in the image blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Now we define B′  N as the block which is the  10  ROBERT CARDONA  image of ˜B by the translation  τ(x, y) := (x + α, y),  where α is the number associate do the left-infinite sequence (a∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=') via a map  like (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' That is, if the alphabet was of two symbols and a∞ = (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='s−2s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='),  then α = �∞  i=1 s−i 2  3−i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  It is clear that B′  N remains disjoint from B′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', B′  N−1, since we just slightly  translated the horizontal Cantor block ˜B horizontally, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' the part of B′  N that  is not in ˜B, lies outside the unit square I2 and hence cannot intersect any other  Cantor block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We define F|BN to be a composition of the trivial translation of BN  into B′  N and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Observe that if we start with a sequence (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='˜q0t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=') in ˜AZ,  it is encoded to BN ⊂ ˜C, and applying F it is sent to the point whose associated  sequence is (a∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='˜q1t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tn0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In other words, given an input (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='˜q0t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='), in  one step all the advice of the machine is included in the sequence, and then we  proceed by computing as specified by φGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Hence F : �N  i=1 Bi → �N  i=1 B′  i induces  on ˜C a map that simulates T in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  □  Observe that even if the initial machine T is already reversible, the proof of  Theorem 5 shows that the simulation is only done in polynomial time, not in real  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This is because one needs to preprocess a∞ by introducing the auxiliary  machine ˜T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  The map F extends to a compactly supported area-preserving diffeomorphism  of the disk arguing exactly as in [15, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Given a polynomial-time Turing machine with polynomial advice  (T, a), there exists a compactly supported area-preserving diffeomorphism of a disk  D that simulates T in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  The simulation is concretely described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The symbolic system φ :  AZ → AZ (that simulates (T, a) in polynomial time) that is constructed in the  proof of Theorem 5 is semiconjugate, along the sequences that encode configura-  tions of (T, a), to the area-preserving diffeomorphism of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Regarding computability, the encoding (1) is of course computable on com-  pactly supported sequences, and so is the map by blocks except along the block  that simulates the first step of the machine (block BN in the proof of Theorem  5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This is necessary, since the dynamics applied to that block adds to the com-  putational process the possibly non-computable advice string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Hypercomputation in ideal fluid motion  In this section, we introduce a notion of simulation of Turing machines by  volume-preserving autonomous flows that admits a well-defined notion of time  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We show that there exist stationary solutions to the Euler equations  on compact three-dimensional geometric domains equipped with some Riemann-  ian metric that are capable of simulating a given Turing machine with advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION  11  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Simulation and complexity with conservative ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='2,  we introduced a computational model that computes beyond Turing machines,  following [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We showed that countable generalized shifts are capable of sim-  ulating polynomial-time Turing machines with polynomial advice in real-time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  a step of the Turing machine corresponds to an iteration of a countable general-  ized shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' However, if we are interested in showing that a continuous dynamical  system simulates a polynomial-time Turing machine, we need to specify how to  measure “time” to compare it to a step of the algorithm of the Turing machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Through this section, every object we consider, like a manifold or a vector field,  will be assumed to be smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let X be an autonomous (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' time-independent)  vector field on a manifold M (for concreteness, one might think of an ordinary  differential equation defined on Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Given a point p ∈ M the integral curve of X  through p is the solution to the system  �  y′(t) = X(y(t)),  y(0) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  (2)  There are several ways to define continuous-time simulation of a Turing machine T  by a vector field X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' One approach, introduced in [31] and followed in [15, 12, 14], is  to define simulation by requiring that the halting (perhaps with a prescribed finite  part of the output) of the machine for a given input is equivalent to the integral  curve of X through a computable point p ∈ M intersecting a computable open  set U ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This definition does not impose a “step-by-step” simulation, as it is  more customary in previous works [8, 17, 18, 21], although the constructions in  [31, 15, 12, 14] do in fact provide step-by-step simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Inspired by [17, 18], we  consider the following definition of step-by-step simulation that takes into account  the behavior of the time parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let X be a vector field on a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let T be a Turing  machine a denote by ∆ its global transition function, and by ϕ : P ֒→ M an  injective encoding of the configurations P of T in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We say that X simulates T  if for each initial configuration (q0, s) ∈ P, there is a constant Ks such that the  solution y(t) to y′(t) = X(y(t)) with initial condition y(0) = ϕ(q0, s) satisfies  y(Ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='n) = ϕ(∆n(q0, s)),  (3)  for each n ≤ N, where N ∈ N ⊔ {+∞} is the halting time of T with input (q0, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  With this definition, the values of t that are integer multiples of Ks measure  the steps of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' So given one computation (an initial configuration),  each step of the algorithm is performed in the same amount of continuous time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Definition 7 defines real-time simulation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' one step of the ma-  chine corresponds to one step (measured by time multiples of Ks) of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Polynomial-time simulation can be defined analogously, by replacing y(Ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='n) by  y(Ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='Q(n)) in Equation (3) for some polynomially-bounded function Q(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  12  ROBERT CARDONA  We will next see why this definition behaves well for conservative autonomous  flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We can further require that the positive trajectory y(t) as above  intersects some open set Us (for example, a ε-neighborhood of the image by ϕ  of the halting configurations) if and only if T halts with input (q0, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This is  interesting because the halting of T with a given input is equivalent to an integral  curve reaching an explicit open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In addition, we can require that the orbit y(t)  either intersects Us or remains at a positive distance (bounded from below) from  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' It is also possible to use an encoding ϕ that assigns an open set to  each configuration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' as in [14]), and then require that y(Kti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='n) lies in the open  set that encodes ∆n(q0, tin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Definition 7 readily generalizes to robust simulations  as in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Given an autonomous vector field simulating a Turing machine as in Definition  7, one is tempted to use the time-parameter t of an integral curve as a measure  of time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' According to the definition of step-by-step simulation, the  solution at time t = k corresponds to the kth step of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' However, as  classically treated in the literature [6, 7], the parameter t is not a well-defined  measure of time for a very simple reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' One can rescale the vector field X  defined in M by considering ˜X = fX for some positive function f ∈ C∞(M), so  that the orbit can compute faster or slower depending on the step of the algorithm  that is being simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The same happens if we consider a vector field simulating  a Turing machine as in Definition 7, taking as the time step the values of the  continuous-time parameter given by integer multiples of Ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' However, if X is  assumed to preserve some volume form µ ∈ Ωn(M) (where n = dim M), the  following proposition shows that there is a well-defined notion of time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let X be a vector field on M, preserving some volume form µ ∈  Ωn(M), and simulating a Turing machine T as in Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let �  X = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='X, with  f ∈ C∞(M) a positive function, be a reparametrization of X that also preserves  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Then �  X also satisfies Definition 7 with the same encoding and constants �Ks =  Ks  f(ϕ(q0,s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Assume that X satisfies Definition 7 with encoding ϕ and constant Ks  for a given initial configuration (q0, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let �  X = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='X be a reparametrization of  the vector field X that preserves µ, for some positive function f ∈ C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This  implies that L ˜  Xµ = 0, and is equivalent to  dιfXµ = df ∧ ιXµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Contracting this equation with X again, we see that a necessary and sufficient  condition is that ιXdf = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' that f is a first integral of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let us show that  ˜X satisfies Definition 7 with the same encoding as X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let (q0, s) be an initial  HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION  13  configuration of the machine T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let ˜y(t) be the integral curve of ˜X with initial  condition ϕ(q0, s), hence the solution to the system  �  ˜y′(t) = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='X(˜y(t)),  ˜y(0) = ϕ(q0, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  (4)  Since f is a first integral of X, it is constant along any orbit of X, so we might  replace f by the constant Cs = f(ϕ(q0, s)) in Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let y(t) denote the  solution to the system (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' It is standard that ˜y(t) satisfies  ˜y(t) = y(Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Consider the constant ˜Ks = Ks  Cs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Using that X simulates T with constants Ks and  encoding ϕ, we deduce that  ˜y( ˜Ksn) = y(Ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='n) = ϕ(∆n(q0, s)),  for each n ≤ N, where N is the halting time of T with input (q0, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  □  It is clear that if a vector field X satisfies the property described in Remark  9, any reparametrization also satisfies it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The time complexity of a computation  can be measured in a well-defined way by using the values of t that are integer  multiples of Ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' From a physical point of view, the value of f along an orbit of  X measures the norm of X along that orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' It is reasonable that computations  along an orbit where X has greater norm (though of as a measure of the “energy”  of the system along that orbit) occur faster in terms of the continuous measure of  time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' However, the time complexity measured discretely is invariant under these  reparametrizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Stationary Euler flows computing P/poly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Having introduced a well-  defined notion of time complexity for conservative vector fields, we will prove in  this subsection that given any polynomial-time Turing machine with polynomial  advice, there exists a solution to the stationary Euler equations in some compact  Riemannian three-manifold that simulates it (polynomially in time) according to  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  The Euler equations model the dynamics of an ideal (incompressible and with-  out viscosity) fluid on a Riemannian manifold (M, g) where they take the form  �  ∂tu + ∇u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='u = −∇p,  div u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Here u is the velocity field of the fluid, the scalar function p is the pressure  function, and all the differential operators are defined with the ambient metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' A  stationary solution is a solution satisfying ∂tu = 0, it is hence a time-independent  vector field whose integral curves define the particle paths of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The second  equation ensures that u is always a volume-preserving vector field with respect  to the Riemannian volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In order to prove that there exist stationary solutions  that simulate polynomial-time Turing machines with polynomial advice, our main  14  ROBERT CARDONA  tool will be the connection between Euler flows and Reeb flows in contact geometry  established by Etnyre and Ghrist [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This connection was used in [15] to prove  that there exists Turing complete steady Euler flows in three-dimensional compact  manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Let us recall that on a three-dimensional manifold M, a (cooriented) contact  structure is a plane distribution ξ defined as the kernel of a one-form α ∈ Ω1(M)  that satisfies the non-integrability condition α∧dα ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We call α a contact form,  and any positive multiple γ = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='α with f ∈ C∞(M) is another contact form  defining ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Each contact form γ uniquely defines a vector field R called the Reeb  vector field, which is determined by the equations  �  γ(R) = 1,  ιRdγ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Reeb fields will play a role in the proof of the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let (T, a) be a polynomial-time Turing machine with polynomial  advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' There exists a metric g in S3 and a stationary solution to the Euler equa-  tions X in (S3, g) that simulates T polynomially in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  The simulation will be according to Definition 7 and Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let (T, a) be a polynomial-time Turing machine with polynomial advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  By Corollary 6 there exists a compactly supported area-preserving diffeomorphism  of a disk  H : D −→ D  that simulates (T, a) in polynomial-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Concretely, as done in Theorem 5, there  exists a symbolic system φ : AZ → AZ and a computable map E : P → AZ  encoding the configurations of the machine such that φ simulates (T, a) in poly-  nomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Then H satisfies H(e(s)) = e(φ(s)) for every s ∈ E(P) ⊂ AZ, where  e denotes an encoding as in Equation (1) (perhaps using an expansion in base k  instead of two) into some square Cantor set ˜C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Fix the contact manifold (S3, ξstd), where S3 is the three-sphere and ξstd the  standard tight contact structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' By [15, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='1], there exists a contact  form α whose Reeb field R exhibits a Poincar´e disk-like section DM ⊂ M whose  first-return map is conjugate to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This means that DM is an embedded disk  transverse to the flow, and that there exists a smooth function τ : DM → R such  that the flow of R, that we denote by ϕt : M −→ M, satisfies  ϕτ(p)(p) = ψ ◦ H ◦ ψ−1(p), for p ∈ DM  where ψ : D −→ DM is a chart identifying D with the disk-like section DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  We will now choose a suitable positive rescaling function h ∈ C∞(M) so that the  first-return time of the flow of X = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='R at DM is constant and equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' First,  up to multiplying R by a small enough constant, we can assume that τ(p) < 1 for  all p ∈ DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Choose flow-box coordinates (x, y, z) of U = {ϕz(D′) | z ∈ [−ε, ε]} ∼=  D2 ×[−ε, ε], where D′ is a slightly bigger disk-like section containing DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Denote  HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION  15  by F the first-return map on D′, it satisfies F|DM = ϕτ(p)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In these coordinates  R =  ∂  ∂z, and the integral curve with initial condition (x, y, −ε) takes exactly time  ε to hit DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Construct a smooth function g : D′ × [−ε, ε] → R constantly equal to 1 near  (D′ × ({−ε, ε}) ∪ (D′ × [0, ε)) and such that  � 0  −ε  1  1 + g(x, y, z)dz = 1 − τ(F −1(x, y)) + ε, for (x, y) ∈ DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  It is clear that such a function exists: for a fixed point (x, y), we are choosing a  function depending on z with a given integral value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Varying smoothly the value  of the integral we can smoothly vary the function with respect to z, parametrically  with respect to two parameters x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Consider the vector field X = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We claim that the first-return time of X  to DM is constant and equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Consider p ∈ DM, then the solution to the  ODE defined by X and initial condition p hits DM × {−ε} at a point (x, y, −ε)  after time τ(p) − ε, since X = R along the piece of orbit outside of D′ × [−ε, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  In particular, we have (x, y) = F(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' On the other hand, the solution u(t) to the  ODE defined by X and initial condition (x, y, −ε) satisfies  t =  � z  −ε  1  1 + g(x, y, z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  It follows that the solution intersects DM = {z = 0} when t = 1− τ(F −1(x, y)) =  1− τ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Hence the time that the flow of X takes to send a point p back to DM is  ˜τ(p) = τ(p) − ε + 1 − τ(p) + ε = 1  as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  We have thus constructed a reparametrized Reeb field X = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='R that has a  disk-like Poincar´e section, with first-return time equal to one, and conjugated  to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' It was proved in [16] that any such vector field is a stationary solution  to the Euler equations for some Riemannian metric in the ambient manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' It  only remains to check that X does simulate the Turing machine T according to  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Given a configuration c ∈ P of the machine, it is mapped to an  element of AZ by the map E : P −→ AZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The set of sequences AZ in the image of  E is injectively mapped to the square Cantor set ˜C2 by the map e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' As encoding,  we choose ˜e = ψ ◦ e ◦ E and as constants we take Ks = 1 for all s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The first-  return map FX of X at any point p ∈ DM is given by the flow of X at time  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Hence, given an initial configuration (q0, s) of the machine T, we consider the  solution y(t) to the ODE defined by X and initial condition ˜e(q0, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Using that  FX = ψ ◦ H ◦ ψ−1(p) and that y(k) = F k  X(p), we deduce that  y(Q(n)) = ˜e(∆n(q0, s)),  16  ROBERT CARDONA  for each n smaller than the halting time of T with input (q0, s), where Q(n) is a  polynomially-bounded function that comes from the polynomial-time simulation  of (T, a) by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This concludes the proof that X simulates T according to Definition  7, polynomially in time as described in Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  □  We point out that, as mentioned in the context of neural networks [27], a given  computation requires only polynomial time with respect to the size of the input  and hence is simulated by a finite portion of the associated integral curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Hence,  a given computation is robust to perturbations of some size that depends on the  size of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Indeed, only finitely many positions of the sequence need to be  read to simulate a finite number of iterations of the symbolic system in ˜C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Remark 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The reparametrization argument used in the proof of Theorem 12  can be applied to the Turing complete Reeb flows constructed in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This yields a  stationary solution to the Euler equations in some Riemannian three-sphere that  simulates a universal Turing machine according both to the definition used in [15]  and to Definition 7 that takes into account time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='1 in [15] can be applied to any fixed closed contact three-manifold  (M, ξ), or open contact three-manifold such as (R3, ξstd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Furthermore, observe  that if the machine is taken to be reversible, then no increase of time is required  for the simulation and the Euler flow simulates in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This corresponds to  the statement of our main Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  In addition, the strong property described in Remark 9 is also true, namely  that there exists an open set U such that for any initial configuration (q0, s), the  orbit of X through the explicit point ps ∈ M associated with (q0, s) intersects U  if and only if T halts with input (q0, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The orbit either intersects U or stays at  a positive distance from U uniformly bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Even if a polynomial-  time Turing machine halts in every input, the last property is not meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  One could, for example, modify the Turing machine so that halting only occurs  on accepted inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Then the previous property would mean that the trajectory  associated to an input that is accepted intersects some domain U, but if the input  is not accepted then the associated trajectory remains at a positive distance of  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This is seen as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Let y(t) be the trajectory associated with an initial configuration (q0, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' All  the points representing a halting configuration are encoded in a finite collection of  blocks of the square Cantor set contained in DM, and no non-halting configuration  is encoded there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let UD be a small enough neighborhood of those blocks not  intersecting any other block, and U be defined as  U = {φX  t (p) | p ∈ UD, t ∈ (−ε, ε)},  where ϕX  t : M −→ M denotes the flow defined by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' It is clear that y(t) intersects  U if and only if T halts with input (q0, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In particular, since H(e(p)) = e(φ(p))  is satisfied for all the points encoding configurations of the machine (T, a), the  HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION  17  orbit of the initial configuration by the first-return map will always remain in the  blocks with non-halting configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' These blocks are distance greater than δ,  for some δ > 0, of the halting configuration blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This shows that the orbit  of a non-halting initial configuration stays at a positive distance (bounded from  above) of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Variations of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We end up this section by discussing other mod-  els, either of complexity or of ideal fluids, that could be considered in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Time complexity via orbit lenght.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In [6], another possible measure of the  time complexity of simulations with ODEs was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' When there is a metric  in the ambient space (for example, the Euclidean metric for an ODE defined in  Rm), a measure of time that is invariant with respect to reparametrizations is the  length of the orbit with respect to the Riemannian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This approach is also  reasonable in the context of hydrodynamics, since the space is endowed with a  natural Riemannian metric, the one for which the flow solves the Euler equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  This point of view can as well be taken in our construction in Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Indeed,  the vector field X has no zeroes, and the ambient manifold is compact, hence there  are c, C ∈ R constants such that  c < g(X, X) < C,  (5)  where g denotes the metric for which X is a stationary solution to the Euler  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In particular, the length of an injective piece of an integral curve grows  linearly with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' For the flow X, the computational steps are given by Ks = 1  (as in Definition 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' By hypothesis, given an input of size n the machine halts  after P(n) steps (where P is a polynomially-bounded function and n the size of  the input), the integral curve simulates the process in time P(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' By Equation  (5), the length of the curve up to time P(n) is polynomial as well, so polynomial  complexity is well-defined using the approach proposed in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Note that it is also  possible to construct a metric ˜g for which X is a stationary solution to the Euler  equations as well, and X has constant norm equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In that case, the length  coincides with the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The construction of ˜g is done using the arguments ex-  plained in [11, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='4 page 85], by considering a one-form α (which is not  anymore of contact type everywhere, but is instead closed in the solid torus) such  that α(X) = 1 in the solid torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Other hydrodynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Ideal fluid flows capable of universal com-  putation have been constructed in other situations, besides stationary flows on  geometric three-dimensional domains endowed with an adapted (not fixed a pri-  ori) Riemannian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Indeed, a natural requirement is to impose that the metric  is a fixed natural one, such as the flat metric on the three-torus or the Euclidean  metric in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In [14], it was shown that at the high cost of losing compactness, one  can construct stationary solutions to the Euler equation in R3 with the Euclidean  18  ROBERT CARDONA  metric that can simulate a universal Turing machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' One can check that the sim-  ulation is not as good as the one defined in Definition 7, because the simulation  has an exponential slow-down in terms of the continuous-time parameter of the  ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Even if we use the orbit-length approach to time complexity, one cannot  simulate polynomial-time Turing machines in polynomial time using the construc-  tion done in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' A natural question is then whether there is another construction  of stationary solutions to the Euler equations in Euclidean space that simulate  either in real-time or in polynomial time any Turing machine (with or without  advice), according to some natural definition of simulation and time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Similarly, it was proved in [12] that there are time-dependent solutions to the  Euler equations in some high enough dimensional closed manifold that are capa-  ble of simulating a universal Turing machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Those solutions not only have an  exponential slow-down as well but also rely on constructions of polynomial ODEs  [18] that simulate any Turing machine which might not hold for Turing machines  with advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Hence another question is if there are time-dependent solutions to the  Euler equations modeling hypercomputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Perhaps this can be shown as well  by doing a construction that can use the embedding results in [33, 32], as done  in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The question of whether viscous fluids, as modeled by the Navier-Stokes  equation, can simulate a universal Turing remains open in any possible context  [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Of course, the same question can be asked about the class P/poly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Countable generalized shifts and P/poly  In this section, we introduce a class of symbolic dynamical systems that contains  in particular generalized shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' As we will see, our generalization is different from  the analog shift map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Countable generalized shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The broader class of symbolic systems that  we introduce in this section should be thought of as a countable version of gen-  eralized shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Instead of changing the sequence according to a finite portion of  it of fixed size (like DF and DG), we change it according to a finite portion of a  variable but arbitrarily large size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This is a different generalization than analog  shift maps, where only a finite portion of fixed size determines the image of the  sequence, but infinitely many symbols can be changed in one step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  A countable generalized shift φ : AZ → AZ is defined by the following informa-  tion:  (1) a set P of pairs {(nj, Ij) ∈ Z × Amj}, with j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', N} or j ∈ N, such  that for each s ∈ AZ there is at most one j such that snj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='snj+mj−1 = Ij,  (2) a map J assigning to each element of p = (nj, Ij) ∈ P a word J(p) = I′  j ∈  Amj,  (3) a map H : P → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION  19  We denote by SP ⊂ AZ the set of sequences s ∈ AZ such that there is some  (nj, Ij) ∈ P such that snj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='snj+mj−1 = Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  The dynamical system is described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Given some s ∈ AZ, if s ̸∈ SP  then φ(s) := s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Otherwise, let p = (nj, Ij) be the only pair assigned to s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The  sequence φ(s) is obtained by changing the symbols in positions nj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', nj + mj − 1  by J(p), and then shifting by H(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' To simplify notation, given a pair (nj, Ij) we say that the symbols of  the word Ij are at positions nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='nj + mj − 1 of Ij (instead of positions 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='., mj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Furthermore, given a pair p ∈ P we say that a sequence s ∈ AZ coincides with  p ∈ P (or with Ij) if  snj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='snj+mj−1 = Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  (6)  Similarly, if we denote a finite word with indices w = wnwn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='wm−1wm with  n, m ∈ Z, we say that s coincides with w if  sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='sm = wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='wm,  (7)  where the left hand side denotes the symbols in position n, n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', m of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  It is easy to see that a generalized shift is, in particular, a countable generalized  shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Any generalized shift is a countable generalized shift with a finite set  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let φ : AZ −→ AZ be a generalized shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Without loss of generality, we can  assume that DF = DG, simply by taking the union of both domains and redefining  F and G appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Hence φ is defined by F : Al → Z and G : Al → Al, where  DF = DG = {i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', i + l − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let us define a countable generalized shift that is  equal to φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' As space of pairs, we choose P = {(i, (t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tl)) | (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', tl) ∈ Al}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Now  we define J((i, (t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tl))) = G(t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tl) and H((i, (t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tl)) = F(t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We obtain a  countable generalized shift ψ such that ψ(s) = φ(s) for each s ∈ AZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  □  An interesting property of the set SP ⊂ AZ is that it can never be equal to AZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  The diagonal argument used in the proof of this lemma will be useful throughout  the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' There is no countable generalized shift satisfying that P is not a  finite set and SP = AZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let φ be a countable generalized shift such that P is not a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Then  there is an infinite sequence pik = (nik, Iik) ∈ P, with Iik ∈ A⋗ℶℸ such that |mik|  or |nik| go to infinity as k goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' To simplify, assume that we found a  family Iik such that mik → ∞, an analogous argument works for the other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Choose a family of sequences sk coinciding with each pik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Endow AZ with the  20  ROBERT CARDONA  metric  d(t, t′) =  k  �  i=0  (2N)−k(|tk − t′  k| + |t−k − t′  −k|),  where N is the cardinality of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Then AZ is compact with this metric and the  sequence sk admits a convergent subsequence skr such that skr → ˜s as kr → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  We will show that ˜s ̸∈ SP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Indeed, assume that there is some ˜p = (˜n, ˜I) ∈ P (with  ˜I of size ˜m) such that ˜s coincides with ˜p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Choose some M such that |˜n| < M  and | ˜m| < M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Since skr → ˜s, there is some K0 such that |skr − ˜s| < N −M for  every kr > K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This implies that skr and ˜s are equal for symbols in the positions  −M, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', M, and hence skr coincides with ˜p for every kr > K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We deduce that  skr coincides both with ˜p and pkr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' For a big enough kr the element pik is such  that mik > M, and hence ˜p ̸= pkr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This is a contradiction with the definition of  a countable generalized shift: every sequence coincides with at most one element  in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  □  It is possible to characterize, although with a property that is difficult to verify  for a given example, those countable generalized shifts that are generalized shifts  too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let φ be a countable generalized shift, it is a generalized shift if  and only if there is some N ∈ N for which we can associate to each word w =  w−N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='wN ∈ A2N+1 an integer kw ∈ Z and a word w′ = w′  −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='w′  N satisfying  the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Given s ∈ AZ, whose symbols at positions −N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='., N  determines a unique word ws, we have:  If s ∈ SP coincides with some p = (nj, Ij), then H(p) = kws and J(p) is  such that for every position r ∈ {−N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', N}, either r ∈ {nj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', nj+mj−1}  and J(p)r = (ws)r or r ̸∈ {nj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', nj + mj − 1} and then (w′  s)r = (ws)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  If s ̸∈ SP , then changing the symbols of s at positions −N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', N by w′  s  and shifting by kws recovers s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let φ : AZ −→ AZ be a countable generalized satisfying the property in  the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Define a generalized shift ˆφ : AZ −→ AZ by taking DF = DG =  {−N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', N} and functions  F(w) = kw, for each w ∈ A2N+1  and  G(w) = w′, for each w ∈ A2N+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  It follows that if s ∈ SP then it follows from the first item above that ˆφ(s) = φ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Furthermore if s ̸∈ SP then φ(s) = s, but by the second item above ˆφ(s) = s too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Conversely, let φ : AZ −→ AZ be a countable generalized such that there is  some generalized shift ˜φ : AZ −→ AZ, defined by functions  F ′ : Ab−a → Z,  HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION  21  where DF ′ = {a, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', b} and  G′ : Ad−c → Ad−c,  where DG′ = {c, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', d} and such that φ(s) = ˜φ(s) for every s ∈ AZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  First, to simplify, we can easily construct another generalized shift ˆφ : AZ −→  AZ with associated functions F, G such that DF , DG = {−N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', N} for some N  and ˆφ(s) = ˜φ(s) for every s ∈ AZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' To do so, let N denote the greatest absolute  value of the elements of DF ′ and DG′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' To define F, given a word w−N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='wN ∈  A2N+1, we define F(w) := F ′(wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='wb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' To define G, given a word w−N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='wN if  G′(wc, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', wd) = w′  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='w′  d then we define  G(w−N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='wN) := w−N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='wc−1w′  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='w′  awa+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='wN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Let us now show that for this N the claimed property is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Fix any word  w = w−N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='wN, and denote F(w) and G(w) by kw ∈ Z and ω′ = w′  −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='w′  N  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Given any sequence s that coincides with w, since φ(s) = ˆφ(s), the  image of s is obtained by replacing the symbols at positions −N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', N by w′ and  shifting by kw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' If s ̸∈ SP, then φ(s) = s and item two above is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' If s ∈ SP ,  then there is some p = (nj, Ij) such that s coincides with p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' If Ij = s′  nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='s′  nj+mj−1,  let s′ be the sequence  s′ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0w−N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='wnj−1s′  nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='s′  nj+mj−1wnj+mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='wN10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  which also coincides with p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Now the image of s′ can be computed either by φ  or by ˆφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' By looking at the one placed at the last position of s′ which is not  zero, we deduce that necessarily H(p) = kw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This immediately implies that the  sequence obtained from s′ either by changing symbols at position −N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', N by  w′, or symbols at position nj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', nj +mj −1 by J(p) are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In other words,  J(p) and w′ are the same in the positions they have in common, and w′ coincides  with w at those positions which are not in {nj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', nj + mj − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This finishes the  proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  □  Let us give a sufficient criterion that is easy to check in practice to determine  when a countable generalized shift is not a generalized shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Definition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We asy that a countable generalized shift φ “modifies at infinity”  if there is a sequence of numbers |rik| → ∞ with rik ∈ {nik, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', nik + mik − 1}  for some pk = (nik, Iik) ∈ P, such that the symbol at position rik of Iik does not  coincide with the symbol at position rik of J(pk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' If a countable generalized shift φ modifies at infinity, then φ|SP is  not induced by a generalized shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let φ be a countable GS defined by P, J and H, and assume that it modifies  at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  There is a sequence of numbers |rik| → ∞ with rik ∈ {nik, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', nik + mik − 1} for  some pk = (nik, Iik) ∈ P satisfying Definition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' By Lemma 3, there is always a  shifted version of φ(s) that coincides with s except maybe along positions in DG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  22  ROBERT CARDONA  We will prove that this is not the case under our hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Assume that φ is  a generalized shift given by P, J and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' For a fixed k, let sk be a sequence that  has zeroes everywhere except in positions nik, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', nik + mik − 1 where it coincides  with Iik = anik .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='anik +mik−1, and also has a 1 at position nik + mik, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='e sk is of  the form  sk = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0anik .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='ak  nik +mik −110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  The image φ(sk) is obtained by changing the symbols in position nik, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', nik +  mik − 1 and shifting by H(pk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Let us show that for any integer r, the sequence sk and the sequence φ(sk)  do not coincide in some element in position greater or equal to rik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' To see this,  given an arbitrary r let φ(sk)r be the r-shifted sequence of φ(sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' If r > 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  left shift), then the symbol in position nik + mik > rik of sk (which is a one) does  not coincide with that of φ(sk)r, which is a zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' If r < 0 (right shift) then the  symbol in position nik +mik +r > rik (which is a zero) does not coincide with the  symbol of φ(sk)r in that position (which is a one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Finally when r = 0, we have  that the symbol in position rik does not coincide with that of φ(sk) by hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  We conclude that some symbol at a position greater or equal than rik of sk and  of any shifted version of φ(sk) are not equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Since rik is arbitrarily large choosing  an arbitrarily large k, this gives a contradiction Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We conclude that φ is  not a generalized shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  □  Lemma 18 can be used to easily construct examples of countable generalized  shifts that are not generalized shifts, even bijective ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We shall call a countable  generalized shift that is not a generalized shift an “infinite generalized shift”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We  will refer to a countable generalized shift that is a generalized shift as a “finite”  generalized shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Remark 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' As for generalized shifts [23, Lemma 1], any countable generalized  shift is conjugate (perhaps injectively semi-conjugate depending on the cardinality  of the alphabet) to another one whose alphabet is Σ = {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This is done by  identifying the symbols of the alphabet with large enough blocks of zeroes and  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Computational power of countable generalized shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In [23], Moore  showed that generalized shifts are equivalent to Turing machines, both from a  dynamical and computational point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In this section, we analyze the com-  putational power of countable generalized shifts and show that they can simu-  late Turing machines with advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' As done in previous sections, we restrict to  polynomial-time Turing machines with polynomial advice, which define the com-  plexity class P/poly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Technical assumptions on Turing machines with advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Given a Turing  machine with advice (T, a), we will assume that the first symbol of each advice  string is always a zero, and we will only consider inputs tin of size n such that  HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION  23  ti ̸= 0 for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='., n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' It is clear that this does not restrict the computational power  of the resulting polynomial-time Turing machines with polynomial advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' When  the Turing machine T that we consider is assumed to be reversible, we will assume  as well that q0 is not in the image of the transition function δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This can easily be  assumed, as discussed in previous sections, even if we restrict to reversible Turing  machines [24, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The latter assumption ensures as well that if we take  any Turing machine that is reversible, then adding advice to it keeps the global  transition function injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Let us show how to simulate polynomial-time Turing machines with polynomial  advice using countable generalized shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Theorem 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let (T, a) be a polynomial-time Turing machine with polynomial  advice a = {an}n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Then there is a countable generalized shift φ with some  alphabet A and an injective map ϕ : P −→ SP ⊂ AZ such that ∆ = ϕ−1 ◦φ|SP ◦ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  If ∆ is injective, then we can assume that φ|SP is injective in all SP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let (T, a) be a polynomial-time Turing machine with polynomial advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Take the alphabet of the countable generalized shift to be A = Q∪Σ∪{d}, where  d is a symbol disjoint from Q and Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let ϕ be the encoding function, which maps  injectively the configurations of T to sequences in AZ, defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  ϕ : P −→ AZ  (8)  (q0, tin = (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=')) �−→ (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='q0t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tnd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='), ti ̸= 0, n ≥ 0  (9)  (q, (ti)) �−→ (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='qt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=') otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  (10)  Let us define φ in terms of the space of pairs P and the maps J and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The space  of pairs P is defined by P1 ⊔ P2, where P1 is infinite and given by  P1 = {(−1, (dq0t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tnd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0  ����  p(n)-1  )) | n ∈ N and ti ̸= 0 for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', n},  where p(n) is the polynomially-bounded function assigning to an input of size n  its advice of size p(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The second set of pairs P2 is given by  P2 = {(−1, (t−1qt0) | (t−1qt0) ∈ Σ × Q × Σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  We define then the map J on P1 as  J((−1, (dq0t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tnd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0  ����  p(n)-1  )) = (0q0t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tnan  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='an  p(n)),  where an  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='an  p(n) is the advice string assigned to inputs of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' To define it on  P2, for any (q, t0) let δ(q, t0) = (q′, t′, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  J((−1, (t−1qt0)) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  (t−1t′q′) if ε = +1  (q′t−1t′) if ε = −1  (t−1q′t′) if ε = 0  24  ROBERT CARDONA  Finally, define H as H(x) = 0 for each x ∈ P1 and H((−1, (t−1qt0)) = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Observe  that a given sequence s ∈ AZ coincides with at most one element in P = P1 ∪ P2,  so P can be used to define a countable generalized shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The countable general-  ized shift φ is such that ∆ = ϕ−1 ◦ φSP ◦ ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Assume that ∆ is injective, and by contradiction assume further that φ|SP  is not injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Then there are two sequences s, t ∈ SP such that s ̸= t and  φ(s) = φ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The countable generalized shift φ changes at most a finite number  of symbols of each sequence, which is then shifted by at most 1 position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Assume  to simplify that both sequences are not shifted (an analogous argument works if  they are shifted in any direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Take the two unique pairs p1 = (−1, I1) and  p2 = (−1, I2) such that s, t coincide respectively with p1 and p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Then φ(s) is equal  to s except maybe at those symbols in positions D1 = {−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', m1 − 2} and φ(t)  coincides with t except maybe at those symbols in position D2 = {−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', m2 −2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  In particular, for each k ∈ D1 \\ D2 we deduce that tk is equal to the symbol in  position k of J(p1), and for each r ∈ D2 \\ D1 we deduce that sr is equal to the  symbol in position r of J(p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' For each position j ∈ D1 ∩ D2, we must have that  the symbol at position j of J(p1) is equal to the symbol in position j of J(I2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Consider the sequence s′ defined by  s′  i =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  0 if i ̸∈ D1 ∪ D2,  si if i ∈ D1  J(p2)i if i ∈ D2 \\ D1,  (11)  and the sequence t′ defined by  t′  i =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  0 if i ̸∈ D1 ∪ D2,  ti if i ∈ D2  J(p1)i if i ∈ D1 \\ D2  (12)  Our previous discussion shows that φ(s′) = φ(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' On the other hand, we know  that  s and t are equal in any position away from D1 ∪ D2,  s and t are equal to s′ and t′ respectively in positions D1 ∪ D2,  s ̸= t,  so we deduce that we must have s′ ̸= t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Using the description of φ and the fact that ∆ is injective, we will reach a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let us analyze case by case depending on p1 and P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' If p1, p2 ∈  P2, then D1 = D2 = {−1, 0, 1} and we deduce that s′ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='q0s10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' and  t′ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='q0t10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='. It follows that s′, t′ ∈ ϕ(P) as per equation (10), which is a  contradiction with the fact that ∆ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' If p1 ∈ P1 and p2 ∈ P2, then  s′ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='q0s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='sjd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  t′ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='qt1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' with t−1, t1 ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION  25  The sequence φ(s′) has a q0 in the zero position, while φ(t′) has a q ̸= q0 in the  zero position (since we assumed that q0 is not in the image of δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We reached a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  The last case is when p1, p2 ∈ P1, then p1 = (−1, I1) and p2 = (−1, I2) with  I1 = (dq0s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='sm1d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0) and I2 = (dq0t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tm20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Assume that m1 ≥ m2 and  then s′ and t′ are of the form  s′ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='q0s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='sjd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0  ����  p(j)-1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=',  and t′ is of the form  t′ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='q0t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='trd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0  ����  p(r)-1  sp(r)+r+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='ska1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='ap(j)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  if r + 2 + p(r) ≤ j  or  t′ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='q0t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='trd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0  ����  p(r)-1  ar+2+p(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='ap(j)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  if r + 2 + p(r) > j,  where the input (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='sj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=') has an advice (a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='ap(j)) and the input (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='tr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=')  has an advice (b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='bp(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Then φ(s′) = φ(t′) implies that (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='q0s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='sja1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='ap(j)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=')  is equal to  (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='q0t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='trb1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='bp(r)ap(r+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='ap(j)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='),  if p(r) ≥ j, or to  (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='q0t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='trb1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='bp(r)sp(r)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='sja0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='.ap(j)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='),  if p(r) < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' If m1 = m2 then we deduce that t′ = s′ which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In  general, we deduce that ti = si for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', r, and that sr+1 = b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' However,  by our technical assumptions, we know that b1 = 0 and that sr+1 ̸= 0, hence  finishing the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  □  The countable generalized shift constructed in the proof of Theorem 20 is an  infinite generalized shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Indeed, the description of J on P1 implies that the  countable generalized shift modifies at infinity, so by Lemma 18 it is an infinite  generalized shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The simulation by a countable generalized shift of a polynomial-  time Turing machine is done in real-time: a step of the countable generalized shift  corresponds to a step of the machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This is an advantage with respect to the  simulation via analog shifts [27], where the first step is necessarily simulated in  polynomial time with respect to the size of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Area-preserving homeomorphisms of the disk  In this last section, we show that some countable generalized shifts can be  embedded, at least partially, in the evolution of a compactly supported area-  preserving homeomorphism of a disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This will be used to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  26  ROBERT CARDONA  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Cantor set and map by blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' From now on, we will make the sim-  plifying assumption that a countable generalized shift is defined on the alphabet  A = {0, 1}, see Remark 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' As done in previous sections we identify sequences  s = (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='s0s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=') in {0, 1}Z with points C2 via the bijection introduced in Equa-  tion (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The following lemma shows that countable generalized shifts are induced  by countably piecewise linear maps of blocks of C2, just as for standard generalized  shift [23, Lemma 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Given a countable generalized shift φ, there exists a piecewise linear  and area-preserving map f defined over a countable set of blocks into another  countable set of blocks of the square Cantor set such that φ|SP = e−1 ◦ f ◦ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The  following are equivalent:  φ|SP : SP → AZ is injective,  the image blocks are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Each element of pj = (j, Ij) ∈ P determines a block Bj of the square Cantor  determined by all those sequences (si) such that snj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='snj+mj−1 coincides with Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Each block Bj is first translated into the block determined by J(pj), then Baker’s  map is applied H(pj) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The block Bj might be cut into rj ≤ 2|H(pj)| connected  pieces when applying the Baker’s map (or its inverse if H(pj) is negative) is applied  |H(pj)| times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let Bk  j denote the preimages of those pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Then define the map  f :  �  j∈N  rj  �  k=0  Bk  j → I2  which coincides in each block Bk  j with the translation and iteration of Baker’s  map associated to Bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Clearly f corresponds to Φ when applied to a point of  the Cantor set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Observe that two image blocks intersect if and only if there is a  point of the Cantor set in both blocks, which happens if and only if f|e(SP ) is not  injective, which happens if and only if φ|SP is not injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  □  It follows from the construction that if there is a finite word w = (wn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', wn+m)  such that any sequence s ∈ AZ that coincides with w is not in SP, then we can  as well define f to be the identity in the block defined by w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Hence the piece-wise  map f and the conjugacy with φ can be extended for other sequences that are not  in SP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The only problem, if we want a piece-wise map defined on blocks, arises  with those sequences that are obtained as the limit of a family of words obtained  from pairs pj = (nj, Ij) ∈ P with a size that tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Extension to area-preserving homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' In the case that φ|SP  is injective, the image blocks are disjoint and we can extend the map to an  area-preserving homeomorphism of a disk containing the unit square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' This fact is  proven by generalizing [15, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='1] to the case of countably many pairwise  disjoint blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Taking a non-finite family of disks will imply a loss of regularity  of the homeomorphism, which is only continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION  27  Proposition 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let φ be a countable generalized shift such that φ|SP is in-  jective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Then there exists a compactly supported area-preserving homeomorphism  H : D −→ D of some disk D strictly containing the unit square such that  H(e(s)) = e(φ(s)) for every s ∈ SP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let φ be a countable generalized shift such that φ|SP is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' By Lemma  21, there exists two countable families of pairwise disjoint blocks �∞  i=1 Bi and  �∞  i=1 B′  i and an area-preserving piece-wise linear bijective map  F :  ∞  �  i=1  Bi −→  ∞  �  i=1  B′  i,  such that F(Bi) = B′  i and e ◦ φSp = F ◦ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let D be a big enough disk containing  in its interior the unit square, for example a disk of radius 2 centered at (1/2, 0)  is enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We will construct an area-preserving continuous isotopy ϕt : D −→ D,  compactly supported, and such that ϕ1|�∞  i=1 Bi = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Since the area of Bi and B′  i  are the same, we can choose two small closed smooth neighborhoods Di and D′  i  of Bi and B′  i respectively, both diffeomorphic to a closed disk and with the same  area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Here the area refers to the integral of the standard area-form ωstd = dx ∧ dy  along that disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Notice that the distance between any two disjoint blocks of the  square Cantor set is always at positive, so we can assume all the Di are pairwise  at a positive distance, and that all the D′  i are pairwise at a positive distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' The  linear area-preserving bijective map F|�∞  i=1 Bi : Bi → �∞  i=1 B′  i can be naturally  extended to an area-preserving diffeomorphism F : �∞  i=1 Di → �∞  i=1 D′  i satisfying  F|�∞  i=1 Bi ≡ F, for example by taking the same linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  The set A = D \\  �  ⊔∞  i=1 Di ∪ D′  i  �  can be assumed to have a non-empty interior,  since the blocks are all contained in I2, and D contains I2 in its interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Since D  is big enough, we can find a countable family of pairwise disjoint disks �Di such  that �Di ⊂ D \\ A and area( ˜Di) = area(Di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Starting with D1 and ˜D1, choose an  embedded arc τ1 such that  τ(0) ∈ D1,  τ(1) ∈ �D1,  τ1(I) ∩  � �∞  i=1 Di ∪ �Di  �  ⊂ D1 ∪ �  D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  This is possible because D \\  � �∞  i=1 Di ∪ �Di  �  is path-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let �U1 be a small  enough connected set (diffeomorphic to a disk) containing D1, �D1 and τ1, and  such that �U1 ∩  � �∞  i=1 Di ∪ �Di  �  ⊂ D1 ∪ �D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Since D \\  �� �∞  i=1 Di ∪ �Di  �  ∪ �U1  �  is  still path-connected, we can find similarly a set �U2 containing D2, �D2 and a path  τ2 connecting them and that only intersects those two disks in the family Di, �Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Inductively, we find a family of embedded closed disks �Ui ⊂ D that are disjoint  and contain Di and �Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We easily find for each �Ui a compactly supported family  of embeddings ei  t : Di → �Ui such that ei  0(Di) = Di and ei  1(Di) = �Di, for example  by compressing Di into a small enough disk, moving it to the center of �Di and  28  ROBERT CARDONA  undoing the compression1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' By the isotopy extension property, we find for each �Ui  a compactly supported isotopy ξi  t : �Ui → �Ui such that ξi  1|Di is a translation with  image �Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Using the relative Moser’s path method, exactly as in [15, Proof of Proposition  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='1], we obtain from ξi  t an area-preserving isotopy ηi  t such that ηi  1|Di is a translation  with image �Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Define the isotopy ϕt : D → D for t ∈ [0, 1/2] as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  ϕt(p) =  �  ηi  2t(p) if p ∈ �Ui,  p otherwise,  t ∈ [0, 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  This is a well-defined continuous area-preserving isotopy of D, since the isotopies  ηi  t restrict as the identity near the boundary of �Ui and each of them preserves  the measure induced by the standard area form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' However, regularity is lost by  the extension in the complement of the countably many domains �Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Indeed, the  derivatives might not be continuous at any point where the family Bi accumulates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  We argue now similarly with the family of disks { �Di | i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='} and {D′  i | i =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='}, which are pairwise at positive distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let γ1 : I −→ D be an embedded  arc such that  γ1(0) ∈ �D1,  γ1(1) ∈ D′  1,  γ1(I) ∩  � �∞  i=1 �Di ∪ D′  i  �  ⊂ �D1 ∪ D′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Choose an open set U1 containing D1, D′  1 and a small enough neighborhood of γ1,  such that D \\  �  U1 ∪  � �∞  i=1 �Di ∪ D′  i  ��  is path-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We will now construct a  compactly supported isotopy of U1  G1  t : U1 → U1, t ∈ [0, 1]  such that G1  1|D1 is area-preserving, G1  0| �  D1 = �D1 and G1  1| �  D1 = D′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Actually we will  impose as well that G1  1 ◦ ϕ1/2|Di = F|Di, hence prescribing the homeomorphism  between �D1 and D′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  To do so, choose small enough disks d1 ⊂ �D1 and d′  1 ⊂ D′  1 obtained by applying  contracting homotheties hs, h′  s centered respectively at the center of �D1 and D′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Define a parametric family of embeddings et : �D1 → U1, t ∈ [0, 4] as follows  (1) For t ∈ [0, 1/4], we shrink �D1 into d1 using hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  (2) For t ∈ [1/4, 1/2], we isotope d1 through the open neighborhood of γ1 up  to a small disk ˜d1 centered at the center of D′  1 obtained by a translation  of d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' That is e1/2 corresponds to a homothety and a translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  (3) For t ∈ [1/2, 3/4] we apply an isotopy that first transforms ˜d1 to d′  1 in a  way that e3/4 = (h′  1)−1F ◦ ϕ−1  1/2,  1To simplify, we could have chosen each �Di to be a translation of Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION  29  (4) For t ∈ [3/4, 1] we expand d′  1 to D′  1 by using the inverse homothety (h′  s)−1,  hence e1 = F ◦ ϕ−1  1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  The family et extends, by the isotopy extension theorem, to a compactly supported  isotopy G1  t of U1 that satisfies the claimed properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  We can iterate this process for each pair of disks �Di, D′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Indeed, D \\  �  U1 ∪  � �∞  i=1 �Di ∪ D′  i  ��  is path-connected, so we can find an arc γ2 connecting �D2 and  D′  2 as before, and a connected open set U2 ⊂ D \\ U1 containing �D2 and D′  2 such  that D \\  �  U1 ∪ U2 ∪  � �∞  i=1 �Di ∪ D′  i  ��  is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We obtain a countable number  of disjoint embedded disks Ui and compactly supported isotopies Gi  t : Ui → Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Arguing as before, these can be assumed to be area-preserving using the relative  Moser’s path method while still satisfying Gi  1 ◦ ϕ1/2|Di = F Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' We extend ϕt for  t ∈ [1/2, 1] as  ϕt(x) =  �  G2(t−1/2)(p) if p ∈ Ui,  p otherwise,  The homeomorphism H = ϕ1 is a area-preserving and satisfies H|Bi = F|Bi for  each i, since H|Di = ϕ1|Di = F|Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Thus H(e(s)) = e(φ(s)) follows from the fact  that e(SP ) ⊂ �∞  i=1 Bi, which is satisfied by the construction of the map by blocks  in Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  □  Remark 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' From the previous proof, one extracts the following general state-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Let Di and D′  i be two families of countably many embedded disks on a  disk D (or any connected surface), such that in each family the elements are pair-  wise at positive distance and D \\ (�∞  i=1 Di ∪ D′  i) has non-empty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Assume  that area(Di) = area(D′  i) and that we are given an area-preserving diffeomor-  phism Fi : Di → D′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Then there exists a compactly supported area-preserving  homeomorphism H of D such that H|Di ≡ Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Finally, Theorem 2 stated in the introduction is obtained by combining Theo-  rem 20 and Proposition 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  References  [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' Computation with perturbed dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' Polynomial time corresponds to solutions of polyno-  mial ordinary differential equations of polynomial length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content='  30  ROBERT CARDONA  [7] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' Springer, Cham, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' Non-computable Julia sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' The geometry and topology of steady Euler flows, integrability and singular  geometric structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' PhD Thesis, Universitat Polit`ecnica de Catalunya, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' Miranda, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Peralta-Salas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Turing universality of the incompressible Euler  equations and a conjecture of Moore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' Miranda, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Peralta-Salas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Looking at Euler flows through a contact mirror:  universality and undecidability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' To appear in the Proceedings of the 8th European Congress  of Mathematics (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' Miranda, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Peralta-Salas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Computability and Beltrami fields in Euclidean  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' Miranda, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Peralta-Salas, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Presas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Constructing Turing complete Euler  flows in dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' Beltrami fields and the Seifert  conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' Robust simulations of Turing machines with  analytic maps and flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' ), CiE 2005: New  Computational Paradigms, in: Lecture Notes Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' 3526, Springer, 2005, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' Computing the exact number of periodic orbits for planar flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' Computation beyond the Turing limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
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+page_content=' The simple dynamics of super Turing theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='2  (1996): 461-472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  [29] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Siegelmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Neural networks and analog computation: beyond the Turing limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Springer Science & Business Media, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  [30] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Siegelmann, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Sontag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' On the computational power of neural nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' 50 (1995) 132–150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  [31] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Tao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' On the universality of potential well dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' PDE 14 (2017) 219-238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  HYDRODYNAMIC AND CONSERVATIVE MODELS OF HYPERCOMPUTATION  31  [32] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Tao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' On the universality of the incompressible Euler equation on compact manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Discrete Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' A 38 (2018) 1553–1565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  [33] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Torres de Lizaur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Chaos in the incompressible Euler equation on manifolds of high di-  mension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' 228 (2022) 687–715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Turing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Systems of logic based on ordinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='45 (1939), 161-  228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  [35] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Weihrauch, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Zhong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Is wave propagation computable or can wave computers beat the  Turing machine?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content=', 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='3 (2002), 312-332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='  Robert Cardona, Laboratory of Geometry and Dynamical Systems, Department  of Mathematics, Universitat Polit`ecnica de Catalunya and BGSMath Barcelona  Graduate School of Mathematics, Avinguda del Doctor Mara˜non 44-50, 08028 ,  Barcelona  Email address: robert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='cardona@upc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
+page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFKT4oBgHgl3EQfdi4B/content/2301.11820v1.pdf'}
diff --git a/IdFJT4oBgHgl3EQfvS1K/content/tmp_files/2301.11625v1.pdf.txt b/IdFJT4oBgHgl3EQfvS1K/content/tmp_files/2301.11625v1.pdf.txt
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+arXiv:2301.11625v1  [math.PR]  27 Jan 2023
+January 27, 2023.
+Some definite integrals arising from selfdecomposable
+characteristic functions.
+Zbigniew J. Jurek
+University of Wrocław, Wrocław, Poland;
+zjjurek@math.uni.wroc.pl and
+www.math.uni.wroc.pl/∼zjjurek
+Abstract.
+In the probability theory selfdecomposable, or class L0 dis-
+tributions play an important role as they are limiting distributions of nor-
+malized partial sums of sequences of independent, not necessarily identically
+distributed, random variables. The class L0 is quite large and includes many
+known classical distributions and statistics. For this note the most important
+feature of the selfdecomposable variables are their random integral represen-
+tation with respect to Lévy process. From those random integral represen-
+tation we get equality of logarithms of some characteristic functions. These
+allows us to get formulas for some definite integrals, some of them probably
+were unknown before.
+2020 Mathematics Subject Classifications: Primary: 60E07,60E10; Sec-
+ondary: 60G51,60H05
+Key words and phrases: infinite divisibility; selfdecomposability; Lévy
+process; Urbanik classes; random integral; hyperbolic characteristic functions;
+log-gamma distribution; generalized logistic distribution; Meixner distribu-
+tions; Feller-Spitzer distributions
+The class L of selfdecomposable distributions appears in the classical prob-
+ability theory as a class of limiting distributions obtained from infinitesimal
+triangular arrays arising from sequences of independent random variables. In
+the case of independent and identically distributed variables we get the class
+S of stable distributions, which contains the normal distributions. On the
+other hand, when one considers arbitrary infinitesimal triangular arrays we
+end-up with the class ID of infinitely divisible distributions. Thus we have
+the inclusions S ⊊ L ⊊ ID.
+Infinitely divisible random variables X ∈ ID are uniquely determined by
+their characteristic functions
+φX(t) := E[exp(itX)] = exp
+�
+ita− 1
+2σ2t2+
+�
+R\{(0)}
+(eitx−1−
+itx
+1 + x2)MX(dx)
+�
+,
+where a ∈ R, σ2 ≥ 0 (Gaussian part),
+�
+R\{(0)} min(1, x2)M(dx) < ∞.
+1
+
+The measure M is called the Lévy measure of X. The above integral
+formula is refered to as the Lévy-Khintchine representation. The triple a, σ2
+and M is uniquely determined by X.
+The classic references to those topics are: Feller (1966), Gnedenko and Kolo-
+mogorov (1954) or Loeve (1963).
+For infinite divisibility for Banach space valued variables we refer to
+Araujo and Gine (1980), Chapter 3.
+The class L is quite large and includes many classical distributions in
+probability and statistics; for examples cf. Jurek (1997) and (2021). More-
+over, during last decades the selfdecompsability appeared in mathematical
+finance (Carr, Geman , Madan and Yor (2007), a book by Schoutens (2003))
+and in statistical physics ( deConinck (1984), deConinck and Jurek (2000),
+and Jurek (2001)).
+The aim of this note, (after the preliminary Proposition 1, in the Intro-
+duction), is to show how some definite integrals can be computed from the
+selfdecomposable characteristic functions; cf. Lemmas 1 , 2 and Corollary 1
+(for the hyperbolic characteristic functions) and Corollaries 2-6, in Sections
+2 and 3 (for the characteristic function expressed via Euler’s Euler’s gamma
+function; in particular, for the Meixner or Feller-Spitzer distributions).
+Introduction.
+A random variable X is called selfdecomoposable(in symbols: X ∈ L0), if
+∀ (t > 0) ∃ (Xt ∈ ID independent of X) X
+d= e−tX + Xt.
+(1)
+And inductively, for k ≥ 1, we define, decreasing sequence of the Urbanik
+classes Lk, as follows:
+X ∈ Lk iff, in (1), Xt ∈ Lk−1, for all t > 0; cf. Urbanik (1972),(1973); (1a)
+The "remainders" (Xt, t > 0) in (1) satisfy the following cocycle equation
+Xt+s
+d= e−tXs + Xt. It allows to construct a cadlag Lévy process and to infer
+that the selfdecomposability of X is equivalent to the claim that there exits
+unique, in distribution, a Lévy process (YX(t), t ≥ 0), such that
+X ∈ L0 iff X
+d=
+� ∞
+0
+e−tdYX(t),
+and E[log(1 + |YX(1)|)] < ∞,
+(2)
+cf. Jurek and Vervaat (1983),Theorem 3.2, pp. 252-253 or Jurek and Mason
+(1993), Theorem 3.6.8, pp. 124-126 and Jurek (1982) for a Banach space
+valued random variables.
+To the random variable YX(1) we refer to as the background driving ran-
+dom variable of X; is short: BDRV.
+2
+
+For this note crucial are the following facts listed in :
+Proposition 1. (a) Let X and YX(1) be a selfdecopmosable variable and its
+BDRV, respectively, in the random integral representation (2). Then for the
+characteristic functions φX(t) := E[exp(itX)] and ψX(t) := E[exp(itYX(1))]
+we have
+ψX(t) = exp(t(log φX(t))′), t ̸= 0, ψX(0) = 1.
+(3)
+(b) Equivalently, if [a, σ2, MX] is the triple in the Lévy-Khintchine formula
+for X and similarly [b, s2, NX] is the triple for YX(1) then
+MX(B) =
+� ∞
+0
+NX(etB)dt,
+Borel B ⊂ R \ {(0)}; MX(R \ {(0)}) = ∞;
+MX(dx) = kX(x)dx,
+NX(dx) = hX(x)dx,
+hX(x) = (−xkX(x))′ ≥ 0,
+s2 = 2σ2,
+b = a +
+�
+R\{((0)}
+x
+1 + x2 − arctan(x))hX(x)dx.
+(4)
+(c) [The condition (−xkX(x))′ ≥ 0 is equivalent to the condition that the
+function xkX(x) is not increasing of both half-lines. This means that
+[a, σ2, kX(x)] is selfdecomposable variable.]
+(d) Hence, by comparing (3) and [b, s2, NX] in (4), we conclude the fol-
+lowing identity for t ∈ R,
+itb − 1
+2s2t2 +
+�
+R\{((0)}
+(eitx − 1 −
+itx
+1 + x2)hX(x)dx = t(log φX(t))′.
+(5)
+For the proof of (3) cf. Jurek (2001b), Proposition 3. For proof of (4)
+cf. Jurek and Mason (1993), p.120, formulae (3.6.9)-(3.6.11) for Q = I, or
+Jurek (1996), p.175. Finally for the equivalences in the square bracket [...],
+cf. Jurek and Mason (1993), Theorem 3.4.4, p. 94 or Steutel and Van Haarn
+(2004), Theorem 6.12, p. 277.
+The above equality (5) is the key identity for all the definite integrals in
+this note.
+Remark 1. (i) If selfdecomposable X has finite second moment then in (5)
+we will use the kernel exp(itx)−1−itx; cf. Bilingsley (1986), Theorem 28.1,
+p. 384. Moreover, in that case, in (5), b = a.
+(ii) From (4) we see that X ∈ L0 has Gaussian part if and only if its
+BDRV YX(1) has a Gaussian part.
+(iii) For the real characteristic functions, as in Section 1 below, we get
+simpler formulas as we can discard the imaginary part.
+3
+
+1. The hyperbolic characteristic function.
+For the information about the hyperbolic-type characteristic function we
+refer to Pitaman and Yor (2003) or Jurek and Yor (2004). Also this section
+may be viewed as a complement to the recent preprint Jurek (2022 a).
+a). Hyperbolic-sine function.
+Lemma 1. From the selfdecomposability of the hyperbolic-sine characteristic
+function φ ˆS(t) =
+t
+sinh(t) we get the following formulas (Lévy exponents):
+(i)
+� ∞
+0
+(cos(tx) − 1)(π/2)csch2(πx/2)dx = 1 − t coth(t);
+(ii)
+� ∞
+0
+(cos(tx)−1)π
+2 csch2(πx
+2 ) (πx coth(πx
+2 )−1)dx = t2csch2(t)−t coth(t);
+(iii)
+� ∞
+0
+(cos(tx) − 1)π
+4csch2(πx/2)
+�
+2π2x2 coth2(πx/2) + π2x2csch2(πx/2)
+− 6πx coth(πx/2) + 2
+�
+dx = 3t2csch2(t) − t coth(t)(2t2csch2(t) + 1).
+Proof. Since
+φ ˆS(t) =
+t
+sinh t = [0, 0, k ˆS(x)] ∈ ID,
+k ˆS(x) = 1
+|x|
+1
+eπ|x| − 1, x ∈ R \ {(0)},
+i.e., in this example in a = 0 and σ2 = 0 for the random variable X := ˆS.
+Since h ˆS(x) := (−xk ˆS(x))′ = (π/4)csch2(πx/2), x ∈ R \ {(0)}, is positive
+thus by part (c) of Proposition 1 we get that ˆS ∈ L0. Its BDRV Y ˆS(1) =
+[0, 0, h ˆS(x)], as in view of symmetry of h ˆS(x) we have b = 0 and s2 = 0 in
+(4) above.
+On the other hand, as ψ ˆS(t) = exp(t(log φ ˆS(t))′) = exp(1 − t coth(t), from
+(6) we get the part (i) of Lemma 1.
+Since
+g ˆS(x) := −(xh ˆS(x))′ = π
+4csch2(πx/2) (πx coth(πx/2) − 1)
+is positive and also symmetric we infer by Proposition 1 that ˆS ∈ L1 (Urbanik
+class). Moreover, g ˆS(x) is the density of the Lévy measure of the BDRV
+Y ˆS(1) = [0, 0, g ˆS].
+On the other hand ,
+t(log ψ ˆS(t))′ = t(1 − t coth(t))′ = t2csch2(t) − t coth(t)
+which together with (5) gives the part (ii) of Lemma 1.
+4
+
+Next, using Wolframalpha, we have that
+r ˆS(x) := −(xg ˆS(x))′ = π
+8 csch2(πx/2)
+[2π2x2 coth2(πx/2) + π2x2csch2(πx/2) − 6πx coth(πx/2) + 2] ≥ 0,
+(6)
+is non-negative function therefore, by the part (c) of Proposition 1, we get
+that ˆS ∈ L2, (Urbanik class).
+Finally, since
+exp(t(t2csch2(t) − t coth(t))′) = exp(3t2csch2(t) − t coth(t)(2t2csch2(t) + 1))
+from (5) we get the part (iii) in Lemma 1.
+NOTE 1: In Jurek (2022a), Theorem 1(a) is proved that ˆS ∈ L2 \ L3.
+So ˆS ∈ L2 and that fact was used in the proof of Lemma 1.
+b). Hyperbolic-cosine function.
+Similarly as in the section a) for the hyperbolic-sine functions, we have
+Lemma 2. From the selfdecomposability of the hyperbolic-cosine character-
+istic function φ ˆC =
+1
+cosh(t) we get the following Lévy exponents:
+(i)
+� ∞
+0
+(cos(tx) − 1)π
+2
+cosh(πx/2)
+sinh2(πx/2)dx = −t tanh(t);
+(ii)
+� ∞
+0
+(cos(tx)−1)π
+4 csch(πx
+2 )[πx coth2(πx
+2 )−2 coth(πx
+2 )+πxcsch2(πx
+2 )]dx
+= −t tanh(t) − t2sech2(t);
+(iii)
+� ∞
+0
+(cos(tx) − 1)π
+8csch(πx/2)[(πx)2 coth3(πx/2)
++coth(πx/2)(5(πx)2csch2(πx/2)+4)−6πx coth2(πx/2)−6πxcsch2(πx/2)]dx
+= −t tanh(t) − t2(3 − 2t tanh t)sech2(t).
+Proof. For the hyperbolic-cosine variable ˆC we have its characteristic func-
+tion
+φ ˆC(t) =
+1
+cosh t ∈ L0; k ˆC(x) =
+e−π|x|/2
+|x|(1 − e−π|x|) =
+1
+2|x| sinh(π|x|/2);
+that is, ˆC = [0, 0, k ˆC] ∈ L0, (by the condition (c) in Proposition 1), with
+Lévy spectral measure M ˆC(dx) := k ˆC(x)dx; cf. Jurek (1996), Pitman-Yor
+(2003) or Jurek - Yor (2004).
+5
+
+Hence for h ˆC(x) := (−xk ˆC(x))′ = π
+4
+cosh(πx/2)
+sinh2(πx/2) is non-negative and sym-
+metric density of Lévy measure of Y ˆC(1) in (2). Thus in (4) we get b = 0,
+that is, Y ˆC(1) = [0, 0, h ˆC(x)].
+On the other hand, by (3), and (5)
+ψ ˆC(t) := exp(t(log(φ ˆC(t))′) = exp(−t tanh(t)) = [0, 0, h ˆC(x)]
+= exp
+� ∞
+0
+(cos(tx) − 1)(π
+2
+cosh(πx/2)
+sinh2(πx/2))dx,
+which gives (i) of Lemma 2.
+As Y ˆC(1) ∈ L0 (is selfdecomposable) we can repeat the procedure from
+the previous step. Since, by WolframAlpha,
+g ˆC(x) := (−xh ˆC(x))′ = π
+8 csch(πx
+2 )[πx coth2(πx
+2 )
+− 2 coth(πx
+2 ) + πxcsch2(πx
+2 )],
+is a positive and symmetric density of a Lévy measure of BDRV of Y ˆC(1) in
+(2).
+On the other hand, by(3), (t(−t tanh(t))′)) = −t tanh(t)−t2sech2(t)), which
+together with the formula for g ˆC(x) proves the equality (ii) in Lemma 2.
+Again, as above, by WolframAlpha,
+(−xg ˆC(x))′ = π
+8 csch(πx/2)[(πx)2 coth3(πx/2) + coth(πx/2)
+(5(πx)2csch2(πx/2) + 4) − 6πx coth2(πx/2) − 6πxcsch2(πx/2)],
+is positive and symmetric density of Lévy measure we have that [0, 0, g ˆC(x)] ∈
+L0, by Proposition 1 c). On the other one, by (3),
+− t(t tanh(t) + t2sech2(t))′ = −t tanh(t) − t2(3 − 2t tanh t)sech2(t),
+which, in view of (5), gives the identity (iii) in Lemma 2.
+NOTE 2: In Jurek (2022a), Theorem 1(a) is proved that ˆC ∈ L2 \ L3.
+So ˆC ∈ L2 and that fact was used in the proof of Lemma 1.
+c). Hyperbolic-tangent function.
+For the hyperbolic tangent variable ˆT = [0, 0, kT(x) where the character-
+istic function and Lévy measure density are:
+φ ˆT(t) = tanh(t)/ and
+k ˆT(x) =
+1
+2|x|
+e−π|x|/4
+cosh(π|x|/4).
+6
+
+respectively. By (e) in Proposition 1 we get that ˆT ∈ L0. Then its BDRV
+YT(1) = [0, 0, hT(x)], where
+h ˆT (t) = (−xk ˆT )′ = π
+8
+1
+cosh2(πx/4);
+ψ ˆT (t) = exp(t(log φ ˆT(t))′) = exp
+�
+2t
+sinh(2t) − 1
+�
+,
+cf. fromula (2) in Proposition 1 (a).
+Hence we get
+Corollary 1. (i) From the selfdecomposability of hyperbolic tangent ˆT we get
+equality
+�
+R\{(0)}
+(cos(tx) − 1)π
+8
+1
+cosh2(πx/4)dx =
+2t
+sinh(2t) − 1; t ∈ R.
+(ii) The characteristic function ψT (t) represents a compound Poisson dis-
+tribution therefore T /∈ L1 Urbanik class.
+For the part (i), also cf. Jurek-Yor (2004), Proposition 1 with Corollary
+1 and the equality (10). Since Lévy measures of class L0 are infinite by (4),
+we get the part (ii) of corollary.
+Remark 2.
+Formulas Lemma 1(i), Lemma 2(i) and the above had already
+appeared in Jurek-Yor (2004). They are added here for the completeness of
+this presentation.
+2.
+Characteristic functions expressed via the Euler’s gamma
+function.
+(a). The log-gamma distribution.
+Log-gamma variables are just the logarithms of the gamma γα,λ variables
+with the parameters λ > 0 (scale) and α > 0 (shape).They are selfdecom-
+posable with characteristic functions
+φlog γα,λ(t) = e−it log(λ) Γ(α + it)
+Γ(α)
+= exp[it(Ψ(0)(α) − log λ) +
+� 0
+−∞
+(eitx − 1 − itx)
+eαx
+|x|(1 − ex)dx],
+klog γα,1(x) :=
+eαx
+|x|(1 − ex)1(−∞,0)(x),
+(7)
+7
+
+where Ψ(0)(z) := d log Γ(z)/dz denotes the digamma function; cf.
+Jurek
+(1997), p. 98 or Jurek (2022), p. 110 ( a comment below the formula (16)).
+And because log γα,λ have finite second moment ( cf. Corollary 2, in Jurek
+(2022)) the kernel under the integrand is from Kolmogorov’s formula; cf.
+Remark 1 (i).
+For the random variable Ylog γα,1(1), in the random integral representation
+(2), we have
+hlog γα,1(x) = (−xklog γα,1(x))′ = eαx(α(1 − ex) + ex)
+(1 − ex)2
+1(−∞,0)(x),
+b = Ψ(0)(α) − log λ and s2 = 0. Hence
+ψlog γα,λ(t) = exp[it(Ψ(0)(α)−log λ)+
+� 0
+−∞
+(eitx−1−itx)eαx(α(1 − ex) + ex)
+(1 − ex)2
+]dx.
+On the other hand, from (3), we get
+ψlog γα,λ(t) = exp(t(log φlog γα,λ(t))′ = exp(it(Ψ(0)(α + it) − log λ).
+All in all, from the identity (5) (taking the Kolmogorov’s kernel) we infer the
+identity
+Corollary 2. From the selfdecomposability property of the log-gamma vari-
+ables, for α > 0, β > 0 and t ∈ R we have
+(1)
+� 0
+−∞(eitx − 1 − itx)eαx α(1−ex)+ex
+(1−ex)2
+]dx = it(Ψ(0)(α + it) − Ψ(0)(α)).
+(2)
+� ∞
+0 (e−itx − 1 + itx)e−βx β(1−e−x)+e−x
+(1−e−x)2
+]dx = it(Ψ(0)(β − it) − Ψ(0)(β)).
+(b). Logistic distribution.
+Let lα denote the logistic distribution; cf. Ushakov (1999), p. 298, Feller
+(1966), p.52, Jurek (2021), p. 101. Then (by (7) we have
+φlα(t) = |Γ(α + it/π)
+Γ(α)
+|2 = φ1/π log γα,1(t) φ1/π log γα,1(−t)
+= exp
+� ∞
+−∞
+(cos(tx) − 1)klα(x)dx;
+with klα(x) = 1
+|x|
+e−απ|x|
+1 − e−π|x|;
+(8)
+and by Proposition 1 (c) we get that lα ∈ L0.
+Furthermore, from
+hlα(x) := (−xklα(x))′
+= π
+4
+1
+sinh2(π|x|/2)e−(α−1)π|x|{α(1 − e−π|x|) + e−π|x|} > 0.
+(9)
+8
+
+we conclude that [0, 0, hlα(x)] is the background driving variable Ylα(1) in the
+integral representation (2).
+On the other hand from (3) we get
+ψlα(t) = exp(t(log(φlα(t)))′) = exp(t/π(iΨ(0)(α + it/π) − iΨ(0)(α − it/π)))
+= exp(t/π[iΨ(0)(α + it/π) + (iΨ(0)(α + it/π))]) = 2t
+π ℜ[iΨ(0)(it/π + α)].
+All in all we get
+Corollary 3. From the selfdecomposability property of the logistic distribu-
+tion lα, α > 0, for t ∈ R, we have
+� ∞
+0
+(cos(tx) − 1) π
+2
+e−(α−1)π|x|(α + (1 − α)e−π|x|)
+sinh2(π|x|/2)
+dx = 2t
+π ℜ[iΨ(0)(it/π + α)].
+c). The generalized z-distribution.
+For positive parameters a, b1, b2, d and m ∈ R the generalized z-distribution
+GZ ≡ GZ(a, b1, b2, d, m) is given by its characteristic function
+φGZ(t) :=
+�B(b1 + iat
+2π , b2 − iat
+2π )
+B(b1, b2)
+�2d eimt =
+�Γ(b1 + iat
+2π )
+Γ(b1)
+Γ(b2 − iat
+2π )
+Γ(b2)
+�2deimt,
+where B(z1, z2) denotes the beta-function; cf. Schoutens (2003) , p. 64, or
+Ushakov (1999) , p. 309.
+(For a particular choices of parameters we get Fisher z-distribution; cf.
+Jurek (2021), the section 3.14, p.105.)
+Let us note that the characteristic function φGZ(t) can be expressed via
+characteristic functions of log-gamma variables. Namely, as we have
+φGZ(t) =
+�
+φlog γb1,1(at/(2π)) φ− log γb2,1(at/(2π))
+�2deimt
+=
+�
+φ(a/2π) log γb1,1(t) φ−(a/2π) log γb2,1(t)
+�2deimt t ∈ R;
+(10)
+cf. the section (b), on log-gamma variables, above.
+Below we will assume a = 2π, d = 1/2 and m = 0 and denote
+˜
+GZ ≡
+GZ(2π, b1, b2, 1/2, 0).
+Hence by (7) and section (a) on the log-gamma variables, the
+˜
+GZ distri-
+bution has Lévy (spectral) measure ν ˜
+GZ(dx) := k ˜
+GZ(x)dx where the density
+is of the from
+k ˜
+GZ(x) =
+eb1x
+|x|(1 − ex)1(−∞,0)(x) +
+e−b2x
+x(1 − e−x)1(0,∞)(x)].
+(11)
+9
+
+Hence,
+h ˜
+GZ(x) := (−xk ˜
+GZ(x))′ = [(
+eb1x
+(1 − ex))′1(−∞,0)(x) − (
+e−b2x
+(1 − e−x))′1(0,∞)(x)]
+= eb1x b1(1 − ex) + ex
+(1 − ex)2
+1(−∞,0)(x)
++ e−b2xb2(1 − e−x) + e−x
+(1 − e−x)2
+1(0,∞)(x) > 0;
+(12)
+is the density of Lévy measure of the BDRV YGZ(1). From Corollary 2, in
+Jurek (2022), we have that log-gamma variables have finite second moment.
+Consequently, in Kolmogorov’s representation, we have
+YGZ(1) = [Ψ(0)(b1) + Ψ(0)(b2), 0, h ˜
+GZ(x)].
+On the other hand, from (5) we get that
+ψ ˜
+GZ(t) := E[exp(itY ˜
+GZ(1)] = exp[t(log φ ˜
+GZ(t))′]
+= exp[t
+�
+log Γ(b1 + it) + log Γ(b2 − it)
+�′]
+exp[it(Ψ(0)(b1 + it) − Ψ(0)(b2 − it)))]
+(13)
+where Ψ(0)(z) := dLogΓ(z)/dz is the digamma function. Hence by (6) we
+infer the following:
+Corollary 4. From the selfdecomposability of the generalized z-distribution
+we have the identity
+�
+R\{(0}
+(eitx − 1 − itx)(eb1x b1(1 − ex) + ex
+(1 − ex)2
+1(−∞,0)(x)
++ e−b2x b2(1 − e−x) + e−x
+(1 − e−x)2
+1(0,∞))dx
+= it[(Ψ(0)(b1 + it) − Ψ(0)(b1)) − (Ψ(0)(b2 − it) − Ψ(0)(b2))],
+for all t ∈ R.
+3. Meixner and Feller-Spitzer ( or Bessel) distributions.
+(a). The Meixner M distribution.
+For the parameters a > 0, −π < b < π, d > 0, m ∈ R, the probability
+density function f(x; a, b, m, d), x ∈ R,
+f(x; a, b, m, d) := (2 cos(b/2))2d
+2aπΓ(2d)
+exp(b(x − m)
+a
+)|Γ(d + i(x − m)
+a
+)|2,
+(14)
+10
+
+is called Meixner distribution. It has all moments finite and the characteristic
+functions is
+φM(t) ≡ φM(a,b,d,m)(t) = ( cos(b/2)
+cosh( at−ib
+2 ))2d exp(imt);
+(15)
+cf. Schoutens (2003), p. 62-63. From above we infer that M are infinitely
+divisible. Furthermore, for our purposes, without the loss of generality we
+assume that m = 0 and d = 1/2.
+Being infinitely divisible, with finite second moment, Meixner distribu-
+tions admit the following representations
+φM(t) = exp(itγ +
+� ∞
+−∞
+(eitx − 1 − itx 1(|x|≤1)(x)kM(dx))), where
+γ := a
+2 tan(b/2) −
+� ∞
+1
+sinh(bx/a)
+sinh(πx/a)dx, kM(dx) :=
+ebx/a
+2 x sinh(πx/a)dx;
+(16)
+cf. Schoutens (2003), p. 153 for the shift parameter γ and p. 154 for the
+density kM(x) of Lévy measure.
+This can be extended to Kolmogorov’s type kernel as follows:
+φM(t) = exp(it˜γ +
+� ∞
+−∞
+(eitx − 1 − itx)kM(dx))), where
+˜γ := γ +
+� ∞
+−∞
+(1 − 1|x|<1(x))
+ebx/a
+2 sinh(πx/a)dx = (a/2) tan(b/2)
+−
+� ∞
+1
+ebx/a − e−bx/a
+2 sinh(πx/a) dx +
+�
+(|x|≥1)
+ebx/a
+2 sinh(πx/a)dx = a/2 tan(b/2),
+i.e., ˜γ = a
+2 tan(b/2) and M = [a/2 tan(b/2), 0, kM(x)] in Kolmogorov’s repre-
+sentation. Hence, in the random integral representation (2), we have
+YM(1) = [a
+2 tan(b/2), 0, hM(x)],
+hM(x) = (−xkM(x))′ = 1
+4aebx/a[eπx/a(π − b) + e−πx/a(b + π)]
+sinh2(πx/a)
+,
+is positive as |b| < π. Hence we infer that Meixner M ∈ L0.
+On the other hand, from (5)
+E[exp(itYM(1))] = ψM(t) := exp[t(log φM(t))′] = exp[−t(log cosh((at−ib)/2))′]
+= exp[−at/2 tanh((at − ib)/2)].
+All in all we get
+11
+
+Corollary 5. From the selfdecomposability property of Meixner M(a, b, 1/2, 0)
+variable with constants a > 0, |b| < π we have the identity
+�
+R\{(0)}
+(eitx − 1 − itx) 1
+4aebx/aeπx/a(π − b) + e−πx/a(b + π)
+sinh2(πx/a)
+dx
+= −iat
+2 tan(b/2) − at
+2 tanh(at − ib
+2
+) = −iat
+2 tan(b/2) − at
+2
+sinh(at) − i sin(b)
+cosh(at) + cos(b)
+= −at
+2
+sinh(at)
+cosh(at) + cos(b) − i at
+2 (tan(b/2) +
+sin(b)
+cosh(at) + cos(b)).
+Remark 3. As an addition to the above :
+(i) tanh[(at − ib)/2] = at
+2
+sinh(at)−i sin(b)
+cosh(at)+cos(b) , for real a, b, t.
+(ii) In particular, for the real and imaginary parts we have
+�
+R
+(cos(tx)−1) 1
+4aebx/aeπx/a(π − b) + e−πx/a(π + b)
+sinh2(πx/a)
+dx = −at
+2
+sinh(at)
+cosh(at) + cos(b),
+and
+�
+R
+(sin(tx) − tx)) 1
+4aebx/aeπx/a(π − b) + e−πx/a(π + b)
+sinh2(πx/a)
+dx
+= − at
+2 (tan(b/2) +
+sin(b)
+cosh(at) + cos(b)).
+(b). The Feller-Spitzer FS distribution.
+For ν > 0 and the modified Bessel function Iν(x) we define the probability
+density function pν(x) := e−x νIν(x)
+x
+, 0 < x < ∞, which has the Feller-Spitzer
+characteristic function
+φF S(ν)(t) :=
+� ∞
+0
+eitxpν(x)dx = [1 − it −
+�
+(1 − it)2 − 1]ν, t ∈ R,
+(17)
+cf. Feller p. 414 and p.476 or Ushakov p.283; it is called there Bessel distri-
+bution. For further generalizations of FS distributions cf. Vinogradov and
+Paris (2021).
+From above we get that FS variable is in ID class and it has Lévy-
+Khintchine representation
+φF S(ν)(t) = exp ν[ita +
+� ∞
+0
+(eitx − 1 −
+itx
+1 + x2)kF S(x)dx],
+where kF S(x) := e−xI0(x)
+x
+1(0,∞)(x); a :=
+� ∞
+0
+e−xI0(x)
+1 + x2 dx
+(18)
+12
+
+cf. Jurek (2021) p. 103 and FS = [a, 0, kF S] ∈ L0. Note that the above Lévy
+measure density kF S(x) coincides with the density τ1(x), in Vinogradov and
+Paris (2021), Definition 2..
+For the BDRV YF S(1) = [b, 0, hF S(x)] we get that
+hF S(x) := (−xkF S(x))′ = (−e−xI0(x))′ = e−x(I0(x) − I1(x)) > 0,
+is positive (by Jones (1968) or Paris and Vinogradov (2021) and the inequality
+(110)) and integrable to 1. For the shift parameter, by (4), we have
+b = a +
+�
+R
+(
+x
+1 + x2 − arctan(x))hF S(x)dx
+=
+� ∞
+0
+e−xI0(x)
+1 + x2 dx +
+� ∞
+0
+(
+x
+1 + x2 − arctan(x))e−x(I0(x) − I1(x))dx
+=
+� ∞
+0
+e−xI0(x)(arctan(x))′dx −
+� ∞
+0
+arctan(x))(−e−xI0(x))′dx
++
+� ∞
+0
+x
+1 + x2e−x(I0(x) − I1(x))dx = e−xI0(x) arctan(x)|x=∞
+x=0
++
+� ∞
+0
+x
+1 + x2e−x(I0(x) − I1(x))dx =
+� ∞
+0
+x
+1 + x2e−x(I0(x) − I1(x))dx,
+(19)
+because limx→0 e−xI0(x) arctan(x) = limx→∞ e−xI0(x) arctan(x) = 0.
+Consequently, from above and (4) we have
+ψF S(t) = exp(it(
+� ∞
+0
+x
+1 + x2e−x(I0(x) − I1(x))dx)
++
+� ∞
+0
+(eitx − 1 − it
+x
+1 + x2)e−x(I0(x) − I1(x))dx)
+=
+� ∞
+0
+(eitx − 1)e−x(I0(x) − I1(x))dx,
+which is the characteristic function of the compound Poisson distribution,
+thus ψF S(t) /∈ L0, and FS ∈ L0 \ L1.
+On the other hand, by (5) we have that
+ψF S(t) = exp(t(log(φF S(t)))′)
+= exp(t(log(1 − it −
+�
+(1 − it)2 − 1))′) = exp(
+it
+�
+−t(t + 2i)
+which gives
+13
+
+Corollary 6. From the selfdecomposability property of the Feller-Spitzer dis-
+tribution we have the identity
+� ∞
+0
+(eitx − 1)µ(dx) =
+it
+�
+−t(t + 2i)
+, t ∈ R.
+where µ(dx) := e−x(I0(x) − I1(x))1(0,∞)(x)dx is a probability measure. Both
+the right and the left expressions are logarithms of infinitely divisible charac-
+teristic functions (Poisson compound distributions) .
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+Z. J. Jurek (2021), On background driving distribution functions (BDDF)
+for some selfdecomposable variables, Mathematica Applicanda , vol 49(2),
+p.85-109.
+Z. J. Jurek (2022), Background driving distribution functions and series
+representations for log-gamma self-decomposable random variables, Theor.
+Probab. Appl. vol.67, no 1, pp. 105-117.
+Z. J. Jurek (2023), Which Urbanik class Lk, do the hyperbolic and the
+generalized logistic characteristic functions belong to?, math. arXiv:2211.17064v1
+[math.PR] 30Nov 2022.
+Z. J. Jurek and J.D. Mason (1993), Operator-limit distributions in prob-
+ability theory, John Wiley & Sons, Inc., New York
+Z.J. Jurek and W. Vervaat (1983), An integral representation for selfde-
+composable Banach space valued random variables, Z. Wahrscheinlichkeits-
+theorie und verw. Gebiete, vol.62, pp.51-62.
+Z. J. Jurek and Yor (2004), Selfdecomposable laws associated with hy-
+perbolic functions, Probab. Math. Stat. vol.24 Fasc. 1, pp. 181-190.
+M. Loeve (1963), Probability theory, 3rd Edition, D.Van Nostrand Com-
+pany, Inc. Princeton.
+J. Pitman and M. Yor (2003), Infinitely divisible laws associated with
+hyperbolic functions, Canad. J. Math. vol. 55 (2), pp. 292-330.
+W. Schoutens, (2003), Lévy processes in finance. Pricing financial deriva-
+tives, J. Wiley and Sons, England.
+F. Steutel and K. van Harn (2004), Infinite divisibility of probability dis-
+tributions on the real line , Marcel Dekker Inc.
+J. Wiley and Sons, England.
+15
+
+K. Urbanik (1972), Slowly varying sequences of random variables, Bull.
+de L’Acad. Polon. Sciences; Ser. Math. Astr. Phys. 28:2, pp. 679-682.
+K. Urbanik (1973), Limit laws for sequences of normed sums satisfying
+some stability conditions; Proc. 3rd Internstional Symp. Multivar. Analysis,
+Wright State University, Dayton Ohio, USA, June 19-24, 1972; Academic
+Press 1973.
+[ Also on: www.math.uni.wroc.pl/ zjjurek/urb-limitLawsOhio1973.pdf]
+V. V. Vinogradov and R. B. Paris (2021), On two extensions of the canon-
+ical Feller-Spitzer distribution, J. Stat. Distributions and Appl., (2021)8:3,
+https://doi.org/101186/s40488-021-0013-4
+16
+
diff --git a/IdFJT4oBgHgl3EQfvS1K/content/tmp_files/load_file.txt b/IdFJT4oBgHgl3EQfvS1K/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf,len=478
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='11625v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='PR]  27 Jan 2023  January 27, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Some definite integrals arising from selfdecomposable  characteristic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Zbigniew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Jurek  University of Wrocław, Wrocław, Poland;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  zjjurek@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='wroc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='pl and  www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='wroc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='pl/∼zjjurek  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  In the probability theory selfdecomposable, or class L0 dis-  tributions play an important role as they are limiting distributions of nor-  malized partial sums of sequences of independent, not necessarily identically  distributed, random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' The class L0 is quite large and includes many  known classical distributions and statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' For this note the most important  feature of the selfdecomposable variables are their random integral represen-  tation with respect to Lévy process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' From those random integral represen-  tation we get equality of logarithms of some characteristic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' These  allows us to get formulas for some definite integrals, some of them probably  were unknown before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  2020 Mathematics Subject Classifications: Primary: 60E07,60E10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Sec-  ondary: 60G51,60H05  Key words and phrases: infinite divisibility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' selfdecomposability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Lévy  process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Urbanik classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' random integral;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' hyperbolic characteristic functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  log-gamma distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' generalized logistic distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Meixner distribu-  tions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Feller-Spitzer distributions  The class L of selfdecomposable distributions appears in the classical prob-  ability theory as a class of limiting distributions obtained from infinitesimal  triangular arrays arising from sequences of independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' In  the case of independent and identically distributed variables we get the class  S of stable distributions, which contains the normal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' On the  other hand, when one considers arbitrary infinitesimal triangular arrays we  end-up with the class ID of infinitely divisible distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Thus we have  the inclusions S ⊊ L ⊊ ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Infinitely divisible random variables X ∈ ID are uniquely determined by  their characteristic functions  φX(t) := E[exp(itX)] = exp  �  ita− 1  2σ2t2+  �  R\\{(0)}  (eitx−1−  itx  1 + x2)MX(dx)  �  ,  where a ∈ R, σ2 ≥ 0 (Gaussian part),  �  R\\{(0)} min(1, x2)M(dx) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  1  The measure M is called the Lévy measure of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' The above integral  formula is refered to as the Lévy-Khintchine representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' The triple a, σ2  and M is uniquely determined by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  The classic references to those topics are: Feller (1966), Gnedenko and Kolo-  mogorov (1954) or Loeve (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  For infinite divisibility for Banach space valued variables we refer to  Araujo and Gine (1980), Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  The class L is quite large and includes many classical distributions in  probability and statistics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' for examples cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Jurek (1997) and (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' More-  over, during last decades the selfdecompsability appeared in mathematical  finance (Carr, Geman , Madan and Yor (2007), a book by Schoutens (2003))  and in statistical physics ( deConinck (1984), deConinck and Jurek (2000),  and Jurek (2001)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  The aim of this note, (after the preliminary Proposition 1, in the Intro-  duction), is to show how some definite integrals can be computed from the  selfdecomposable characteristic functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Lemmas 1 , 2 and Corollary 1  (for the hyperbolic characteristic functions) and Corollaries 2-6, in Sections  2 and 3 (for the characteristic function expressed via Euler’s Euler’s gamma  function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' in particular, for the Meixner or Feller-Spitzer distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  A random variable X is called selfdecomoposable(in symbols: X ∈ L0), if  ∀ (t > 0) ∃ (Xt ∈ ID independent of X) X  d= e−tX + Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (1)  And inductively, for k ≥ 1, we define, decreasing sequence of the Urbanik  classes Lk, as follows:  X ∈ Lk iff, in (1), Xt ∈ Lk−1, for all t > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Urbanik (1972),(1973);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' (1a)  The "remainders" (Xt, t > 0) in (1) satisfy the following cocycle equation  Xt+s  d= e−tXs + Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' It allows to construct a cadlag Lévy process and to infer  that the selfdecomposability of X is equivalent to the claim that there exits  unique, in distribution, a Lévy process (YX(t), t ≥ 0), such that  X ∈ L0 iff X  d=  � ∞  0  e−tdYX(t),  and E[log(1 + |YX(1)|)] < ∞,  (2)  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Jurek and Vervaat (1983),Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 252-253 or Jurek and Mason  (1993), Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 124-126 and Jurek (1982) for a Banach space  valued random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  To the random variable YX(1) we refer to as the background driving ran-  dom variable of X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' is short: BDRV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  2  For this note crucial are the following facts listed in :  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' (a) Let X and YX(1) be a selfdecopmosable variable and its  BDRV, respectively, in the random integral representation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Then for the  characteristic functions φX(t) := E[exp(itX)] and ψX(t) := E[exp(itYX(1))]  we have  ψX(t) = exp(t(log φX(t))′), t ̸= 0, ψX(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (3)  (b) Equivalently, if [a, σ2, MX] is the triple in the Lévy-Khintchine formula  for X and similarly [b, s2, NX] is the triple for YX(1) then  MX(B) =  � ∞  0  NX(etB)dt,  Borel B ⊂ R \\ {(0)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' MX(R \\ {(0)}) = ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  MX(dx) = kX(x)dx,  NX(dx) = hX(x)dx,  hX(x) = (−xkX(x))′ ≥ 0,  s2 = 2σ2,  b = a +  �  R\\{((0)}  x  1 + x2 − arctan(x))hX(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (4)  (c) [The condition (−xkX(x))′ ≥ 0 is equivalent to the condition that the  function xkX(x) is not increasing of both half-lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' This means that  [a, σ2, kX(x)] is selfdecomposable variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=']  (d) Hence, by comparing (3) and [b, s2, NX] in (4), we conclude the fol-  lowing identity for t ∈ R,  itb − 1  2s2t2 +  �  R\\{((0)}  (eitx − 1 −  itx  1 + x2)hX(x)dx = t(log φX(t))′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (5)  For the proof of (3) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Jurek (2001b), Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' For proof of (4)  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Jurek and Mason (1993), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='120, formulae (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='9)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='11) for Q = I, or  Jurek (1996), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Finally for the equivalences in the square bracket [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='],  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Jurek and Mason (1993), Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 94 or Steutel and Van Haarn  (2004), Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  The above equality (5) is the key identity for all the definite integrals in  this note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' (i) If selfdecomposable X has finite second moment then in (5)  we will use the kernel exp(itx)−1−itx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Bilingsley (1986), Theorem 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='1,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Moreover, in that case, in (5), b = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (ii) From (4) we see that X ∈ L0 has Gaussian part if and only if its  BDRV YX(1) has a Gaussian part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (iii) For the real characteristic functions, as in Section 1 below, we get  simpler formulas as we can discard the imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' The hyperbolic characteristic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  For the information about the hyperbolic-type characteristic function we  refer to Pitaman and Yor (2003) or Jurek and Yor (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Also this section  may be viewed as a complement to the recent preprint Jurek (2022 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Hyperbolic-sine function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' From the selfdecomposability of the hyperbolic-sine characteristic  function φ ˆS(t) =  t  sinh(t) we get the following formulas (Lévy exponents):  (i)  � ∞  0  (cos(tx) − 1)(π/2)csch2(πx/2)dx = 1 − t coth(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (ii)  � ∞  0  (cos(tx)−1)π  2 csch2(πx  2 ) (πx coth(πx  2 )−1)dx = t2csch2(t)−t coth(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (iii)  � ∞  0  (cos(tx) − 1)π  4csch2(πx/2)  �  2π2x2 coth2(πx/2) + π2x2csch2(πx/2)  − 6πx coth(πx/2) + 2  �  dx = 3t2csch2(t) − t coth(t)(2t2csch2(t) + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Since  φ ˆS(t) =  t  sinh t = [0, 0, k ˆS(x)] ∈ ID,  k ˆS(x) = 1  |x|  1  eπ|x| − 1, x ∈ R \\ {(0)},  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=', in this example in a = 0 and σ2 = 0 for the random variable X := ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Since h ˆS(x) := (−xk ˆS(x))′ = (π/4)csch2(πx/2), x ∈ R \\ {(0)}, is positive  thus by part (c) of Proposition 1 we get that ˆS ∈ L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Its BDRV Y ˆS(1) =  [0, 0, h ˆS(x)], as in view of symmetry of h ˆS(x) we have b = 0 and s2 = 0 in  (4) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  On the other hand, as ψ ˆS(t) = exp(t(log φ ˆS(t))′) = exp(1 − t coth(t), from  (6) we get the part (i) of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Since  g ˆS(x) := −(xh ˆS(x))′ = π  4csch2(πx/2) (πx coth(πx/2) − 1)  is positive and also symmetric we infer by Proposition 1 that ˆS ∈ L1 (Urbanik  class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Moreover, g ˆS(x) is the density of the Lévy measure of the BDRV  Y ˆS(1) = [0, 0, g ˆS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  On the other hand ,  t(log ψ ˆS(t))′ = t(1 − t coth(t))′ = t2csch2(t) − t coth(t)  which together with (5) gives the part (ii) of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  4  Next, using Wolframalpha, we have that  r ˆS(x) := −(xg ˆS(x))′ = π  8 csch2(πx/2)  [2π2x2 coth2(πx/2) + π2x2csch2(πx/2) − 6πx coth(πx/2) + 2] ≥ 0,  (6)  is non-negative function therefore, by the part (c) of Proposition 1, we get  that ˆS ∈ L2, (Urbanik class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Finally, since  exp(t(t2csch2(t) − t coth(t))′) = exp(3t2csch2(t) − t coth(t)(2t2csch2(t) + 1))  from (5) we get the part (iii) in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  NOTE 1: In Jurek (2022a), Theorem 1(a) is proved that ˆS ∈ L2 \\ L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  So ˆS ∈ L2 and that fact was used in the proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Hyperbolic-cosine function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Similarly as in the section a) for the hyperbolic-sine functions, we have  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' From the selfdecomposability of the hyperbolic-cosine character-  istic function φ ˆC =  1  cosh(t) we get the following Lévy exponents:  (i)  � ∞  0  (cos(tx) − 1)π  2  cosh(πx/2)  sinh2(πx/2)dx = −t tanh(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (ii)  � ∞  0  (cos(tx)−1)π  4 csch(πx  2 )[πx coth2(πx  2 )−2 coth(πx  2 )+πxcsch2(πx  2 )]dx  = −t tanh(t) − t2sech2(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (iii)  � ∞  0  (cos(tx) − 1)π  8csch(πx/2)[(πx)2 coth3(πx/2)  +coth(πx/2)(5(πx)2csch2(πx/2)+4)−6πx coth2(πx/2)−6πxcsch2(πx/2)]dx  = −t tanh(t) − t2(3 − 2t tanh t)sech2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' For the hyperbolic-cosine variable ˆC we have its characteristic func-  tion  φ ˆC(t) =  1  cosh t ∈ L0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' k ˆC(x) =  e−π|x|/2  |x|(1 − e−π|x|) =  1  2|x| sinh(π|x|/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  that is, ˆC = [0, 0, k ˆC] ∈ L0, (by the condition (c) in Proposition 1), with  Lévy spectral measure M ˆC(dx) := k ˆC(x)dx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Jurek (1996), Pitman-Yor  (2003) or Jurek - Yor (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  5  Hence for h ˆC(x) := (−xk ˆC(x))′ = π  4  cosh(πx/2)  sinh2(πx/2) is non-negative and sym-  metric density of Lévy measure of Y ˆC(1) in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Thus in (4) we get b = 0,  that is, Y ˆC(1) = [0, 0, h ˆC(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  On the other hand, by (3), and (5)  ψ ˆC(t) := exp(t(log(φ ˆC(t))′) = exp(−t tanh(t)) = [0, 0, h ˆC(x)]  = exp  � ∞  0  (cos(tx) − 1)(π  2  cosh(πx/2)  sinh2(πx/2))dx,  which gives (i) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  As Y ˆC(1) ∈ L0 (is selfdecomposable) we can repeat the procedure from  the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Since, by WolframAlpha,  g ˆC(x) := (−xh ˆC(x))′ = π  8 csch(πx  2 )[πx coth2(πx  2 )  − 2 coth(πx  2 ) + πxcsch2(πx  2 )],  is a positive and symmetric density of a Lévy measure of BDRV of Y ˆC(1) in  (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  On the other hand, by(3), (t(−t tanh(t))′)) = −t tanh(t)−t2sech2(t)), which  together with the formula for g ˆC(x) proves the equality (ii) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Again, as above, by WolframAlpha,  (−xg ˆC(x))′ = π  8 csch(πx/2)[(πx)2 coth3(πx/2) + coth(πx/2)  (5(πx)2csch2(πx/2) + 4) − 6πx coth2(πx/2) − 6πxcsch2(πx/2)],  is positive and symmetric density of Lévy measure we have that [0, 0, g ˆC(x)] ∈  L0, by Proposition 1 c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' On the other one, by (3),  − t(t tanh(t) + t2sech2(t))′ = −t tanh(t) − t2(3 − 2t tanh t)sech2(t),  which, in view of (5), gives the identity (iii) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  NOTE 2: In Jurek (2022a), Theorem 1(a) is proved that ˆC ∈ L2 \\ L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  So ˆC ∈ L2 and that fact was used in the proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Hyperbolic-tangent function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  For the hyperbolic tangent variable ˆT = [0, 0, kT(x) where the character-  istic function and Lévy measure density are:  φ ˆT(t) = tanh(t)/ and  k ˆT(x) =  1  2|x|  e−π|x|/4  cosh(π|x|/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  6  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' By (e) in Proposition 1 we get that ˆT ∈ L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Then its BDRV  YT(1) = [0, 0, hT(x)], where  h ˆT (t) = (−xk ˆT )′ = π  8  1  cosh2(πx/4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  ψ ˆT (t) = exp(t(log φ ˆT(t))′) = exp  �  2t  sinh(2t) − 1  �  ,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' fromula (2) in Proposition 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Hence we get  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' (i) From the selfdecomposability of hyperbolic tangent ˆT we get  equality  �  R\\{(0)}  (cos(tx) − 1)π  8  1  cosh2(πx/4)dx =  2t  sinh(2t) − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (ii) The characteristic function ψT (t) represents a compound Poisson dis-  tribution therefore T /∈ L1 Urbanik class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  For the part (i), also cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Jurek-Yor (2004), Proposition 1 with Corollary  1 and the equality (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Since Lévy measures of class L0 are infinite by (4),  we get the part (ii) of corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Formulas Lemma 1(i), Lemma 2(i) and the above had already  appeared in Jurek-Yor (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' They are added here for the completeness of  this presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Characteristic functions expressed via the Euler’s gamma  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' The log-gamma distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Log-gamma variables are just the logarithms of the gamma γα,λ variables  with the parameters λ > 0 (scale) and α > 0 (shape).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='They are selfdecom-  posable with characteristic functions  φlog γα,λ(t) = e−it log(λ) Γ(α + it)  Γ(α)  = exp[it(Ψ(0)(α) − log λ) +  � 0  −∞  (eitx − 1 − itx)  eαx  |x|(1 − ex)dx],  klog γα,1(x) :=  eαx  |x|(1 − ex)1(−∞,0)(x),  (7)  7  where Ψ(0)(z) := d log Γ(z)/dz denotes the digamma function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Jurek  (1997), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 98 or Jurek (2022), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 110 ( a comment below the formula (16)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  And because log γα,λ have finite second moment ( cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Corollary 2, in Jurek  (2022)) the kernel under the integrand is from Kolmogorov’s formula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Remark 1 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  For the random variable Ylog γα,1(1), in the random integral representation  (2), we have  hlog γα,1(x) = (−xklog γα,1(x))′ = eαx(α(1 − ex) + ex)  (1 − ex)2  1(−∞,0)(x),  b = Ψ(0)(α) − log λ and s2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Hence  ψlog γα,λ(t) = exp[it(Ψ(0)(α)−log λ)+  � 0  −∞  (eitx−1−itx)eαx(α(1 − ex) + ex)  (1 − ex)2  ]dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  On the other hand, from (3), we get  ψlog γα,λ(t) = exp(t(log φlog γα,λ(t))′ = exp(it(Ψ(0)(α + it) − log λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  All in all, from the identity (5) (taking the Kolmogorov’s kernel) we infer the  identity  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' From the selfdecomposability property of the log-gamma vari-  ables, for α > 0, β > 0 and t ∈ R we have  (1)  � 0  −∞(eitx − 1 − itx)eαx α(1−ex)+ex  (1−ex)2  ]dx = it(Ψ(0)(α + it) − Ψ(0)(α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (2)  � ∞  0 (e−itx − 1 + itx)e−βx β(1−e−x)+e−x  (1−e−x)2  ]dx = it(Ψ(0)(β − it) − Ψ(0)(β)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Logistic distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Let lα denote the logistic distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Ushakov (1999), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 298, Feller  (1966), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='52, Jurek (2021), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Then (by (7) we have  φlα(t) = |Γ(α + it/π)  Γ(α)  |2 = φ1/π log γα,1(t) φ1/π log γα,1(−t)  = exp  � ∞  −∞  (cos(tx) − 1)klα(x)dx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  with klα(x) = 1  |x|  e−απ|x|  1 − e−π|x|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (8)  and by Proposition 1 (c) we get that lα ∈ L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Furthermore, from  hlα(x) := (−xklα(x))′  = π  4  1  sinh2(π|x|/2)e−(α−1)π|x|{α(1 − e−π|x|) + e−π|x|} > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (9)  8  we conclude that [0, 0, hlα(x)] is the background driving variable Ylα(1) in the  integral representation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  On the other hand from (3) we get  ψlα(t) = exp(t(log(φlα(t)))′) = exp(t/π(iΨ(0)(α + it/π) − iΨ(0)(α − it/π)))  = exp(t/π[iΨ(0)(α + it/π) + (iΨ(0)(α + it/π))]) = 2t  π ℜ[iΨ(0)(it/π + α)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  All in all we get  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' From the selfdecomposability property of the logistic distribu-  tion lα, α > 0, for t ∈ R, we have  � ∞  0  (cos(tx) − 1) π  2  e−(α−1)π|x|(α + (1 − α)e−π|x|)  sinh2(π|x|/2)  dx = 2t  π ℜ[iΨ(0)(it/π + α)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' The generalized z-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  For positive parameters a, b1, b2, d and m ∈ R the generalized z-distribution  GZ ≡ GZ(a, b1, b2, d, m) is given by its characteristic function  φGZ(t) :=  �B(b1 + iat  2π , b2 − iat  2π )  B(b1, b2)  �2d eimt =  �Γ(b1 + iat  2π )  Γ(b1)  Γ(b2 − iat  2π )  Γ(b2)  �2deimt,  where B(z1, z2) denotes the beta-function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Schoutens (2003) , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 64, or  Ushakov (1999) , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (For a particular choices of parameters we get Fisher z-distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Jurek (2021), the section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='14, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=')  Let us note that the characteristic function φGZ(t) can be expressed via  characteristic functions of log-gamma variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Namely, as we have  φGZ(t) =  �  φlog γb1,1(at/(2π)) φ− log γb2,1(at/(2π))  �2deimt  =  �  φ(a/2π) log γb1,1(t) φ−(a/2π) log γb2,1(t)  �2deimt t ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (10)  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' the section (b), on log-gamma variables, above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Below we will assume a = 2π, d = 1/2 and m = 0 and denote  ˜  GZ ≡  GZ(2π, b1, b2, 1/2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Hence by (7) and section (a) on the log-gamma variables, the  ˜  GZ distri-  bution has Lévy (spectral) measure ν ˜  GZ(dx) := k ˜  GZ(x)dx where the density  is of the from  k ˜  GZ(x) =  eb1x  |x|(1 − ex)1(−∞,0)(x) +  e−b2x  x(1 − e−x)1(0,∞)(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (11)  9  Hence,  h ˜  GZ(x) := (−xk ˜  GZ(x))′ = [(  eb1x  (1 − ex))′1(−∞,0)(x) − (  e−b2x  (1 − e−x))′1(0,∞)(x)]  = eb1x b1(1 − ex) + ex  (1 − ex)2  1(−∞,0)(x)  + e−b2xb2(1 − e−x) + e−x  (1 − e−x)2  1(0,∞)(x) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (12)  is the density of Lévy measure of the BDRV YGZ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' From Corollary 2, in  Jurek (2022), we have that log-gamma variables have finite second moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Consequently, in Kolmogorov’s representation, we have  YGZ(1) = [Ψ(0)(b1) + Ψ(0)(b2), 0, h ˜  GZ(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  On the other hand, from (5) we get that  ψ ˜  GZ(t) := E[exp(itY ˜  GZ(1)] = exp[t(log φ ˜  GZ(t))′]  = exp[t  �  log Γ(b1 + it) + log Γ(b2 − it)  �′]  exp[it(Ψ(0)(b1 + it) − Ψ(0)(b2 − it)))]  (13)  where Ψ(0)(z) := dLogΓ(z)/dz is the digamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Hence by (6) we  infer the following:  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' From the selfdecomposability of the generalized z-distribution  we have the identity  �  R\\{(0}  (eitx − 1 − itx)(eb1x b1(1 − ex) + ex  (1 − ex)2  1(−∞,0)(x)  + e−b2x b2(1 − e−x) + e−x  (1 − e−x)2  1(0,∞))dx  = it[(Ψ(0)(b1 + it) − Ψ(0)(b1)) − (Ψ(0)(b2 − it) − Ψ(0)(b2))],  for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Meixner and Feller-Spitzer ( or Bessel) distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' The Meixner M distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  For the parameters a > 0, −π < b < π, d > 0, m ∈ R, the probability  density function f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' a, b, m, d), x ∈ R,  f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' a, b, m, d) := (2 cos(b/2))2d  2aπΓ(2d)  exp(b(x − m)  a  )|Γ(d + i(x − m)  a  )|2,  (14)  10  is called Meixner distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' It has all moments finite and the characteristic  functions is  φM(t) ≡ φM(a,b,d,m)(t) = ( cos(b/2)  cosh( at−ib  2 ))2d exp(imt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (15)  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Schoutens (2003), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 62-63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' From above we infer that M are infinitely  divisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Furthermore, for our purposes, without the loss of generality we  assume that m = 0 and d = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Being infinitely divisible, with finite second moment, Meixner distribu-  tions admit the following representations  φM(t) = exp(itγ +  � ∞  −∞  (eitx − 1 − itx 1(|x|≤1)(x)kM(dx))), where  γ := a  2 tan(b/2) −  � ∞  1  sinh(bx/a)  sinh(πx/a)dx, kM(dx) :=  ebx/a  2 x sinh(πx/a)dx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (16)  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Schoutens (2003), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 153 for the shift parameter γ and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 154 for the  density kM(x) of Lévy measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  This can be extended to Kolmogorov’s type kernel as follows:  φM(t) = exp(it˜γ +  � ∞  −∞  (eitx − 1 − itx)kM(dx))), where  ˜γ := γ +  � ∞  −∞  (1 − 1|x|<1(x))  ebx/a  2 sinh(πx/a)dx = (a/2) tan(b/2)  −  � ∞  1  ebx/a − e−bx/a  2 sinh(πx/a) dx +  �  (|x|≥1)  ebx/a  2 sinh(πx/a)dx = a/2 tan(b/2),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=', ˜γ = a  2 tan(b/2) and M = [a/2 tan(b/2), 0, kM(x)] in Kolmogorov’s repre-  sentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Hence, in the random integral representation (2), we have  YM(1) = [a  2 tan(b/2), 0, hM(x)],  hM(x) = (−xkM(x))′ = 1  4aebx/a[eπx/a(π − b) + e−πx/a(b + π)]  sinh2(πx/a)  ,  is positive as |b| < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Hence we infer that Meixner M ∈ L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  On the other hand, from (5)  E[exp(itYM(1))] = ψM(t) := exp[t(log φM(t))′] = exp[−t(log cosh((at−ib)/2))′]  = exp[−at/2 tanh((at − ib)/2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  All in all we get  11  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' From the selfdecomposability property of Meixner M(a, b, 1/2, 0)  variable with constants a > 0, |b| < π we have the identity  �  R\\{(0)}  (eitx − 1 − itx) 1  4aebx/aeπx/a(π − b) + e−πx/a(b + π)  sinh2(πx/a)  dx  = −iat  2 tan(b/2) − at  2 tanh(at − ib  2  ) = −iat  2 tan(b/2) − at  2  sinh(at) − i sin(b)  cosh(at) + cos(b)  = −at  2  sinh(at)  cosh(at) + cos(b) − i at  2 (tan(b/2) +  sin(b)  cosh(at) + cos(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' As an addition to the above :  (i) tanh[(at − ib)/2] = at  2  sinh(at)−i sin(b)  cosh(at)+cos(b) , for real a, b, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (ii) In particular, for the real and imaginary parts we have  �  R  (cos(tx)−1) 1  4aebx/aeπx/a(π − b) + e−πx/a(π + b)  sinh2(πx/a)  dx = −at  2  sinh(at)  cosh(at) + cos(b),  and  �  R  (sin(tx) − tx)) 1  4aebx/aeπx/a(π − b) + e−πx/a(π + b)  sinh2(πx/a)  dx  = − at  2 (tan(b/2) +  sin(b)  cosh(at) + cos(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' The Feller-Spitzer FS distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  For ν > 0 and the modified Bessel function Iν(x) we define the probability  density function pν(x) := e−x νIν(x)  x  , 0 < x < ∞, which has the Feller-Spitzer  characteristic function  φF S(ν)(t) :=  � ∞  0  eitxpν(x)dx = [1 − it −  �  (1 − it)2 − 1]ν, t ∈ R,  (17)  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Feller p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 414 and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='476 or Ushakov p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='283;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' it is called there Bessel distri-  bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' For further generalizations of FS distributions cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Vinogradov and  Paris (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  From above we get that FS variable is in ID class and it has Lévy-  Khintchine representation  φF S(ν)(t) = exp ν[ita +  � ∞  0  (eitx − 1 −  itx  1 + x2)kF S(x)dx],  where kF S(x) := e−xI0(x)  x  1(0,∞)(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' a :=  � ∞  0  e−xI0(x)  1 + x2 dx  (18)  12  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Jurek (2021) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 103 and FS = [a, 0, kF S] ∈ L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Note that the above Lévy  measure density kF S(x) coincides with the density τ1(x), in Vinogradov and  Paris (2021), Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='.  For the BDRV YF S(1) = [b, 0, hF S(x)] we get that  hF S(x) := (−xkF S(x))′ = (−e−xI0(x))′ = e−x(I0(x) − I1(x)) > 0,  is positive (by Jones (1968) or Paris and Vinogradov (2021) and the inequality  (110)) and integrable to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' For the shift parameter, by (4), we have  b = a +  �  R  (  x  1 + x2 − arctan(x))hF S(x)dx  =  � ∞  0  e−xI0(x)  1 + x2 dx +  � ∞  0  (  x  1 + x2 − arctan(x))e−x(I0(x) − I1(x))dx  =  � ∞  0  e−xI0(x)(arctan(x))′dx −  � ∞  0  arctan(x))(−e−xI0(x))′dx  +  � ∞  0  x  1 + x2e−x(I0(x) − I1(x))dx = e−xI0(x) arctan(x)|x=∞  x=0  +  � ∞  0  x  1 + x2e−x(I0(x) − I1(x))dx =  � ∞  0  x  1 + x2e−x(I0(x) − I1(x))dx,  (19)  because limx→0 e−xI0(x) arctan(x) = limx→∞ e−xI0(x) arctan(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Consequently, from above and (4) we have  ψF S(t) = exp(it(  � ∞  0  x  1 + x2e−x(I0(x) − I1(x))dx)  +  � ∞  0  (eitx − 1 − it  x  1 + x2)e−x(I0(x) − I1(x))dx)  =  � ∞  0  (eitx − 1)e−x(I0(x) − I1(x))dx,  which is the characteristic function of the compound Poisson distribution,  thus ψF S(t) /∈ L0, and FS ∈ L0 \\ L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  On the other hand, by (5) we have that  ψF S(t) = exp(t(log(φF S(t)))′)  = exp(t(log(1 − it −  �  (1 − it)2 − 1))′) = exp(  it  �  −t(t + 2i)  which gives  13  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' From the selfdecomposability property of the Feller-Spitzer dis-  tribution we have the identity  � ∞  0  (eitx − 1)µ(dx) =  it  �  −t(t + 2i)  , t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  where µ(dx) := e−x(I0(x) − I1(x))1(0,∞)(x)dx is a probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Both  the right and the left expressions are logarithms of infinitely divisible charac-  teristic functions (Poisson compound distributions) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
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+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
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+page_content=' Jour-  nal, 39(1), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 33-41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Grigelionis (2000), Generalized z-distributions and related stochastic  processes, Mathematikos Ir Informaticos Institutas Preprintas, Nr 2000-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Vilnius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 47, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='220-221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Jurek (1982), An integral representation of operator-selfdecomposable  random variables, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Polon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' 30, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='385-393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  14  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Jurek (1983), The classesLm(Q) of probability measures on Banach  spaces, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
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+page_content=' Wiley and Sons, England.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
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+page_content=' Academic  Press 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='  [ Also on: www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
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+page_content='uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
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+page_content='pl/ zjjurek/urb-limitLawsOhio1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content='pdf]  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Vinogradov and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Paris (2021), On two extensions of the canon-  ical Feller-Spitzer distribution, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=' Distributions and Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
+page_content=', (2021)8:3,  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfvS1K/content/2301.11625v1.pdf'}
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+manuscript submitted to journal
+Accelerating large-eddy simulations of clouds with
+Tensor Processing Units
+Sheide Chammas1, Qing Wang1, Tapio Schneider1,2, Matthias Ihme1,3, Yi-fan
+Chen1, and John Anderson1
+1Google LLC
+2California Institute of Technology
+3Stanford University
+Key Points:
+• We introduce a large-eddy simulation (LES) framework that runs on Tensor Pro-
+cessing Units (TPUs, accelerators designed for machine learning)
+• The fidelity of the LES is established by reproducing aircraft observations of noc-
+turnal stratocumulus clouds over the Pacific
+• The LES exhibit unprecedented scalability on TPUs, enabling the large-scale gen-
+eration of training data for cloud parameterizations
+Corresponding author: Sheide Chammas, sheide@google.com
+–1–
+arXiv:2301.04698v1  [physics.ao-ph]  11 Jan 2023
+
+manuscript submitted to journal
+Abstract
+Clouds, especially low clouds, are crucial for regulating Earth’s energy balance and me-
+diating the response of the climate system to changes in greenhouse gas concentrations.
+Despite their importance for climate, they remain relatively poorly understood and are
+inaccurately represented in climate models. A principal reason is that the high compu-
+tational expense of simulating them with large-eddy simulations (LES) has inhibited broad
+and systematic numerical experimentation and the generation of large datasets for train-
+ing parametrization schemes for climate models. Here we demonstrate LES of low clouds
+on Tensor Processing Units (TPUs), application-specific integrated circuits that were orig-
+inally developed for machine learning applications. We show that TPUs in conjunction
+with tailored software implementations can be used to simulate computationally chal-
+lenging stratocumulus clouds in conditions observed during the Dynamics and Chem-
+istry of Marine Stratocumulus (DYCOMS) field study. The TPU-based LES code suc-
+cessfully reproduces clouds during DYCOMS and opens up the large computational re-
+sources available on TPUs to cloud simulations. The code enables unprecedented weak
+and strong scaling of LES, making it possible, for example, to simulate stratocumulus
+with 10× speedup over real-time evolution in domains with a 34.7 km×53.8 km hori-
+zontal cross section. The results open up new avenues for computational experiments
+and for substantially enlarging the sample of LES available to train parameterizations
+of low clouds.
+Plain Language Summary
+The study of clouds has been impeded by, among other factors, limitations in our
+ability to simulate them rapidly and on sufficiently large domains. In particular, com-
+putational limitations in simulating low clouds are among the reasons for the difficul-
+ties of representing them accurately in climate models; this is one of the dominant un-
+certainties in climate predictions. This paper demonstrates how the large computing power
+available on Tensor Processing Units (integrated circuits originally designed for machine
+learning applications) can be harnessed for simulating low clouds. We demonstrate the
+largest simulations of low clouds to date, with hundreds of billions of variables, and we
+document their fidelity to aircraft observations. The results open up the large compu-
+tational resources available on TPUs, and hitherto primarily used for machine learning,
+to the study of clouds in the climate system.
+1 Introduction
+Scientific progress accelerates when it is possible to cycle rapidly through the knowl-
+edge discovery loop: design and conduct experiments, learn from the experiments, and
+design and conduct new experiments to test and refine models and hypotheses with the
+information obtained from them (National Academies of Sciences, Engineering, and Medicine,
+2022). In the computational sciences, experiments are conducted numerically, and the
+ability to cycle through the knowledge discovery loop has advanced hand-in-hand with
+the evolution of computer hardware. The atmospheric sciences represent a prime exam-
+ple of advances in computer hardware enabling and accelerating scientific progress. The
+first experiments with two-dimensional atmosphere models (Charney et al., 1950) and,
+soon thereafter, with quasigeostrophic two-layer models (Phillips, 1954, 1956) only al-
+lowed simulations that were slower than or comparable with the real-time evolution of
+the atmosphere. The first experiments using general circulation models similarly pushed
+the envelope of what was computationally feasible at the time (Smagorinsky, 1963; Smagorin-
+sky et al., 1965; Manabe et al., 1965). Once such simulations of atmospheric flows, al-
+beit at coarse resolution, became routine and rapidly executable, systematic exploration
+and experimentation followed, enabling rapid progress in our understanding of the at-
+mosphere’s general circulation, from its dependence on planetary characteristics such as
+–2–
+
+manuscript submitted to journal
+planetary radius and rotation rate (Williams, 1988a, 1988b), over the nature of atmo-
+spheric turbulence (Rhines, 1975, 1979; Held & Larichev, 1996; Held, 1999; Schneider
+& Walker, 2006; Schneider, 2006), to elucidating the hydrologic cycle (Manabe & Wether-
+ald, 1975; Rind et al., 1992; Held & Soden, 2006; Allen & Ingram, 2002; Chou & Neelin,
+2004; O’Gorman & Schneider, 2008; Schneider et al., 2010). Similarly, our understand-
+ing of deep convective clouds advanced substantially once deep-convection resolving sim-
+ulations in limited areas became routinely feasible (Held et al., 1993; Tompkins & Craig,
+1998; T. Cronin, 2014; T. W. Cronin et al., 2015; Wing et al., 2018). In contrast, our
+understanding of the dynamics of low clouds is in its infancy. We do not have quanti-
+tative theories of their response to climate change (Bretherton, 2015), and shortcomings
+in their representation in climate models have long dominated uncertainties in climate
+projections (Cess et al., 1990, 1996; Bony & Dufresne, 2005; Dufresne & Bony, 2008; Vial
+et al., 2013; Brient & Schneider, 2016; Brient et al., 2016; Webb et al., 2006, 2013; Zelinka
+et al., 2017). Numerical experiments have been limited to studies that have explored a
+few dozen canonical situations, mostly in the tropics (Siebesma et al., 2003; Stevens et
+al., 2005; Rauber et al., 2007; Caldwell & Bretherton, 2009; Sandu & Stevens, 2011; Zhang
+et al., 2012, 2013; Blossey et al., 2013, 2016; Schalkwijk et al., 2015; Tan et al., 2016, 2017).
+Broader exploration has been limited by the computational expense necessary to resolve
+the meter-scale dynamics of low clouds in large-eddy simulations (LES).
+Here we take the next step in the co-evolution of science and computing hardware
+by demonstrating LES of low clouds on Tensor Processing Units (TPUs). TPUs are application-
+specific integrated circuits (ASICs), originally developed for machine learning applica-
+tions, which are dominated by dense vector and matrix computations (Jouppi et al., 2017).
+The current TPU architecture integrates 4,096 chips into a so-called TPU Pod, which
+achieves 1.1 exaflops in aggregate at half precision. TPUs are publicly available for cloud
+computing and can be leveraged for fluids simulations (Wang et al., 2022) and other sci-
+entific computing tasks (Belletti et al., 2019; Lu et al., 2020; Pederson et al., 2022), with
+remarkable computational throughput and scalability. Large, high-bandwidth memory
+and fast chip-to-chip interconnects (currently 1.1 PB/s) contribute to the performance
+of TPUs and alleviate bottlenecks that computational fluid dynamics (CFD) applica-
+tions typically face on accelerator platforms (Balaji, 2021). However, the native half- or
+single-precision arithmetic of TPUs can also create challenges in CFD applications (Wang
+et al., 2022).
+The objective of this study is to evaluate the throughput and scalability achiev-
+able on TPUs in simulations of subtropical stratocumulus clouds under conditions en-
+countered during the Dynamics and Chemistry of Marine Stratocumulus (DYCOMS)
+field study (Stevens et al., 2005). Stratocumulus clouds are a particularly good testbed
+for low-cloud simulations for two reasons: First, they are the most frequent cloud type
+on Earth, covering about 20% of tropical oceans, with an outsize impact on Earth’s en-
+ergy balance (Wood, 2012). Reductions or increases in the area they cover by a mere 4%
+can have an impact on Earth’s surface temperature comparable to doubling or halving
+atmospheric carbon dioxide concentrations (Randall et al., 1984). Second, they are no-
+toriously difficult to simulate, even in LES, because key processes responsible for their
+maintenance, such as turbulent entrainment of air across the often sharp temperature
+inversions at their tops, occur on scales of meters (Mellado, 2016). The resulting numer-
+ical challenges lead to large differences among various LES codes owing to differences in
+the numerical discretizations (Stevens et al., 2005; Pressel et al., 2017). For example, weighted
+essentially non-oscillatory (WENO) advection schemes at resolutions of O(10 m) lead
+to more faithful simulations—relative to field measurements—than centered difference
+advection schemes at resolutions of O(1 m) (Schneider et al., 2019, their supplementary
+Fig. 3). These two reasons make progress in simulating subtropical stratocumulus both
+important and challenging.
+–3–
+
+manuscript submitted to journal
+This paper is structured as follows. Section 2 describes the governing equations,
+numerical methods, and TPU-specific implementation decisions in our LES code. Sec-
+tion 3 presents a dry buoyant bubble and a density current as validation examples of the
+code. Section 4 presents the DYCOMS simulations, including comparisons with field data
+and a scaling analysis of the simulations. Section 5 summarizes the conclusions and new
+opportunities afforded by this TPU-enabled cloud-simulation capability.
+2 Model Formulation, Numerics, and TPU implementation
+2.1 Governing Equations
+Our LES simulates the anelastic equations for moist air, understood to be an ideal
+admixture of dry air, water vapor, and any condensed water that is suspended in and
+moves with the air. Precipitating condensate (e.g., rain and snow) is not considered part
+of the working fluid, and the suspended constituents of the moist air are taken to be in
+local thermodynamic equilibrium. By Gibbs’ phase rule, then, a complete thermodynamic
+description of this system with two components (dry air and water) and three phases (wa-
+ter vapor, liquid water, ice) requires specification of two thermodynamic variables, in ad-
+dition to the density ρ and pressure p of the moist air. We choose the total water spe-
+cific humidity qt (total mass of water per unit mass of moist air) and liquid-ice poten-
+tial temperature θl (Tripoli & Cotton, 1981). This choice of thermodynamic variables
+is advantageous because both the total specific humidity qt and (approximately) the liquid-
+ice potential temperature θl are materially conserved even in the presence of reversible
+phase transitions of water. The temperature T and specific humidities ql and qi of cloud
+liquid and ice can be computed from the other thermodynamic variables.
+The anelastic approximation eliminates physically insignificant acoustic waves by
+linearizing the density ρ(x, y, z, t) = ρ0(z) + ρ′(x, y, z, t) and pressure p(x, y, z, t) =
+p0(z)+p′(x, y, z, t) around a dry reference state with density ρ0(z) and hydrostatic pres-
+sure p0(z), which depend only on altitude z. Here, reference state variables are indicated
+by a subscript 0, and perturbation variables by primes. Perturbation variables are re-
+tained only where they affect accelerations. The reference density and pressure depend
+only on the vertical coordinate z and are in hydrostatic balance,
+∂p0(z)
+∂z
+= −ρ0(z)g.
+(1)
+For energetic consistency, the reference state must be adiabatic, i.e., the reference po-
+tential temperature θ0 needs to be constant. Therefore,
+T0 = θ0
+� p0
+p00
+�Rd/cpd
+,
+(2)
+p0 = p00
+�
+1 −
+gz
+cpdθ0
+�cpd/Rd
+,
+(3)
+ρ0 =
+p0
+RdT0
+.
+(4)
+Table 1 summarizes the thermodynamic constants and other parameters used in the present
+study.
+Thermodynamic consistency of the anelastic system requires that thermodynamic
+quantities are evaluated with the reference pressure p0(z) (Pauluis, 2008). Therefore, the
+liquid-ice potential temperature we use is
+θl(T, ql, qi; p0) = T
+Π
+�
+1 − Lv,0ql + Ls,0qi)
+cpmT
+�
+,
+(5)
+where
+Π =
+�p0(z)
+p00
+�Rm/cpm
+(6)
+–4–
+
+manuscript submitted to journal
+Table 1: Thermodynamic constants and other parameters used in this study.
+Symbol
+Name
+Value
+p00
+Constant reference pressure
+1017.8 hPa
+θ0
+Reference potential temperature
+290 K
+Rd
+Gas constant of dry air
+287 J (kg K)−1
+Rv
+Gas constant of water vapor
+461.89 J (kg K)−1
+cpd
+Isobaric specific heat capacity of dry air
+1004.5 J (kg K)−1
+cpv
+Isobaric specific heat capacity of water vapor
+1859.5 J (kg K)−1
+cl
+Specific heat capacity of liquid water
+4181 J (kg K)−1
+ci
+Specific heat capacity of ice
+2100 J (kg K)−1
+Lv,0
+Specific latent heat of vaporization
+2.5 MJ kg−1
+Ls,0
+Specific latent heat of sublimation
+2.83 MJ kg−1
+Tf
+Freezing point temperature
+273.15 K
+f
+Coriolis parameter
+7.62 × 10−5 s−1
+g
+Gravitational acceleration
+9.81 m s−2
+cs
+Smagorinsky constant
+0.18
+Pr
+Turbulent Prandtl number
+0.4
+Scqt
+Turbulent Schmidt number of water
+0.4
+is the Exner function, evaluated with the altitude-dependent reference pressure p0(z) and
+the constant pressure p00. We take water vapor and suspended cloud condensate into
+account in the gas “constant“ Rm = (1−qt)Rd+(qt−qc)Rv (which is not constant be-
+cause it depends on the total specific humidity qt and condensate specific humidity qc =
+ql + qi) and in the isobaric specific heat cpm = (1 − qt)cpd + (qt − qc)cpv + qlcl + qici.
+With these definitions, the anelastic governing equations in conservation form are
+∇ · (ρ0u) = 0,
+(7)
+∂(ρ0u)
+∂t
++ ∇ · (ρ0u ⊗ u) = −ρ0∇ (α0p′) + ρ0bk − fk × ρ0(u − ug) + ∇ · (ρ0σ),
+(8)
+∂(ρ0θl)
+∂t
++ ∇ · (ρ0uθl) = −
+1
+cpmΠ∇ · (ρ0FR) + ρ0wsub
+∂θl
+∂z + 1
+Pr∇ ·
+�
+ρ0νt∇θl
+�
+,
+(9)
+∂(ρ0qt)
+∂t
++ ∇ · (ρ0uqt) = ρ0wsub
+∂qt
+∂z +
+1
+Scqt
+∇ · (ρ0νt∇qt).
+(10)
+Here,
+b = g α(θl, qt, p0) − α0(z)
+α0(z)
+(11)
+is the buoyancy, and α0 = 1/ρ0 and α = 1/ρ are specific volumes. The specific vol-
+ume α(θl, qt, p0) is calculated from the approximate equation of state, again with the ref-
+erence pressure p0 in place of the total pressure,
+α = RmT
+p0
+.
+Neglected in these equations is differential settling of condensate relative to the surround-
+ing air and all precipitation processes. Table 2 lists the variables we use. The pertur-
+bation pressure p′ is obtained as solution to a Poisson equation, which follows by tak-
+ing the divergence of the momentum equation. The numerical algorithm for solving eqs. (7)
+to (10) is discussed in section 2.4.
+–5–
+
+manuscript submitted to journal
+Table 2: Definitions of Variables
+Variable
+Definition
+Units
+ρ
+Density of moist air
+kg m−3
+α
+Specific volume of moist air
+m3 kg−1
+u
+Velocity of moist air
+m s−1
+ug
+Prescribed geostrophic velocity
+m s−1
+wsub
+Prescribed subsidence velocity
+m s−1
+p
+Pressure
+Pa
+b
+Buoyancy
+m s−2
+k
+Vertical unit vector
+T
+Temperature
+K
+Rm
+Specific gas “constant” of moist air
+J kg−1 K−1
+cpm
+Isobaric specific heat of moist air
+J kg−1 K−1
+σ
+Subgrid-scale stress per unit mass
+m2 s−2
+FR
+Radiative energy flux
+W m kg−1
+qt
+Total water specific humidity
+kg/kg
+qv
+Water vapor specific humidity
+kg/kg
+ql
+Liquid water specific humidity
+kg/kg
+qi
+Ice specific humidity
+kg/kg
+νt
+Turbulent viscosity
+m2 s−1
+z
+Altitude
+m
+2.2 Saturation Adjustment
+The temperature T and the partitioning of total water mass into the liquid phase
+(specific humidity ql) and ice phase (specific humidity qi) are obtained from θl, qt, and
+the reference pressure p0 by a saturation adjustment procedure (Tao et al., 1989). This
+amounts to solving
+θ∗
+l − θl = 0,
+(12)
+where θ∗
+l (T; p0) = θl(T, q∗
+l , q∗
+i ; p0) is the liquid-ice potential temperature at saturation,
+that is, with
+q∗
+l = max
+�
+0, qt − q∗
+v(T, p0)
+�
+H(T − Tf)
+(13)
+and
+q∗
+i = max
+�
+0, qt − q∗
+v(T, p0)
+�
+H(Tf − T).
+(14)
+Here, q∗
+v is the saturation vapor pressure, calculated as in Sridhar et al. (2022), H is the
+Heaviside step function, and Tf is the freezing point temperature. (However, in the ex-
+amples in this paper, we only obtain liquid clouds without ice.) We solve the resulting
+nonlinear problem (12) with the secant method.
+2.3 Subgrid-scale Models
+We model subgrid-scale fluxes with the turbulent viscosity model of Lilly (1962)
+and Smagorinsky (1963). In this model, the turbulent viscosity is represented as
+νt = (cs∆)2fBS.
+(15)
+where S = ∥S∥2 is the 2-norm of the strain rate tensor S = 0.5
+�
+∇u + (∇u)T �
+for the
+resolved velocities u; cs is the Smagorinsky constant (Table 1); and ∆ = (∆x1∆x2∆x3)1/3
+is the geometric mean of the grid spacings in the three space directions. The buoyancy
+–6–
+
+manuscript submitted to journal
+factor 0 ≤ fB ≤ 1 limits the mixing length in the vertical in the case of stable strati-
+fication; it is computed from the moist buoyancy frequency (Durran & Klemp, 1982) as
+described in Pressel et al. (2017). The diffusivities of the liquid-ice potential tempera-
+ture and total specific humidity are obtained from the turbulent viscosity νt by division
+by constant turbulent Prandtl and Schmidt numbers (Table 1).
+To emulate a radiation condition at the upper boundary, we include a sponge layer
+that occupies the top 5% of the domain and absorbs upward propagating waves. The
+sponge is implemented as a linear Rayleigh damping layer (Durran & Klemp, 1983), in
+which the horizontal velocity is relaxed toward the geostrophic wind velocity and the ver-
+tical velocity is relaxed to zero. To avoid reflections at the interface between the sponge
+layer and the undamped flow outside, we use a relaxation coefficient that ensures a grad-
+ual onset of the sponge layer (Klemp & Lilly, 1978), reaching 0.25 s−1 at the top of the
+domain.
+2.4 Numerical Solution
+We discretize the governing equations with a second-order finite-difference method.
+All discrete operators are expressed on a collocated mesh. The third-order QUICK scheme
+is used for the discretization of the advection terms in eqs. (8) to (10). All diffusion terms
+are computed with a second-order central difference scheme.
+An explicit iterative scheme (Wang et al., 2022) is employed for the time advance-
+ment of the numerical solutions. This scheme provides an iterative representation to the
+Crank-Nicolson method, which avoids the computational complexity of solving a high-
+dimensional linear system of equations. Specifically, the momentum equation (8) is solved
+with a predictor-corrector approach. At the prediction step of sub-iteration k+1, the
+momentum equation is solved in discrete form as
+�
+ρ0u − (ρ0u)n
+∆t
+= −ρ0∇
+�
+α0p′k�
++ Rn+ 1
+2 ,
+(16)
+with
+Rn+ 1
+2 = −∇ · [(ρ0u)n+ 1
+2 ⊗ un+ 1
+2 ] + ∇ ·
+�
+ρ0σn+ 1
+2
+�
++ ρ0bn+ 1
+2 k − fk × ρ0
+�
+un+ 1
+2 − ug
+�
+, (17)
+where �
+(·) denotes a prediction of a variable at step n+1 in sub-iteration k +1; (·)k is
+the solution of a variable at step n+1 obtained from sub-iteration k. Variables at state
+(·)n+ 1
+2 are estimated as (·)n+ 1
+2 = [(·)k + (·)n]/2. Note that the prediction of the mo-
+mentum �
+ρ0u in sub-iteration k+1 is evaluated with the pressure from the previous sub-
+iteration. The correct momentum (ρ0u)k+1 needs to be computed with the pressure at
+sub-iteration k + 1, which can be expressed similarly to eq. (16) as
+(ρ0u)k+1 − (ρ0u)n
+∆t
+= −ρ0∇
+�
+α0p′k+1�
++ Rn+ 1
+2 .
+(18)
+Subtracting eq. (16) from eq. (18) yields
+(ρ0u)k+1 − �
+ρ0u
+∆t
+= −ρ0∇
+�
+α0p′k+1 − α0p′k�
+= −ρ0∇ (α0δp) ,
+(19)
+where δp = p′k+1 − p′k is the pressure correction from sub-iteration k to k + 1.
+Taking the divergence of eq. (19) and applying mass conservation at sub-iteration
+k + 1 leads to a generalized Poisson equation for the pressure correction:
+∇2 (α0δp) = α0
+∆t
+�
+∇ · (�
+ρ0u) − ∇ · (ρ0u)k+1�
+= α0
+∆t∇ · (�
+ρ0u).
+(20)
+–7–
+
+manuscript submitted to journal
+To ensure numerical consistency and eliminate the checkerboard effect due to the col-
+located mesh representation, we introduce an additional correction term when solving eq. (20),
+which is described in Appendix A.
+We apply homogeneous Neumann boundary conditions on the pressure, assuming
+a vanishing correction of the mass flux: (ρ0u)k+1−�
+ρ0u = 0. Solving the Poisson equa-
+tion (20) subject to the boundary conditions provides the pressure correction. We solve
+the Poisson equation iteratively with the weighted Jacobi method. The momentum and
+pressure at sub-iteration k + 1 are then updated as
+(ρ0u)k+1 = �
+ρ0u − ∆tρ0∇(α0δp),
+(21)
+pk+1 = pk + δp.
+(22)
+2.5 TPU Implementation
+The discrete formulations are implemented in TensorFlow, to support execution
+on different computing architectures and integration with machine learning approaches.
+In the present study, all computations are performed on TPUs; the host CPU are used
+for data input and output only.
+At the beginning of each simulation, the simulator code is compiled by the Accel-
+erated Linear Algebra (XLA) compiler with the just-in-time (JIT) approach, which builds
+a TensorFlow graph. This approach reduces the computational cost at runtime signif-
+icantly, which is particularly beneficial for simulations with repeated steps. The repre-
+sentations of the three-dimensional data structure and numerical operators are designed
+to optimize the performance within the TensorFlow programming paradigm (Wang et
+al., 2022). The graph is subsequently replicated onto each TPU for computation. The
+initial flow field data are distributed onto each TPU as input to the distributed graph.
+On TPUs, the efficiency of partitioning is anisotropic along different spatial dimen-
+sions. This behavior results from the data structures that are designed for optimal com-
+putational efficiency. With this programming strategy, partitioning in different directions
+leads to different TensorFlow graph structures. As a result, partitioning along the first
+dimension of the allocated 3D tensors is more efficient than along the other two dimen-
+sions (Wang et al., 2022). We investigate the scaling of our simulation framework for dif-
+ferent partitions in section 4.2, with an assessment of implications for cloud simulations.
+3 Validation Study
+To validate the numerical scheme and model formulation, we consider two test cases
+that are widely used for validation and are relevant to the buoyancy-driven dynamics
+prevalent in the atmosphere. The first case is a density current consisting of a two-dimensional
+negatively buoyant dry bubble impinging on a surface (Straka et al., 1993); the second
+case is a rising buoyant bubble (Bryan & Fritsch, 2004).
+3.1 Density Current
+The density current configuration consists of an initial perturbation to a uniform
+potential temperature field. The initial perturbation’s amplitude peaks at −15 K and
+has a horizontal radius of 4 km and a vertical radius of 2 km. The two-dimensional do-
+main is 51.2 km wide and 6.4 km high. As in Pressel et al. (2015), we use periodic hor-
+izontal boundary conditions instead of the no-flux boundary conditions in Straka et al.
+(1993). This benchmark case has an added significance in stratocumulus simulations be-
+cause the density current’s perturbation amplitude is of the same magnitude as the jump
+in temperature observed across the entrainment interfacial layer at the cloud top.
+–8–
+
+manuscript submitted to journal
+Figure 1 shows the potential temperature at t = 900 s for varying resolutions rang-
+ing from a homogeneous resolution of 200 m to 10 m. A uniform kinematic viscosity of
+10 m2 s−1 is used to make the simulations comparable across the wide range of resolu-
+tions. Since the solutions are nearly horizontally symmetric about the center of the do-
+main, only the right half of the bubble is shown. The flow exhibits Kelvin-Helmholtz in-
+stabilities that generate small scales.
+The numerical solutions exhibit increasingly detailed small-scale features as the res-
+olution is increased. Even at the coarsest resolution (200 m), the large-scale flow features
+are preserved, and there are no signs of spurious small-scale oscillations associated with
+numerical dispersion errors. These results suggest the robustness of the numerical scheme
+in capturing sharp gradients and turbulence, even at coarser resolutions.
+3.2 Rising Bubble
+The second test case is a rising dry bubble. The bubble is initialized as a pertur-
+bation to a uniform potential temperature field, following Bryan and Fritsch (2004), with
+a peak amplitude of 2 K. As in the first test case, we use periodic horizontal boundary
+conditions. The domain is 20 km wide and 10 km high.
+Figure 2 shows the potential temperature at t = 1000 s for varying homogeneous
+resolutions ranging from 200 m to 10 m. For this case, a uniform kinematic viscosity of
+1 m2 s−1 was found to be adequate to ensure the simulations are comparable across the
+different resolutions.
+As in the density current case, there are no spurious oscillations even for the sim-
+ulations with coarser resolutions. The numerical solution is essentially converged at 50
+m. The vertical velocity contours are nearly unchanged from the finest resolution down
+to 100 m resolution. This observed stability is due in great part to the QUICK scheme
+used in the scalar and momentum advection. Although the QUICK scheme achieves only
+a second-order accurate approximation of the advective flux, the solutions seen in this
+case suggest a quality and fidelity of simulation comparable to that of the WENO schemes
+on staggered grids (Pressel et al., 2015).
+4 DYCOMS Simulation
+The first nocturnal research flight (RF01) of the Dynamics and Chemistry of Ma-
+rine Stratocumulus (DYCOMS-II) field study (Stevens et al., 2003) serves as the testbed
+of our low-cloud simulations. Among the attractive characteristics of this test case are
+the relative homogeneity of the environmental conditions, the absence of significant driz-
+zle, and the persistence of a stable cloud layer. The basic state for RF01 is idealized as
+a quasi-two-layer structure in potential temperature θl and total-water specific humid-
+ity qt (Stevens et al., 2005). Forcings include geostrophic winds, large-scale subsidence,
+a simple parameterization of longwave radiation, and surface fluxes of latent and sen-
+sible heat.
+We choose the initial liquid-ice potential temperature θl = 289 K and initial to-
+tal specific humidity in the mixed layer to be qt = 8.7 g kg−1, which is slightly lower
+than the DYCOMS default value of 9 g kg−1. This ensures that with our thermodynam-
+ics formulation and constants, we obtain a cloud layer between 600 and 840 m. The ver-
+tical domain extends to 1.5 km, with a no-slip, zero-flux lid at the top. The horizontal
+domain in the default case covers an area of (4 km)2, with periodic horizontal bound-
+ary conditions.
+The default simulation runs for 4 simulated hours on a grid of 128 × 128 × 256
+points with a uniform horizontal grid spacing of 32 m and a uniform vertical grid spac-
+ing of 6 m. Although a vertical resolution of 5 m or less is often desirable to capture the
+–9–
+
+manuscript submitted to journal
+Figure 1: Contours of potential temperature [K] in the density current simulation at
+900 s at mesh resolutions of 200 m, 100 m, 50 m, and 10 m. Contours of potential tem-
+perature are shown at increments of 0.2 K.
+sharp temperature gradient at the inversion above the cloud top without generating spu-
+rious entrainment (Mellado, 2016; Pressel et al., 2017; Mellado et al., 2018), our simu-
+lation did not change materially as we increased the vertical resolution to finer than 6 m.
+A physical time step of 0.3 s (Courant number 0.3) is used in the default configuration.
+4.1 Fidelity of Simulation
+The mean vertical profiles and vertical velocity statistics closely match the obser-
+vations from the research flight. Both the liquid potential temperature profile and the
+total-water specific humidity profile maintain their two-layer structure, with a well-mixed
+–10–
+
+6
+10 m
+4
+km
+0
+0
+5
+10
+15
+20
+[kmmanuscript submitted to journal
+Figure 2: Contours of potential temperature [K] (left) and vertical velocity [m s−1]
+(right) in the rising bubble simulation for mesh resolutions of 200 m, 100 m, 50 m, and
+10 m. Contours of potential temperature and velocity are shown for increments of 0.2 K
+and 2 m s−1, respectively.
+–11–
+
+manuscript submitted to journal
+Figure 3: Profile of mean state of specific humidity and temperature in DYCOMS as ob-
+served (points), from our simulation averaged over the 4th hour (red solid lines), and from
+an implicit LES (Pressel et al., 2017) using a nominally fifth-order WENO scheme (blue
+dashed lines).
+Figure 4: Profile of the variance and skewness of the vertical velocity in DYCOMS as
+observed (points), from our simulation averaged over the 4th hour (red solid lines), and
+from an implicit LES (Pressel et al., 2017) using a nominally fifth-order WENO scheme
+(blue dashed lines).
+boundary layer below a cloud top (fig. 3). For comparison, we also show the vertical pro-
+files for an implicit LES using a nominally fifth-order WENO scheme on a staggered grid,
+a configuration that has been shown to perform well for stratocumulus simulations (Pressel
+et al., 2017); in fact, this configuration at the resolution we use here performs favorably
+relative to simulations with oscillatory numerical schemes for the momentum equation
+on much higher-resolution (meter-scale) isotropic grids (Schneider et al., 2019; Matheou,
+2018; Mellado et al., 2018). (The WENO scheme uses a fifth-order stencil for the flux
+reconstruction but is strictly only of second-order accuracy for nonlinear problems on
+a staggered grid (Mishra et al., 2021); hence, it is only nominally fifth order.)
+The turbulent structure in the boundary layer becomes evident in the variance and
+skewness profiles of the vertical velocity (fig. 4). The variance peaks near the cloud base,
+consistent with the research flight observations and turbulence generation by latent heat
+release at that altitude. The skewness reveals preferential directions of turbulent verti-
+cal velocities. For example, positive vertical velocity skewness near the bottom is con-
+sistent with the presence of significant heat fluxes at the sea surface, which drive con-
+vection. On the other hand, the negative skewness near the cloud top is consistent with
+the presence of downdrafts driven by radiative cooling. Like the variance, the skewness
+in our simulations is consistent with the research flight observations. By contrast, most
+LES in the DYCOMS intercomparison study (Stevens, 2005) are unable to capture the
+negative skewness near the cloud top, likely because of excessive spurious mixing across
+–12–
+
+manuscript submitted to journal
+the inversion. The vertical velocity statistics are an indication that our numerics with
+the QUICK advection scheme avoid the excessive generation of spurious mixing across
+the inversion at the cloud top, which occurs in many other LES. The fidelity of the ver-
+tical velocity statistics to observations is similar to that obtained with WENO schemes
+(Pressel et al., 2017).
+Our simulation maintains more liquid water in the cloud (fig. 3) than most other
+LES in the DYCOMS intercomparison study (Stevens, 2005). LES often have difficul-
+ties maintaining a cloud layer with sufficient liquid water because of spurious numeri-
+cal mixing of dry air across the inversion at the cloud top (Pressel et al., 2017), which
+warms and dries the cloud layer.
+These results indicate a high fidelity of our LES to the observed flow statistics. Our
+LES does not suffer from the shortcomings in many LES that lead to spurious turbu-
+lent entrainment at the inversion and a decoupling boundary layer; it performs similarly
+well as implicit LES with WENO schemes (Pressel et al., 2017). Therefore, it can ad-
+equately capture low clouds and enable the investigation of the feedbacks that make low
+clouds such an important regulator of the strength of greenhouse warming.
+4.2 Time to Solution and Scaling Analysis
+The discretization schemes described above lend themselves to parallelization al-
+gorithms that are well suited for the TPU infrastructure. However, increased parallelism
+generally comes at the expense of greater communication between processors. As the com-
+munication overhead begins to dominate, the marginal benefit from increased parallelism
+diminishes. To assess the appropriateness of the TPU simulation framework for this class
+of problems, it is thus imperative to measure how well the simulation runtime scales with
+increased parallelism.
+We examine the scalability of the solver using the DYCOMS case as a testbed. The
+method for doing so is to measure the mean turnaround time for a time step under dif-
+ferent mesh configurations. For a fair comparison between different configurations, we
+keep the spatial resolution at 35 m × 6 m and the Courant number at approximately
+0.3 for all cases. We find impressive scaling, notwithstanding that each simulation step
+involves the solution of an elliptic (globally nonlocal) problem for the dynamic pressure
+correction.
+4.2.1
+Weak Scaling
+We demonstrate weak scalability by fixing the local grid size per processor and con-
+sidering an ever-growing computational domain. We use Nk to denote the global grid
+size along dimension k, �
+Nk to denote the local per processor subgrid size along dimen-
+sion k, and Pk to denote the number of processors assigned to dimension k in the com-
+putational topology. For the first analysis, the computational domain per TPU core is
+fixed with a size of �
+Nx × �
+Ny × �
+Nz = 1024 × 36 × 1024 grid points, which is about the
+largest partition size that can fit in the TPU RAM considering the data requirements
+of this simulation. Table 3 shows that the turnaround time remains virtually unchanged
+as the number of TPU cores grows from 16 to 2048, corresponding to an increase in to-
+tal number of grid points from 533M to 68.2B and in physical domain size from 36 km×
+18 km to (286 km)2.
+We repeat this weak scaling analysis using a smaller partition size that is more com-
+monly encountered in atmospheric simulations. Table 4 demonstrates weak scalability
+when the computational domain per TPU core is fixed with a size of �
+Nx × �
+Ny × �
+Nz =
+128 × 10 × 256. With about 10× speedup over real-time evolution (10 simulated days
+per day, SDPD) and a 35-meter horizontal resolution, the largest physical domain at-
+tainable with 2048 TPU cores in this configuration is 34.7 km×53.8 km. (Since we are
+–13–
+
+manuscript submitted to journal
+using only a quarter TPU pod, the largest domain size attainable on a full TPU pod,
+with 8192 cores, would be 69.4 km × 107.4 km.)
+Figure 5 shows the efficiency curves for these two cases, normalized relative to the
+smallest simulation. It is worth noting that in the weak scaling analysis with the large
+partitions, the two smallest simulations are in fact less efficient than the larger ones. This
+may seem surprising at first, as efficiency normally decreases with the problem size. How-
+ever, this behavior is most likely a consequence of saturating the partition memory, which
+leads to variations in memory bandwidth utilization, which seem to penalize the perfor-
+mance of smaller TPU configurations more severely. This behavior is not seen in the sec-
+ond weak scaling analysis, which uses a significantly smaller partition size.
+4.2.2
+Strong Scaling
+We now consider how the solver scales when we increase parallelism for a fixed global
+problem size. Throughout this analysis, the vertical dimension has only a single parti-
+tion with a total of 128 levels. The total number of grid points is fixed at 134M. Three
+cases are considered: (i) 2 partitions in the x direction, (ii) 4 partitions in the x direc-
+tion, and (iii) 8 partitions in the x direction. In each of the three cases, we try multi-
+ple partitions in the y direction, starting with 16 and scaling all the way up to 128 par-
+titions. As seen in table 5, the analysis consists of increasing parallelism while propor-
+tionately reducing the workload per processor. Each subsequent row doubles the num-
+ber of cores assigned to the y dimension while simultaneously halving the number of grid
+points per core along that dimension. The measured speedup relative to real time reaches
+a maximum of 14.08 in the configuration with 1024 cores and the smallest partition size.
+The speedup is illustrated in fig. 6. In all cases, the speedup curve shows clear evidence
+of linear (i.e., perfect) strong scaling.
+Table 3: Simulation configurations for weak scalability analysis using large partitions of
+size 1024 × 36 × 1024 grid points per TPU core. The grid dimensions indicated in the
+middle columns do not include ghost points. The last column shows the simulated time
+relative to real time in simulated days per day (SDPD).
+Number of cores
+Grid size
+SDPD
+Ptot
+Px
+Py
+Pz
+Ntot
+Nx (Lx)
+Ny (Ly)
+Nz (Lz)
+16
+1
+16
+1
+533M
+1020 (35.7 km)
+512 (17.9 km)
+1020 (6.1 km)
+0.19
+32
+1
+32
+1
+1.1B
+1020 (35.7 km)
+1024 (35.8 km)
+1020 (6.1 km)
+0.19
+64
+2
+32
+1
+2.1B
+2040 (71.4 km)
+1024 (35.8 km)
+1020 (6.1 km)
+0.20
+128
+2
+64
+1
+4.3B
+2040 (71.4 km)
+2048 (71.7 km)
+1020 (6.1 km)
+0.20
+256
+4
+64
+1
+8.5B
+4080 (142.8 km)
+2048 (71.7 km)
+1020 (6.1 km)
+0.20
+512
+4
+128
+1
+17.1B
+4080 (142.8 km)
+4096 (143.4 km)
+1020 (6.1 km)
+0.20
+1024
+8
+128
+1
+34.1B
+8160 (285.6 km)
+4096 (143.4 km)
+1020 (6.1 km)
+0.20
+2048
+8
+256
+1
+68.2B
+8160 (285.6 km)
+8192 (286.7 km)
+1020 (6.1 km)
+0.19
+4.3 Taking LES of Clouds to the Macroscale
+To demonstrate the TPU framework’s capabilities for simulating clouds on large
+scales, we simulated 4 hours of the DYCOMS conditions on a domain of size 285 km×
+285 km×2 km using a typical DYCOMS resolution of 35 m×6 m. The simulation runs
+on a mesh of 32 billion grid points and requires a little less than 20 wallclock hours to
+simulate 4 hours on 1024 TPU cores (70.4 petaflops at single precision). Analysis of the
+mean profiles of the conserved variables and flow field reveals dynamics that are nearly
+identical to those shown in section 4.1 for the much smaller DYCOMS domain, so we
+–14–
+
+manuscript submitted to journal
+Table 4: Weak scalability analysis with a more typical partition of size 128 × 10 × 256 per
+TPU core.
+Number of cores
+Grid size
+SDPD
+Ptot
+Px
+Py
+Pz
+Ntot
+Nx (Lx)
+Ny (Ly)
+Nz (Lz)
+16
+1
+16
+1
+3M
+124 (4.3 km)
+96 (3.4 km)
+252 (1.5 km)
+10.56
+32
+1
+32
+1
+6M
+124 (4.3 km)
+192 (6.7 km)
+252 (1.5 km)
+10.27
+64
+2
+32
+1
+12M
+248 (8.7 km)
+192 (6.7 km)
+252 (1.5 km)
+10.10
+128
+2
+64
+1
+24M
+248 (8.7 km)
+384 (13.4 km)
+252 (1.5 km)
+10.03
+256
+4
+64
+1
+48M
+496 (17.4 km)
+384 (13.4 km)
+252 (1.5 km)
+10.00
+512
+4
+128
+1
+96M
+496 (17.4 km)
+768 (26.9 km)
+252 (1.5 km)
+10.00
+1024
+8
+128
+1
+192M
+992 (34.7 km)
+768 (26.9 km)
+252 (1.5 km)
+10.03
+2048
+8
+256
+1
+384M
+992 (34.7 km)
+1536 (53.8 km)
+252 (1.5 km)
+10.07
+Figure 5: Normalized efficiency with respect to the weak scaling analysis.
+Table 5: Partitions for strong scalability analysis (Ntot = 134M).
+Number of cores
+Partition size
+SDPD
+Ptot
+Px
+Py
+Pz
+�
+Nx
+�
+Ny
+�
+Nz
+32
+2
+16
+1
+512
+64
+128
+0.83
+64
+2
+32
+1
+512
+32
+128
+1.68
+128
+2
+64
+1
+512
+16
+128
+3.97
+256
+2
+128
+1
+512
+8
+128
+7.71
+64
+4
+16
+1
+256
+64
+128
+1.19
+128
+4
+32
+1
+256
+32
+128
+2.34
+256
+4
+64
+1
+256
+16
+128
+5.62
+512
+4
+128
+1
+256
+8
+128
+10.99
+128
+8
+16
+1
+128
+64
+128
+1.45
+256
+8
+32
+1
+128
+32
+128
+3.27
+512
+8
+64
+1
+128
+16
+128
+7.19
+1024
+8
+128
+1
+128
+8
+128
+14.08
+omit them here. The cloud layer is visualized in fig. 7 at different scales. The visualiza-
+tion reveals large spatial variability of cloud water fraction, with occasional open cells.
+–15–
+
+manuscript submitted to journal
+Figure 6: Strong scaling for different partitions (as in table 5).
+These simulations demonstrate that low-cloud resolving LES are possible in domains
+the size of a grid box in a typical coarse-resolution climate model, which has a footprint
+of around (100 km)2. It enables three-dimensional LES to be embedded in climate model
+grid boxes, to provide high-fidelity representations of cloud dynamics locally in them.
+5 Discussion and Conclusions
+We have demonstrated that LES of low clouds are possible on TPUs and achieve
+unprecedented weak and strong scaling at high numerical fidelity. Our LES code with
+a QUICK advection scheme for momentum and tracers demonstrates a fidelity to air-
+craft observations that is comparable with that obtained with WENO schemes at the
+same resolution, exceeding the fidelities achievable with oscillatory, numerical schemes,
+or combinations of oscillatory schemes for momentum and non-oscillatory schemes for
+tracers (Pressel et al., 2017; Schneider et al., 2019). At the meter-scale resolutions needed
+for resolving the computationally challenging stratocumulus clouds, we have shown that
+the code scales strongly and weakly up to 1024 and 2048 TPU cores, respectively, cor-
+responding to a computational throughput of 70.4 and 140.8 petaflops. This opens up
+the large compute resources with fast chip-to-chip interconnects available on TPUs for
+low-cloud LES. For example, it means that LES with horizontal resolutions around 30 m
+and vertical resolutions around 5 m are achievable at 10 simulated days per wallclock
+day in domains the size of what is becoming a typical climate model grid column (25–
+50 km wide). Thus, it is possible to generate LES with an outer horizontal scale that is
+the same as the inner horizontal scale of climate models (Schneider et al., 2017).
+Our LES code and the compute resources available on TPUs enable the genera-
+tion of large libraries of low-cloud simulations (Shen et al., 2022). These can be used both
+for quantitatively studying mechanisms underlying low-cloud feedbacks to climate change
+(Bretherton, 2015) and as training data for parameterizations of low clouds for coarse-
+resolution climate models (Couvreux et al., 2021; Hourdin et al., 2021; Lopez-Gomez et
+al., 2022). The LES code described here is publicly available for this and similar pur-
+poses.
+6 Open Research
+The source code for all simulations described in this paper is available at https://github.com/google-
+research/swirl-lm. (We will provide a DOI prior to acceptance of this paper.)
+–16–
+
+manuscript submitted to journal
+Figure 7: Volume rendering of the instantaneous cloud water specific humidity qc of a
+simulated stratocumulus cloud covering a horizontal (285 km)2 footprint after 4 simulated
+hours. (left column) Oblique view and (right column) normal view. (top) Entire domain;
+(middle) closeup of a corner: (oblique) 26 km
+×
+26 km and (normal) 52 km
+×
+26 km;
+(bottom) further closeup of the same corner: (oblique) 13 km
+×
+13 km, and (normal)
+26 km × 13 km.
+Acknowledgments
+We thank Jason Hickey for his guidance in the early stages of this research and Tian-
+jian Lu for his thorough review of this article.
+Appendix A Numerically Consistent Poisson Equation on Collocated
+Grids
+To eliminate the discrepancy between the numerical representation of the gradi-
+ent and Laplacian operators in the Poisson eq. (20), and to introduce coupling between
+nodes with odd and even indices, we add an additional correction term that takes the
+form of a fourth-order difference of the pressure correction δp on the right-hand side of eq. (20).
+Specifically, applying the discrete divergence operator to eq. (19) with the enforcement
+–17–
+
+manuscript submitted to journal
+of mass conservation at sub-iteration k + 1, we have
+∇ · ∇(α0δp) = α0
+∆t∇ · (�
+ρ0u).
+(A1)
+Subtracting eq. (A1) from eq. (20) results in a correction term that takes the form:
+C = (∇2 − ∇ · ∇)(α0δp).
+(A2)
+In a discrete representation in which the divergence operator is expressed by the second-
+order central difference scheme,
+∇(·) = (·)l+1 − (·)l−1
+2∆l
+,
+(A3)
+and the Laplacian operator is expressed as
+∇2(·) = (·)l+1 − 2(·)l + (·)l−1
+∆2
+l
+,
+(A4)
+the correction term in eq. (A2) is computed numerically as
+C =
+1
+2∆l
+[∇(α0δp))l+1 − (∇(α0δp))l−1] − 1
+∆2
+l
+[α0δp)l+1 − 2(α0δp)l + (α0δp)l−1]
+=
+1
+4∆2
+l
+[(α0δp)l+2 − 2(α0δp)l + (α0δp)l−2] − 1
+∆2
+l
+[(α0δp)l+1 − 2(α0δp)l + (α0δp)l−1]
+=
+1
+4∆2
+l
+[(α0δp)l+2 − 4(α0δp)l+1 + 6(α0δp)l − 4(α0δp)l−1 + (α0δp)l−2].
+(A5)
+To ensure eq. (20) is solved with numerical consistency, eq. (A5) is added to the diver-
+gence of the momentum on the right-hand side of the equation, which is:
+∇2(α0δp)k+1 = α0
+∆t∇ · (�
+ρ0u) − Ck.
+(A6)
+This ensures a numerically consistent treatment of the derivative operators and leads to
+coupling of nodes with even and odd indices.
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new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf,len=1578
+page_content='manuscript submitted to journal  Accelerating large-eddy simulations of clouds with  Tensor Processing Units  Sheide Chammas1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Qing Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Tapio Schneider1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Matthias Ihme1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Yi-fan  Chen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' and John Anderson1  1Google LLC  2California Institute of Technology  3Stanford University  Key Points:  We introduce a large-eddy simulation (LES) framework that runs on Tensor Pro-  cessing Units (TPUs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' accelerators designed for machine learning)  The fidelity of the LES is established by reproducing aircraft observations of noc-  turnal stratocumulus clouds over the Pacific  The LES exhibit unprecedented scalability on TPUs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' enabling the large-scale gen-  eration of training data for cloud parameterizations  Corresponding author: Sheide Chammas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' sheide@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='com  –1–  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='04698v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='ao-ph]  11 Jan 2023  manuscript submitted to journal  Abstract  Clouds, especially low clouds, are crucial for regulating Earth’s energy balance and me-  diating the response of the climate system to changes in greenhouse gas concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Despite their importance for climate, they remain relatively poorly understood and are  inaccurately represented in climate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' A principal reason is that the high compu-  tational expense of simulating them with large-eddy simulations (LES) has inhibited broad  and systematic numerical experimentation and the generation of large datasets for train-  ing parametrization schemes for climate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Here we demonstrate LES of low clouds  on Tensor Processing Units (TPUs), application-specific integrated circuits that were orig-  inally developed for machine learning applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' We show that TPUs in conjunction  with tailored software implementations can be used to simulate computationally chal-  lenging stratocumulus clouds in conditions observed during the Dynamics and Chem-  istry of Marine Stratocumulus (DYCOMS) field study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The TPU-based LES code suc-  cessfully reproduces clouds during DYCOMS and opens up the large computational re-  sources available on TPUs to cloud simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The code enables unprecedented weak  and strong scaling of LES, making it possible, for example, to simulate stratocumulus  with 10× speedup over real-time evolution in domains with a 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='7 km×53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='8 km hori-  zontal cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The results open up new avenues for computational experiments  and for substantially enlarging the sample of LES available to train parameterizations  of low clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Plain Language Summary  The study of clouds has been impeded by, among other factors, limitations in our  ability to simulate them rapidly and on sufficiently large domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' In particular, com-  putational limitations in simulating low clouds are among the reasons for the difficul-  ties of representing them accurately in climate models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' this is one of the dominant un-  certainties in climate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' This paper demonstrates how the large computing power  available on Tensor Processing Units (integrated circuits originally designed for machine  learning applications) can be harnessed for simulating low clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' We demonstrate the  largest simulations of low clouds to date, with hundreds of billions of variables, and we  document their fidelity to aircraft observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The results open up the large compu-  tational resources available on TPUs, and hitherto primarily used for machine learning,  to the study of clouds in the climate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  1 Introduction  Scientific progress accelerates when it is possible to cycle rapidly through the knowl-  edge discovery loop: design and conduct experiments, learn from the experiments, and  design and conduct new experiments to test and refine models and hypotheses with the  information obtained from them (National Academies of Sciences, Engineering, and Medicine,  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' In the computational sciences, experiments are conducted numerically, and the  ability to cycle through the knowledge discovery loop has advanced hand-in-hand with  the evolution of computer hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The atmospheric sciences represent a prime exam-  ple of advances in computer hardware enabling and accelerating scientific progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The  first experiments with two-dimensional atmosphere models (Charney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 1950) and,  soon thereafter, with quasigeostrophic two-layer models (Phillips, 1954, 1956) only al-  lowed simulations that were slower than or comparable with the real-time evolution of  the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The first experiments using general circulation models similarly pushed  the envelope of what was computationally feasible at the time (Smagorinsky, 1963;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Smagorin-  sky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 1965;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Manabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Once such simulations of atmospheric flows, al-  beit at coarse resolution, became routine and rapidly executable, systematic exploration  and experimentation followed, enabling rapid progress in our understanding of the at-  mosphere’s general circulation, from its dependence on planetary characteristics such as  –2–  manuscript submitted to journal  planetary radius and rotation rate (Williams, 1988a, 1988b), over the nature of atmo-  spheric turbulence (Rhines, 1975, 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Held & Larichev, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Held, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Schneider  & Walker, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Schneider, 2006), to elucidating the hydrologic cycle (Manabe & Wether-  ald, 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Rind et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Held & Soden, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Allen & Ingram, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Chou & Neelin,  2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' O’Gorman & Schneider, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Similarly, our understand-  ing of deep convective clouds advanced substantially once deep-convection resolving sim-  ulations in limited areas became routinely feasible (Held et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Tompkins & Craig,  1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Cronin, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Cronin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Wing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' In contrast, our  understanding of the dynamics of low clouds is in its infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' We do not have quanti-  tative theories of their response to climate change (Bretherton, 2015), and shortcomings  in their representation in climate models have long dominated uncertainties in climate  projections (Cess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 1990, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Bony & Dufresne, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Dufresne & Bony, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Vial  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Brient & Schneider, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Brient et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Webb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2006, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Zelinka  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Numerical experiments have been limited to studies that have explored a  few dozen canonical situations, mostly in the tropics (Siebesma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Stevens et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Rauber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Caldwell & Bretherton, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Sandu & Stevens, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Zhang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2012, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Blossey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2013, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Schalkwijk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2016, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Broader exploration has been limited by the computational expense necessary to resolve  the meter-scale dynamics of low clouds in large-eddy simulations (LES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Here we take the next step in the co-evolution of science and computing hardware  by demonstrating LES of low clouds on Tensor Processing Units (TPUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' TPUs are application-  specific integrated circuits (ASICs), originally developed for machine learning applica-  tions, which are dominated by dense vector and matrix computations (Jouppi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  The current TPU architecture integrates 4,096 chips into a so-called TPU Pod, which  achieves 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1 exaflops in aggregate at half precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' TPUs are publicly available for cloud  computing and can be leveraged for fluids simulations (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2022) and other sci-  entific computing tasks (Belletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Pederson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2022), with  remarkable computational throughput and scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Large, high-bandwidth memory  and fast chip-to-chip interconnects (currently 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1 PB/s) contribute to the performance  of TPUs and alleviate bottlenecks that computational fluid dynamics (CFD) applica-  tions typically face on accelerator platforms (Balaji, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' However, the native half- or  single-precision arithmetic of TPUs can also create challenges in CFD applications (Wang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  The objective of this study is to evaluate the throughput and scalability achiev-  able on TPUs in simulations of subtropical stratocumulus clouds under conditions en-  countered during the Dynamics and Chemistry of Marine Stratocumulus (DYCOMS)  field study (Stevens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Stratocumulus clouds are a particularly good testbed  for low-cloud simulations for two reasons: First, they are the most frequent cloud type  on Earth, covering about 20% of tropical oceans, with an outsize impact on Earth’s en-  ergy balance (Wood, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Reductions or increases in the area they cover by a mere 4%  can have an impact on Earth’s surface temperature comparable to doubling or halving  atmospheric carbon dioxide concentrations (Randall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Second, they are no-  toriously difficult to simulate, even in LES, because key processes responsible for their  maintenance, such as turbulent entrainment of air across the often sharp temperature  inversions at their tops, occur on scales of meters (Mellado, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The resulting numer-  ical challenges lead to large differences among various LES codes owing to differences in  the numerical discretizations (Stevens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Pressel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' For example, weighted  essentially non-oscillatory (WENO) advection schemes at resolutions of O(10 m) lead  to more faithful simulations—relative to field measurements—than centered difference  advection schemes at resolutions of O(1 m) (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2019, their supplementary  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' These two reasons make progress in simulating subtropical stratocumulus both  important and challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  –3–  manuscript submitted to journal  This paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Section 2 describes the governing equations,  numerical methods, and TPU-specific implementation decisions in our LES code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Sec-  tion 3 presents a dry buoyant bubble and a density current as validation examples of the  code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Section 4 presents the DYCOMS simulations, including comparisons with field data  and a scaling analysis of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Section 5 summarizes the conclusions and new  opportunities afforded by this TPU-enabled cloud-simulation capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  2 Model Formulation, Numerics, and TPU implementation  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1 Governing Equations  Our LES simulates the anelastic equations for moist air, understood to be an ideal  admixture of dry air, water vapor, and any condensed water that is suspended in and  moves with the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Precipitating condensate (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', rain and snow) is not considered part  of the working fluid, and the suspended constituents of the moist air are taken to be in  local thermodynamic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' By Gibbs’ phase rule, then, a complete thermodynamic  description of this system with two components (dry air and water) and three phases (wa-  ter vapor, liquid water, ice) requires specification of two thermodynamic variables, in ad-  dition to the density ρ and pressure p of the moist air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' We choose the total water spe-  cific humidity qt (total mass of water per unit mass of moist air) and liquid-ice poten-  tial temperature θl (Tripoli & Cotton, 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' This choice of thermodynamic variables  is advantageous because both the total specific humidity qt and (approximately) the liquid-  ice potential temperature θl are materially conserved even in the presence of reversible  phase transitions of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The temperature T and specific humidities ql and qi of cloud  liquid and ice can be computed from the other thermodynamic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  The anelastic approximation eliminates physically insignificant acoustic waves by  linearizing the density ρ(x, y, z, t) = ρ0(z) + ρ′(x, y, z, t) and pressure p(x, y, z, t) =  p0(z)+p′(x, y, z, t) around a dry reference state with density ρ0(z) and hydrostatic pres-  sure p0(z), which depend only on altitude z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Here, reference state variables are indicated  by a subscript 0, and perturbation variables by primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Perturbation variables are re-  tained only where they affect accelerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The reference density and pressure depend  only on the vertical coordinate z and are in hydrostatic balance,  ∂p0(z)  ∂z  = −ρ0(z)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (1)  For energetic consistency, the reference state must be adiabatic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', the reference po-  tential temperature θ0 needs to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Therefore,  T0 = θ0  � p0  p00  �Rd/cpd  ,  (2)  p0 = p00  �  1 −  gz  cpdθ0  �cpd/Rd  ,  (3)  ρ0 =  p0  RdT0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (4)  Table 1 summarizes the thermodynamic constants and other parameters used in the present  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Thermodynamic consistency of the anelastic system requires that thermodynamic  quantities are evaluated with the reference pressure p0(z) (Pauluis, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Therefore, the  liquid-ice potential temperature we use is  θl(T, ql, qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' p0) = T  Π  �  1 − Lv,0ql + Ls,0qi)  cpmT  �  ,  (5)  where  Π =  �p0(z)  p00  �Rm/cpm  (6)  –4–  manuscript submitted to journal  Table 1: Thermodynamic constants and other parameters used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Symbol  Name  Value  p00  Constant reference pressure  1017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='8 hPa  θ0  Reference potential temperature  290 K  Rd  Gas constant of dry air  287 J (kg K)−1  Rv  Gas constant of water vapor  461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='89 J (kg K)−1  cpd  Isobaric specific heat capacity of dry air  1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='5 J (kg K)−1  cpv  Isobaric specific heat capacity of water vapor  1859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='5 J (kg K)−1  cl  Specific heat capacity of liquid water  4181 J (kg K)−1  ci  Specific heat capacity of ice  2100 J (kg K)−1  Lv,0  Specific latent heat of vaporization  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='5 MJ kg−1  Ls,0  Specific latent heat of sublimation  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='83 MJ kg−1  Tf  Freezing point temperature  273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='15 K  f  Coriolis parameter  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='62 × 10−5 s−1  g  Gravitational acceleration  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='81 m s−2  cs  Smagorinsky constant  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='18  Pr  Turbulent Prandtl number  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='4  Scqt  Turbulent Schmidt number of water  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='4  is the Exner function, evaluated with the altitude-dependent reference pressure p0(z) and  the constant pressure p00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' We take water vapor and suspended cloud condensate into  account in the gas “constant“ Rm = (1−qt)Rd+(qt−qc)Rv (which is not constant be-  cause it depends on the total specific humidity qt and condensate specific humidity qc =  ql + qi) and in the isobaric specific heat cpm = (1 − qt)cpd + (qt − qc)cpv + qlcl + qici.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  With these definitions, the anelastic governing equations in conservation form are  ∇ · (ρ0u) = 0,  (7)  ∂(ρ0u)  ∂t  + ∇ · (ρ0u ⊗ u) = −ρ0∇ (α0p′) + ρ0bk − fk × ρ0(u − ug) + ∇ · (ρ0σ),  (8)  ∂(ρ0θl)  ∂t  + ∇ · (ρ0uθl) = −  1  cpmΠ∇ · (ρ0FR) + ρ0wsub  ∂θl  ∂z + 1  Pr∇ ·  �  ρ0νt∇θl  �  ,  (9)  ∂(ρ0qt)  ∂t  + ∇ · (ρ0uqt) = ρ0wsub  ∂qt  ∂z +  1  Scqt  ∇ · (ρ0νt∇qt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (10)  Here,  b = g α(θl, qt, p0) − α0(z)  α0(z)  (11)  is the buoyancy, and α0 = 1/ρ0 and α = 1/ρ are specific volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The specific vol-  ume α(θl, qt, p0) is calculated from the approximate equation of state, again with the ref-  erence pressure p0 in place of the total pressure,  α = RmT  p0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Neglected in these equations is differential settling of condensate relative to the surround-  ing air and all precipitation processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Table 2 lists the variables we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The pertur-  bation pressure p′ is obtained as solution to a Poisson equation, which follows by tak-  ing the divergence of the momentum equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The numerical algorithm for solving eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (7)  to (10) is discussed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='–5–  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='manuscript submitted to journal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Table 2: Definitions of Variables  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Variable  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Definition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Units  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='ρ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Density of moist air  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='kg m−3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='α  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Specific volume of moist air  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='m3 kg−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='u  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Velocity of moist air  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='m s−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='ug  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Prescribed geostrophic velocity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='m s−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='wsub  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Prescribed subsidence velocity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='m s−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Pressure  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Pa  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Buoyancy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='m s−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Vertical unit vector  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Temperature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Rm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Specific gas “constant” of moist air  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='J kg−1 K−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='cpm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Isobaric specific heat of moist air  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='J kg−1 K−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='σ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Subgrid-scale stress per unit mass  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='m2 s−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='FR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Radiative energy flux  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='W m kg−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='qt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Total water specific humidity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='kg/kg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='qv  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Water vapor specific humidity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='kg/kg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='ql  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Liquid water specific humidity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='kg/kg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='qi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Ice specific humidity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='kg/kg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='νt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Turbulent viscosity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='m2 s−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='Altitude  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='m  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='2 Saturation Adjustment  The temperature T and the partitioning of total water mass into the liquid phase  (specific humidity ql) and ice phase (specific humidity qi) are obtained from θl, qt, and  the reference pressure p0 by a saturation adjustment procedure (Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' This  amounts to solving  θ∗  l − θl = 0,  (12)  where θ∗  l (T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' p0) = θl(T, q∗  l , q∗  i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' p0) is the liquid-ice potential temperature at saturation,  that is, with  q∗  l = max  �  0, qt − q∗  v(T, p0)  �  H(T − Tf)  (13)  and  q∗  i = max  �  0, qt − q∗  v(T, p0)  �  H(Tf − T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (14)  Here, q∗  v is the saturation vapor pressure, calculated as in Sridhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (2022), H is the  Heaviside step function, and Tf is the freezing point temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (However, in the ex-  amples in this paper, we only obtain liquid clouds without ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=') We solve the resulting  nonlinear problem (12) with the secant method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='3 Subgrid-scale Models  We model subgrid-scale fluxes with the turbulent viscosity model of Lilly (1962)  and Smagorinsky (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' In this model, the turbulent viscosity is represented as  νt = (cs∆)2fBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (15)  where S = ∥S∥2 is the 2-norm of the strain rate tensor S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='5  �  ∇u + (∇u)T �  for the  resolved velocities u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' cs is the Smagorinsky constant (Table 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' and ∆ = (∆x1∆x2∆x3)1/3  is the geometric mean of the grid spacings in the three space directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The buoyancy  –6–  manuscript submitted to journal  factor 0 ≤ fB ≤ 1 limits the mixing length in the vertical in the case of stable strati-  fication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' it is computed from the moist buoyancy frequency (Durran & Klemp, 1982) as  described in Pressel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The diffusivities of the liquid-ice potential tempera-  ture and total specific humidity are obtained from the turbulent viscosity νt by division  by constant turbulent Prandtl and Schmidt numbers (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  To emulate a radiation condition at the upper boundary, we include a sponge layer  that occupies the top 5% of the domain and absorbs upward propagating waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The  sponge is implemented as a linear Rayleigh damping layer (Durran & Klemp, 1983), in  which the horizontal velocity is relaxed toward the geostrophic wind velocity and the ver-  tical velocity is relaxed to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' To avoid reflections at the interface between the sponge  layer and the undamped flow outside, we use a relaxation coefficient that ensures a grad-  ual onset of the sponge layer (Klemp & Lilly, 1978), reaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='25 s−1 at the top of the  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='4 Numerical Solution  We discretize the governing equations with a second-order finite-difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  All discrete operators are expressed on a collocated mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The third-order QUICK scheme  is used for the discretization of the advection terms in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (8) to (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' All diffusion terms  are computed with a second-order central difference scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  An explicit iterative scheme (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2022) is employed for the time advance-  ment of the numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' This scheme provides an iterative representation to the  Crank-Nicolson method, which avoids the computational complexity of solving a high-  dimensional linear system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Specifically, the momentum equation (8) is solved  with a predictor-corrector approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' At the prediction step of sub-iteration k+1, the  momentum equation is solved in discrete form as  �  ρ0u − (ρ0u)n  ∆t  = −ρ0∇  �  α0p′k�  + Rn+ 1  2 ,  (16)  with  Rn+ 1  2 = −∇ · [(ρ0u)n+ 1  2 ⊗ un+ 1  2 ] + ∇ ·  �  ρ0σn+ 1  2  �  + ρ0bn+ 1  2 k − fk × ρ0  �  un+ 1  2 − ug  �  , (17)  where �  (·) denotes a prediction of a variable at step n+1 in sub-iteration k +1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (·)k is  the solution of a variable at step n+1 obtained from sub-iteration k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Variables at state  (·)n+ 1  2 are estimated as (·)n+ 1  2 = [(·)k + (·)n]/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Note that the prediction of the mo-  mentum �  ρ0u in sub-iteration k+1 is evaluated with the pressure from the previous sub-  iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The correct momentum (ρ0u)k+1 needs to be computed with the pressure at  sub-iteration k + 1, which can be expressed similarly to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (16) as  (ρ0u)k+1 − (ρ0u)n  ∆t  = −ρ0∇  �  α0p′k+1�  + Rn+ 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (18)  Subtracting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (16) from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (18) yields  (ρ0u)k+1 − �  ρ0u  ∆t  = −ρ0∇  �  α0p′k+1 − α0p′k�  = −ρ0∇ (α0δp) ,  (19)  where δp = p′k+1 − p′k is the pressure correction from sub-iteration k to k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Taking the divergence of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (19) and applying mass conservation at sub-iteration  k + 1 leads to a generalized Poisson equation for the pressure correction:  ∇2 (α0δp) = α0  ∆t  �  ∇ · (�  ρ0u) − ∇ · (ρ0u)k+1�  = α0  ∆t∇ · (�  ρ0u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (20)  –7–  manuscript submitted to journal  To ensure numerical consistency and eliminate the checkerboard effect due to the col-  located mesh representation, we introduce an additional correction term when solving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (20),  which is described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  We apply homogeneous Neumann boundary conditions on the pressure, assuming  a vanishing correction of the mass flux: (ρ0u)k+1−�  ρ0u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Solving the Poisson equa-  tion (20) subject to the boundary conditions provides the pressure correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' We solve  the Poisson equation iteratively with the weighted Jacobi method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The momentum and  pressure at sub-iteration k + 1 are then updated as  (ρ0u)k+1 = �  ρ0u − ∆tρ0∇(α0δp),  (21)  pk+1 = pk + δp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (22)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='5 TPU Implementation  The discrete formulations are implemented in TensorFlow, to support execution  on different computing architectures and integration with machine learning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  In the present study, all computations are performed on TPUs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' the host CPU are used  for data input and output only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  At the beginning of each simulation, the simulator code is compiled by the Accel-  erated Linear Algebra (XLA) compiler with the just-in-time (JIT) approach, which builds  a TensorFlow graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' This approach reduces the computational cost at runtime signif-  icantly, which is particularly beneficial for simulations with repeated steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The repre-  sentations of the three-dimensional data structure and numerical operators are designed  to optimize the performance within the TensorFlow programming paradigm (Wang et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The graph is subsequently replicated onto each TPU for computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The  initial flow field data are distributed onto each TPU as input to the distributed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  On TPUs, the efficiency of partitioning is anisotropic along different spatial dimen-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' This behavior results from the data structures that are designed for optimal com-  putational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' With this programming strategy, partitioning in different directions  leads to different TensorFlow graph structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' As a result, partitioning along the first  dimension of the allocated 3D tensors is more efficient than along the other two dimen-  sions (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' We investigate the scaling of our simulation framework for dif-  ferent partitions in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='2, with an assessment of implications for cloud simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  3 Validation Study  To validate the numerical scheme and model formulation, we consider two test cases  that are widely used for validation and are relevant to the buoyancy-driven dynamics  prevalent in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The first case is a density current consisting of a two-dimensional  negatively buoyant dry bubble impinging on a surface (Straka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' the second  case is a rising buoyant bubble (Bryan & Fritsch, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1 Density Current  The density current configuration consists of an initial perturbation to a uniform  potential temperature field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The initial perturbation’s amplitude peaks at −15 K and  has a horizontal radius of 4 km and a vertical radius of 2 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The two-dimensional do-  main is 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='2 km wide and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='4 km high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' As in Pressel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (2015), we use periodic hor-  izontal boundary conditions instead of the no-flux boundary conditions in Straka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' This benchmark case has an added significance in stratocumulus simulations be-  cause the density current’s perturbation amplitude is of the same magnitude as the jump  in temperature observed across the entrainment interfacial layer at the cloud top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  –8–  manuscript submitted to journal  Figure 1 shows the potential temperature at t = 900 s for varying resolutions rang-  ing from a homogeneous resolution of 200 m to 10 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' A uniform kinematic viscosity of  10 m2 s−1 is used to make the simulations comparable across the wide range of resolu-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Since the solutions are nearly horizontally symmetric about the center of the do-  main, only the right half of the bubble is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The flow exhibits Kelvin-Helmholtz in-  stabilities that generate small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  The numerical solutions exhibit increasingly detailed small-scale features as the res-  olution is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Even at the coarsest resolution (200 m), the large-scale flow features  are preserved, and there are no signs of spurious small-scale oscillations associated with  numerical dispersion errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' These results suggest the robustness of the numerical scheme  in capturing sharp gradients and turbulence, even at coarser resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='2 Rising Bubble  The second test case is a rising dry bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The bubble is initialized as a pertur-  bation to a uniform potential temperature field, following Bryan and Fritsch (2004), with  a peak amplitude of 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' As in the first test case, we use periodic horizontal boundary  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The domain is 20 km wide and 10 km high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Figure 2 shows the potential temperature at t = 1000 s for varying homogeneous  resolutions ranging from 200 m to 10 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' For this case, a uniform kinematic viscosity of  1 m2 s−1 was found to be adequate to ensure the simulations are comparable across the  different resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  As in the density current case, there are no spurious oscillations even for the sim-  ulations with coarser resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The numerical solution is essentially converged at 50  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The vertical velocity contours are nearly unchanged from the finest resolution down  to 100 m resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' This observed stability is due in great part to the QUICK scheme  used in the scalar and momentum advection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Although the QUICK scheme achieves only  a second-order accurate approximation of the advective flux, the solutions seen in this  case suggest a quality and fidelity of simulation comparable to that of the WENO schemes  on staggered grids (Pressel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  4 DYCOMS Simulation  The first nocturnal research flight (RF01) of the Dynamics and Chemistry of Ma-  rine Stratocumulus (DYCOMS-II) field study (Stevens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2003) serves as the testbed  of our low-cloud simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Among the attractive characteristics of this test case are  the relative homogeneity of the environmental conditions, the absence of significant driz-  zle, and the persistence of a stable cloud layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The basic state for RF01 is idealized as  a quasi-two-layer structure in potential temperature θl and total-water specific humid-  ity qt (Stevens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Forcings include geostrophic winds, large-scale subsidence,  a simple parameterization of longwave radiation, and surface fluxes of latent and sen-  sible heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  We choose the initial liquid-ice potential temperature θl = 289 K and initial to-  tal specific humidity in the mixed layer to be qt = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='7 g kg−1, which is slightly lower  than the DYCOMS default value of 9 g kg−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' This ensures that with our thermodynam-  ics formulation and constants, we obtain a cloud layer between 600 and 840 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The ver-  tical domain extends to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='5 km, with a no-slip, zero-flux lid at the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The horizontal  domain in the default case covers an area of (4 km)2, with periodic horizontal bound-  ary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  The default simulation runs for 4 simulated hours on a grid of 128 × 128 × 256  points with a uniform horizontal grid spacing of 32 m and a uniform vertical grid spac-  ing of 6 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Although a vertical resolution of 5 m or less is often desirable to capture the  –9–  manuscript submitted to journal  Figure 1: Contours of potential temperature [K] in the density current simulation at  900 s at mesh resolutions of 200 m, 100 m, 50 m, and 10 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Contours of potential tem-  perature are shown at increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  sharp temperature gradient at the inversion above the cloud top without generating spu-  rious entrainment (Mellado, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Pressel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Mellado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2018), our simu-  lation did not change materially as we increased the vertical resolution to finer than 6 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  A physical time step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='3 s (Courant number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='3) is used in the default configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1 Fidelity of Simulation  The mean vertical profiles and vertical velocity statistics closely match the obser-  vations from the research flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Both the liquid potential temperature profile and the  total-water specific humidity profile maintain their two-layer structure, with a well-mixed  –10–  6  10 m  4  km  0  0  5  10  15  20  [kmmanuscript submitted to journal  Figure 2: Contours of potential temperature [K] (left) and vertical velocity [m s−1]  (right) in the rising bubble simulation for mesh resolutions of 200 m, 100 m, 50 m, and  10 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Contours of potential temperature and velocity are shown for increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='2 K  and 2 m s−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  –11–  manuscript submitted to journal  Figure 3: Profile of mean state of specific humidity and temperature in DYCOMS as ob-  served (points), from our simulation averaged over the 4th hour (red solid lines), and from  an implicit LES (Pressel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2017) using a nominally fifth-order WENO scheme (blue  dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Figure 4: Profile of the variance and skewness of the vertical velocity in DYCOMS as  observed (points), from our simulation averaged over the 4th hour (red solid lines), and  from an implicit LES (Pressel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2017) using a nominally fifth-order WENO scheme  (blue dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  boundary layer below a cloud top (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' For comparison, we also show the vertical pro-  files for an implicit LES using a nominally fifth-order WENO scheme on a staggered grid,  a configuration that has been shown to perform well for stratocumulus simulations (Pressel  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' in fact, this configuration at the resolution we use here performs favorably  relative to simulations with oscillatory numerical schemes for the momentum equation  on much higher-resolution (meter-scale) isotropic grids (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Matheou,  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Mellado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (The WENO scheme uses a fifth-order stencil for the flux  reconstruction but is strictly only of second-order accuracy for nonlinear problems on  a staggered grid (Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' hence, it is only nominally fifth order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=')  The turbulent structure in the boundary layer becomes evident in the variance and  skewness profiles of the vertical velocity (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The variance peaks near the cloud base,  consistent with the research flight observations and turbulence generation by latent heat  release at that altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The skewness reveals preferential directions of turbulent verti-  cal velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' For example, positive vertical velocity skewness near the bottom is con-  sistent with the presence of significant heat fluxes at the sea surface, which drive con-  vection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' On the other hand, the negative skewness near the cloud top is consistent with  the presence of downdrafts driven by radiative cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Like the variance, the skewness  in our simulations is consistent with the research flight observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' By contrast, most  LES in the DYCOMS intercomparison study (Stevens, 2005) are unable to capture the  negative skewness near the cloud top, likely because of excessive spurious mixing across  –12–  manuscript submitted to journal  the inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The vertical velocity statistics are an indication that our numerics with  the QUICK advection scheme avoid the excessive generation of spurious mixing across  the inversion at the cloud top, which occurs in many other LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The fidelity of the ver-  tical velocity statistics to observations is similar to that obtained with WENO schemes  (Pressel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Our simulation maintains more liquid water in the cloud (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' 3) than most other  LES in the DYCOMS intercomparison study (Stevens, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' LES often have difficul-  ties maintaining a cloud layer with sufficient liquid water because of spurious numeri-  cal mixing of dry air across the inversion at the cloud top (Pressel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2017), which  warms and dries the cloud layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  These results indicate a high fidelity of our LES to the observed flow statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Our  LES does not suffer from the shortcomings in many LES that lead to spurious turbu-  lent entrainment at the inversion and a decoupling boundary layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' it performs similarly  well as implicit LES with WENO schemes (Pressel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Therefore, it can ad-  equately capture low clouds and enable the investigation of the feedbacks that make low  clouds such an important regulator of the strength of greenhouse warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='2 Time to Solution and Scaling Analysis  The discretization schemes described above lend themselves to parallelization al-  gorithms that are well suited for the TPU infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' However, increased parallelism  generally comes at the expense of greater communication between processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' As the com-  munication overhead begins to dominate, the marginal benefit from increased parallelism  diminishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' To assess the appropriateness of the TPU simulation framework for this class  of problems, it is thus imperative to measure how well the simulation runtime scales with  increased parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  We examine the scalability of the solver using the DYCOMS case as a testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The  method for doing so is to measure the mean turnaround time for a time step under dif-  ferent mesh configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' For a fair comparison between different configurations, we  keep the spatial resolution at 35 m × 6 m and the Courant number at approximately  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='3 for all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' We find impressive scaling, notwithstanding that each simulation step  involves the solution of an elliptic (globally nonlocal) problem for the dynamic pressure  correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1  Weak Scaling  We demonstrate weak scalability by fixing the local grid size per processor and con-  sidering an ever-growing computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' We use Nk to denote the global grid  size along dimension k, �  Nk to denote the local per processor subgrid size along dimen-  sion k, and Pk to denote the number of processors assigned to dimension k in the com-  putational topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' For the first analysis, the computational domain per TPU core is  fixed with a size of �  Nx × �  Ny × �  Nz = 1024 × 36 × 1024 grid points, which is about the  largest partition size that can fit in the TPU RAM considering the data requirements  of this simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Table 3 shows that the turnaround time remains virtually unchanged  as the number of TPU cores grows from 16 to 2048, corresponding to an increase in to-  tal number of grid points from 533M to 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='2B and in physical domain size from 36 km×  18 km to (286 km)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  We repeat this weak scaling analysis using a smaller partition size that is more com-  monly encountered in atmospheric simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Table 4 demonstrates weak scalability  when the computational domain per TPU core is fixed with a size of �  Nx × �  Ny × �  Nz =  128 × 10 × 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' With about 10× speedup over real-time evolution (10 simulated days  per day, SDPD) and a 35-meter horizontal resolution, the largest physical domain at-  tainable with 2048 TPU cores in this configuration is 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='7 km×53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='8 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (Since we are  –13–  manuscript submitted to journal  using only a quarter TPU pod, the largest domain size attainable on a full TPU pod,  with 8192 cores, would be 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='4 km × 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='4 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=')  Figure 5 shows the efficiency curves for these two cases, normalized relative to the  smallest simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' It is worth noting that in the weak scaling analysis with the large  partitions, the two smallest simulations are in fact less efficient than the larger ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' This  may seem surprising at first, as efficiency normally decreases with the problem size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' How-  ever, this behavior is most likely a consequence of saturating the partition memory, which  leads to variations in memory bandwidth utilization, which seem to penalize the perfor-  mance of smaller TPU configurations more severely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' This behavior is not seen in the sec-  ond weak scaling analysis, which uses a significantly smaller partition size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='2  Strong Scaling  We now consider how the solver scales when we increase parallelism for a fixed global  problem size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Throughout this analysis, the vertical dimension has only a single parti-  tion with a total of 128 levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The total number of grid points is fixed at 134M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Three  cases are considered: (i) 2 partitions in the x direction, (ii) 4 partitions in the x direc-  tion, and (iii) 8 partitions in the x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' In each of the three cases, we try multi-  ple partitions in the y direction, starting with 16 and scaling all the way up to 128 par-  titions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' As seen in table 5, the analysis consists of increasing parallelism while propor-  tionately reducing the workload per processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Each subsequent row doubles the num-  ber of cores assigned to the y dimension while simultaneously halving the number of grid  points per core along that dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The measured speedup relative to real time reaches  a maximum of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='08 in the configuration with 1024 cores and the smallest partition size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  The speedup is illustrated in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' In all cases, the speedup curve shows clear evidence  of linear (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', perfect) strong scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Table 3: Simulation configurations for weak scalability analysis using large partitions of  size 1024 × 36 × 1024 grid points per TPU core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The grid dimensions indicated in the  middle columns do not include ghost points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The last column shows the simulated time  relative to real time in simulated days per day (SDPD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Number of cores  Grid size  SDPD  Ptot  Px  Py  Pz  Ntot  Nx (Lx)  Ny (Ly)  Nz (Lz)  16  1  16  1  533M  1020 (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='7 km)  512 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='9 km)  1020 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1 km)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='19  32  1  32  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1B  1020 (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='7 km)  1024 (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='8 km)  1020 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1 km)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='19  64  2  32  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1B  2040 (71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='4 km)  1024 (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='8 km)  1020 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1 km)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='20  128  2  64  1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='3B  2040 (71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='4 km)  2048 (71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='7 km)  1020 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1 km)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='20  256  4  64  1  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='5B  4080 (142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='8 km)  2048 (71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='7 km)  1020 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1 km)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='20  512  4  128  1  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1B  4080 (142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='8 km)  4096 (143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='4 km)  1020 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1 km)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='20  1024  8  128  1  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1B  8160 (285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='6 km)  4096 (143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='4 km)  1020 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1 km)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='20  2048  8  256  1  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='2B  8160 (285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='6 km)  8192 (286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='7 km)  1020 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1 km)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='19  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='3 Taking LES of Clouds to the Macroscale  To demonstrate the TPU framework’s capabilities for simulating clouds on large  scales, we simulated 4 hours of the DYCOMS conditions on a domain of size 285 km×  285 km×2 km using a typical DYCOMS resolution of 35 m×6 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The simulation runs  on a mesh of 32 billion grid points and requires a little less than 20 wallclock hours to  simulate 4 hours on 1024 TPU cores (70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='4 petaflops at single precision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Analysis of the  mean profiles of the conserved variables and flow field reveals dynamics that are nearly  identical to those shown in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1 for the much smaller DYCOMS domain, so we  –14–  manuscript submitted to journal  Table 4: Weak scalability analysis with a more typical partition of size 128 × 10 × 256 per  TPU core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Number of cores  Grid size  SDPD  Ptot  Px  Py  Pz  Ntot  Nx (Lx)  Ny (Ly)  Nz (Lz)  16  1  16  1  3M  124 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='3 km)  96 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='4 km)  252 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='5 km)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='56  32  1  32  1  6M  124 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='3 km)  192 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='7 km)  252 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='5 km)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='27  64  2  32  1  12M  248 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='7 km)  192 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='7 km)  252 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='5 km)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='10  128  2  64  1  24M  248 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='7 km)  384 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='4 km)  252 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='5 km)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='03  256  4  64  1  48M  496 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='4 km)  384 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='4 km)  252 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='5 km)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='00  512  4  128  1  96M  496 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='4 km)  768 (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='9 km)  252 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='5 km)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='00  1024  8  128  1  192M  992 (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='7 km)  768 (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='9 km)  252 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='5 km)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='03  2048  8  256  1  384M  992 (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='7 km)  1536 (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='8 km)  252 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='5 km)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='07  Figure 5: Normalized efficiency with respect to the weak scaling analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Table 5: Partitions for strong scalability analysis (Ntot = 134M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Number of cores  Partition size  SDPD  Ptot  Px  Py  Pz  �  Nx  �  Ny  �  Nz  32  2  16  1  512  64  128  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='83  64  2  32  1  512  32  128  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='68  128  2  64  1  512  16  128  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='97  256  2  128  1  512  8  128  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='71  64  4  16  1  256  64  128  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='19  128  4  32  1  256  32  128  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='34  256  4  64  1  256  16  128  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='62  512  4  128  1  256  8  128  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='99  128  8  16  1  128  64  128  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='45  256  8  32  1  128  32  128  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='27  512  8  64  1  128  16  128  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='19  1024  8  128  1  128  8  128  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='08  omit them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The cloud layer is visualized in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' 7 at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The visualiza-  tion reveals large spatial variability of cloud water fraction, with occasional open cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  –15–  manuscript submitted to journal  Figure 6: Strong scaling for different partitions (as in table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  These simulations demonstrate that low-cloud resolving LES are possible in domains  the size of a grid box in a typical coarse-resolution climate model, which has a footprint  of around (100 km)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' It enables three-dimensional LES to be embedded in climate model  grid boxes, to provide high-fidelity representations of cloud dynamics locally in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  5 Discussion and Conclusions  We have demonstrated that LES of low clouds are possible on TPUs and achieve  unprecedented weak and strong scaling at high numerical fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Our LES code with  a QUICK advection scheme for momentum and tracers demonstrates a fidelity to air-  craft observations that is comparable with that obtained with WENO schemes at the  same resolution, exceeding the fidelities achievable with oscillatory, numerical schemes,  or combinations of oscillatory schemes for momentum and non-oscillatory schemes for  tracers (Pressel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' At the meter-scale resolutions needed  for resolving the computationally challenging stratocumulus clouds, we have shown that  the code scales strongly and weakly up to 1024 and 2048 TPU cores, respectively, cor-  responding to a computational throughput of 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='4 and 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='8 petaflops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' This opens up  the large compute resources with fast chip-to-chip interconnects available on TPUs for  low-cloud LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' For example, it means that LES with horizontal resolutions around 30 m  and vertical resolutions around 5 m are achievable at 10 simulated days per wallclock  day in domains the size of what is becoming a typical climate model grid column (25–  50 km wide).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Thus, it is possible to generate LES with an outer horizontal scale that is  the same as the inner horizontal scale of climate models (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Our LES code and the compute resources available on TPUs enable the genera-  tion of large libraries of low-cloud simulations (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' These can be used both  for quantitatively studying mechanisms underlying low-cloud feedbacks to climate change  (Bretherton, 2015) and as training data for parameterizations of low clouds for coarse-  resolution climate models (Couvreux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Hourdin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Lopez-Gomez et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The LES code described here is publicly available for this and similar pur-  poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  6 Open Research  The source code for all simulations described in this paper is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='com/google-  research/swirl-lm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (We will provide a DOI prior to acceptance of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=')  –16–  manuscript submitted to journal  Figure 7: Volume rendering of the instantaneous cloud water specific humidity qc of a  simulated stratocumulus cloud covering a horizontal (285 km)2 footprint after 4 simulated  hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (left column) Oblique view and (right column) normal view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (top) Entire domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (middle) closeup of a corner: (oblique) 26 km  ×  26 km and (normal) 52 km  ×  26 km;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (bottom) further closeup of the same corner: (oblique) 13 km  ×  13 km, and (normal)  26 km × 13 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Acknowledgments  We thank Jason Hickey for his guidance in the early stages of this research and Tian-  jian Lu for his thorough review of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Appendix A Numerically Consistent Poisson Equation on Collocated  Grids  To eliminate the discrepancy between the numerical representation of the gradi-  ent and Laplacian operators in the Poisson eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (20), and to introduce coupling between  nodes with odd and even indices, we add an additional correction term that takes the  form of a fourth-order difference of the pressure correction δp on the right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Specifically, applying the discrete divergence operator to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (19) with the enforcement  –17–  manuscript submitted to journal  of mass conservation at sub-iteration k + 1, we have  ∇ · ∇(α0δp) = α0  ∆t∇ · (�  ρ0u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (A1)  Subtracting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (A1) from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (20) results in a correction term that takes the form:  C = (∇2 − ∇ · ∇)(α0δp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (A2)  In a discrete representation in which the divergence operator is expressed by the second-  order central difference scheme,  ∇(·) = (·)l+1 − (·)l−1  2∆l  ,  (A3)  and the Laplacian operator is expressed as  ∇2(·) = (·)l+1 − 2(·)l + (·)l−1  ∆2  l  ,  (A4)  the correction term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (A2) is computed numerically as  C =  1  2∆l  [∇(α0δp))l+1 − (∇(α0δp))l−1] − 1  ∆2  l  [α0δp)l+1 − 2(α0δp)l + (α0δp)l−1]  =  1  4∆2  l  [(α0δp)l+2 − 2(α0δp)l + (α0δp)l−2] − 1  ∆2  l  [(α0δp)l+1 − 2(α0δp)l + (α0δp)l−1]  =  1  4∆2  l  [(α0δp)l+2 − 4(α0δp)l+1 + 6(α0δp)l − 4(α0δp)l−1 + (α0δp)l−2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (A5)  To ensure eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (20) is solved with numerical consistency, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (A5) is added to the diver-  gence of the momentum on the right-hand side of the equation, which is:  ∇2(α0δp)k+1 = α0  ∆t∇ · (�  ρ0u) − Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (A6)  This ensures a numerically consistent treatment of the derivative operators and leads to  coupling of nodes with even and odd indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Tensor processing units for financial Monte Carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  arXiv preprint,  1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='02818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Blossey, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Bretherton, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Cheng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Endo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Heus, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Lock, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', &  van der Dussen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' CGILS Phase 2 LES intercomparison of response  of subtropical marine low cloud regimes to CO2 quadrupling and a CMIP3  composite forcing change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Earth Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 8, 1714–1726.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1002/2016MS000765  Blossey, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Bretherton, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Cheng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Endo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Heus, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Xu,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Marine low cloud sensitivity to an idealized climate change: The  CGILS LES intercomparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Marine boundary layer clouds at the heart of  tropical cloud feedback uncertainties in climate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=',  32, L20806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  –18–  manuscript submitted to journal  Bretherton, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Insights into low-latitude cloud feedbacks from high-  resolution models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1098/  rsta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', & Schneider, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Constraints on climate sensitivity from space-  based measurements of low-cloud reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Climate, 29, 5821–5835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' doi: 10  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1175/JCLI-D-15-0897.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1  Brient, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Schneider, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Tan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Bony, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', & Hall, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Shallowness  of tropical low clouds as a predictor of climate models’ response to warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='1007/s00382-015-2846-0  Bryan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  A reevaluation of ice–liquid water potential  temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Wea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Caldwell, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Large eddy simulation of the diurnal cycle  in southeast pacific stratocumulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='1175/  2008JAS2785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', Boer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Intercomparison and interpretation of climate  feedback processes in 19 atmospheric general circulation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', 95, 16601–16615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', Potter, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Alekseev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Barker,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Wetherald, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Cloud feedback in atmospheric general  circulation models: An update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 101, 12791–12794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1029/96JD00822  Charney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Fj¨orthoft, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & von Neumann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (1950).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Numerical integration of  the barotropic vorticity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Tellus, 2, 237–254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Chou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Neelin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Mechanisms of global warming impacts on re-  gional tropical precipitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Climate, 17, 2688–2701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Couvreux, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Hourdin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Williamson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Roehrig, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Volodina, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Villefranque,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' others  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Process-based climate model development harnessing  machine learning: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' A calibration tool for parameterization improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Earth Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 13, e2020MS002217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1029/2020MS002217  Cronin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Land-atmosphere interaction and radiative-convective equilib-  rium (Unpublished doctoral dissertation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Massachusetts Institute of Technol-  ogy, Cambridge, MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Cronin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Emanuel, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Island precipitation enhance-  ment and the diurnal cycle in radiative-convective equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Meteor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1002/qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='2443  Dufresne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', & Bony, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  An assessment of the primary sources of spread  of global warming estimates from coupled atmosphere–ocean models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Cli-  mate, 21, 5135–5144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1175/2008JCLI2239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1  Durran, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  On the effects of moisture on the Brunt-  V¨ais¨al¨a frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Durran, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  A compressible model for the simulation of  moist mountain waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' The macroturbulence of the troposphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Tellus, 51A-B, 59–70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Held, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Radiative-convective equilib-  rium with explicit two-dimensional moist convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Held, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  A scaling theory for horizontally homoge-  neous, baroclinically unstable flow on a beta-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Held, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Soden, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Robust responses of the hydrological cycle to  global warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Climate, 19, 5686–5699.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Hourdin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Williamson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Rio, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Couvreux, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Roehrig, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Villefranque, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=',  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Volodina, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Process-based climate model development harnessing  machine learning: II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' model calibration from single column to global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  –19–  manuscript submitted to journal  Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Earth Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 13, e2020MS002225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1029/2020MS002225  Jouppi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Young, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Patil, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Patterson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Agrawal, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Bajwa, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  others  (2017, June).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  In-datacenter performance analysis of a tensor pro-  cessing unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  In Proceedings of ISCA ’17 (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' 1–12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Toronto, ON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1145/3079856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='3080246  Klemp, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Lilly, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Numerical simulation of hydrostatic mountain  waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Lilly, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  On the numerical simulation of buoyant convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Tellus, 2,  148–172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Lopez-Gomez, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Christopoulos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Ervik, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Dunbar, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Cohen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', &  Schneider, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Training physics-based machine-learning parameteriza-  tions with gradient-free ensemble Kalman methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Earth Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='.  (in press)  Lu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Hechtman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Anderson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Large-scale  discrete Fourier transform on TPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' arXiv preprint, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='03260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Manabe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Smagorinsky, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Strickler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Simulated climatology of  a general circulation model with a hydrologic cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Wea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 93, 769–  798.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Manabe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Wetherald, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The effects of doubling the CO2 concentra-  tion on the climate of a general circulation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Matheou, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Turbulence structure in a stratocumulus cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Atmosphere, 9,  392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='3390/atmos9100392  Mellado, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Cloud-top entrainment in stratocumulus clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', 49, 145–169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1146/annurev-fluid-010816-06023  Mellado, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', Stevens, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Wyant, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  DNS and  LES for simulating stratocumulus: Better together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Earth Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=',  10, 1421–1438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1029/2018MS001312  Mishra, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Par´es-Pulido, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Pressel, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Arbitrarily high-order  (weighted) essentially non-oscillatory finite difference schemes for anelastic  flows on staggered meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Physics, 29, 1299–1335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  National Academies of Sciences, Engineering, and Medicine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Automated  research workflows for accelerated discovery: Closing the knowledge discovery  loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Washington, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=': The National Academies Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  O’Gorman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Schneider, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The hydrological cycle over a wide range  of climates simulated with an idealized GCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Climate, 21, 3815–3832.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Pauluis, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Thermodynamic consistency of the anelastic approximation for a  moist atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 65(2719–2729).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Pederson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Kozlowski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Song, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Beall, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Ganahl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Hauru, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Vidal,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Large scale quantum chemistry with Tensor Processing Units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='. (in press)  Phillips, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (1954).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Energy transformations and meridional circulations associ-  ated with simple baroclinic waves in a two-level, quasi-geostrophic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Tel-  lus, 6, 273–286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Phillips, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (1956).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The general circulation of the atmosphere: a numerical exper-  iment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Meteor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', 82, 123–164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Schneider, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Tan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Mishra, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Large-eddy  simulation in an anelastic framework with closed water and entropy balances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', 7, 1425–1456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Numerics  and subgrid-scale modeling in large eddy simulations of stratocumulus clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Coakley Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Fairall, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Outlook for research on subtropical marine stratification clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Meteor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', 65, 1290–1301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  doi: 1520-0477(1984)065⟨1290:  OFROSM⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Stevens, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Ochs, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Knight, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Albrecht, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Blyth, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  others  (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Rain in shallow cumulus over ocean: The RICO campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Meteor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Rhines, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Waves and turbulence on a β-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', 69, 417–  443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Rhines, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Geostrophic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 11, 401–441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Rind, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Modelling the hydrological cycle  in assessments of climate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Nature, 358, 119–122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', & Stevens, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' On the factors modulating the stratocumulus to cu-  mulus transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Schalkwijk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Weather forecasting using GPU-based large-eddy simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Meteor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='1175/BAMS-D-14-00114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1  Schneider, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' The general circulation of the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='1146/annurev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='031405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='125144  Schneider, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Possible climate transitions  from breakup of stratocumulus decks under greenhouse warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Nature  Geosci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Schneider, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Water vapor  and the dynamics of climate changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Climate goals and computing the future of clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', & Schneider, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  A library of  large-eddy simulations forced by global climate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Earth  Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='1029/2021MS002631  Siebesma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Bretherton, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Brown, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Stevens, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  A large eddy simulation intercomparison  study of shallow cumulus convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 60, 1201–1219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  General circulation experiments with the primitive equa-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Wea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', Manabe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Holloway, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Numerical results from  a nine-level general circulation model of the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Wea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 93,  727–768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Sridhar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Tissaoui, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Marras, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Shen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Kawczynski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Byrne, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Large-eddy simulations with ClimateMachine v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Geosci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Model Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 15, 6259–6284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='5194/gmd-15-6259-2022  Stevens, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Atmospheric moist convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Earth Planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=',  33, 604–643.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Stevens, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Lenschow, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Lilly, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=',  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Meteor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Bretherton, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Chlond, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', de  Roode, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Zhu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Evaluation of large-eddy simulations via obser-  vations of nocturnal marine stratocumulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Wea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 133, 1443–1462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='1175/MWR2930.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1  Straka, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Wilhelmson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Wicker, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Anderson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Droegemeier,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Numerical solutions of a non-linear density current: A bench-  mark solution and comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Methods Fluids, 17, 1–22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', Teixeira, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Pressel, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Large-eddy simula-  –21–  manuscript submitted to journal  tion of subtropical cloud-topped boundary layers: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' A forcing framework with  closed surface energy balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', 8, 1565–1585.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', Teixeira, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Pressel, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Large-eddy simulation  of subtropical cloud-topped boundary layers: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Cloud response to climate  change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Earth Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 9, 19–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1002/2016MS000655  Tao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Simpson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & McCumber, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' An ice-water saturation adjust-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Wea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 117, 231–235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Tompkins, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Craig, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Radiative–convective equilibrium in a  three-dimensional cloud-ensemble model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Meteor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 124,  2073–2097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1002/qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='49712455013  Tripoli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Cotton, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  The use of lce-liquid water potential tem-  perature as a thermodynamic variable in deep atmospheric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Wea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 109, 1094–1102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Vial, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Dufresne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Bony, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  On the interpretation of inter-model  spread in CMIP5 climate sensitivity estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 41, 3339–3362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1007/s00382-013-1725-9  Wang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Ihme, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Anderson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2022, May).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  A TensorFlow sim-  ulation framework for scientific computing of fluid flows on tensor processing  units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Webb, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Lambert, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Gregory, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Origins of differences in cli-  mate sensitivity, forcing and feedback in climate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 40, 677–  707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1007/s00382-012-1336-x  Webb, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Senior, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Sexton, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Ingram, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Williams, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=',  Ringer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' others (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' On the contribution of local feedback mech-  anisms to the range of climate sensitivity in two GCM ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=',  27, 17–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Williams, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (1988a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  The dynamical range of global circulations — I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  Climate  Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 2, 205–260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' The dynamical range of global circulations — II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content='  Radiative–convective equilibrium model intercomparison project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=' Clearing clouds  of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' others  (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  CGILS: Results from the first phase of an interna-  tional project to understand the physical mechanisms of low cloud feedbacks  in general circulation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Earth Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 5, 826–842.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1002/2013MS000246  Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Bretherton, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Blossey, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Bony, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', Brient, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', & Golaz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='  The CGILS experimental design to investigate low cloud feedbacks  ingeneral circulation models by using single-column and large-eddysimulation  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' Earth Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=', 4, M12001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
+page_content='1029/2012MS000182  –22–' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE3T4oBgHgl3EQfwAst/content/2301.04698v1.pdf'}
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+arXiv:2301.03159v1  [math.FA]  9 Jan 2023
+NUMERICAL RADIUS INEQUALITIES AND ESTIMATION OF
+ZEROS OF POLYNOMIALS
+SUVENDU JANA1, PINTU BHUNIA2 and KALLOL PAUL2
+Abstract. Let A be a bounded linear operator defined on a complex Hilbert space
+and let |A| = (A∗A)1/2 be the positive square root of A. Among other refinements of
+the well known numerical radius inequality w2(A) ≤ 1
+2∥A∗A + AA∗∥, we show that
+w2(A)
+≤
+1
+4w2 (|A| + i|A∗|) + 1
+8
+��|A|2 + |A∗|2�� + 1
+4w (|A||A∗|)
+≤
+1
+2∥A∗A + AA∗∥.
+Also, we develop inequalities involving numerical radius and spectral radius for the
+sum of the product operators, from which we derive the following inequalities
+wp(A) ≤
+1
+√
+2w(|A|p + i|A∗|p) ≤ ∥A∥p
+for all p ≥ 1. Further, we derive new bounds for the zeros of complex polynomials.
+1. Introduction
+Let H be a complex Hilbert space with usual inner product ⟨·, ·⟩ and the correspond-
+ing norm ∥ · ∥ induced by the inner product. Let B(H ) denote the C∗-algebra of
+all bounded linear operators on H .
+For A ∈ B(H ), |A| = (A∗A)1/2 is the posi-
+tive square root of A. The numerical range of A, denoted as W(A), is defined by
+W(A) = {⟨Ax, x⟩ : x ∈ H , ∥x∥ = 1} . Let ∥A∥, r(A) and w(A) denote the operator
+norm, the spectral radius and the numerical radius of A, respectively.
+Recall that
+w(A) = sup {|⟨Ax, x⟩| : x ∈ H , ∥x∥ = 1} . The numerical radius w(·) defines a norm
+on B(H ), (is equivalent to the operator norm ∥·∥) is satisfying the following inequality
+1
+2∥A∥ ≤ w(A) ≤ ∥A∥.
+(1.1)
+The first inequality becomes equality if A2 = 0 and the second one turns into equality
+if A is normal. Similar as the operator norm, numerical radius also satisfies the power
+2020 Mathematics Subject Classification. Primary 47A12, 26C10 Secondary 47A30, 30C15.
+Key words and phrases. Numerical radius, Operator norm, Frobenius companion matrix, Zeros of
+a polynomial.
+Pintu Bhunia would like to thank UGC, Govt. of India for the financial support in the form of
+SRF under the mentorship of Prof. Kallol Paul.
+1
+
+2
+S. JANA P. BHUNIA AND K. PAUL
+inequality:
+w(An) ≤ wn(A) for every n = 1, 2, 3, . . ..
+(1.2)
+It is well known that for A ∈ B(H ),
+r(A) ≤ w(A).
+(1.3)
+The inequality (1.3) is sharp. In fact, if A is normal, then r(A) = w(A) = ∥A∥. For
+A, B ∈ B(H ), we have r(AB) = r(BA) and r(An) = rn(A) for every positive integer
+n. Over the years many eminent mathematicians have studied various refinements of
+(1.1) and obtained various bounds for the zeros of a complex polynomial, we refer
+the readers to [2, 3, 5, 7, 15, 18, 22, 23] and the references therein. In [14], Kittaneh
+improved the inequalities in (1.1) to prove that
+1
+4∥A∗A + AA∗∥ ≤ w2(A) ≤ 1
+2∥A∗A + AA∗∥.
+(1.4)
+In this article, we develop new refinements of the second inequality in (1.4).
+We
+obtain inequalities involving numerical radius and spectral radius of the sum of the
+product operators, from which we achieve a nice refinement of the classical inequality
+w(A) ≤ ∥A∥. As application of the numerical radius inequalities, we give new bounds
+for the zeros of a complex monic polynomial which improve on the existing ones.
+2. Numerical radius inequalities
+We begin the section with the following lemmas.
+Lemma 2.1. [13](Generalized Cauchy-Schwarz inequality) If A ∈ B(H ) and 0 ≤ α ≤
+1, then
+|⟨Ax, y⟩|2 ≤ ⟨|A|2αx, x⟩⟨|A∗|2(1−α)y, y⟩
+for all x, y ∈ H .
+Lemma 2.2. [21](Holder-McCarthy inequality) Let A ∈ B(H ) be positive. Then the
+following inequalities hold:
+⟨Arx, x⟩ ≥ ∥x∥2(1−r)⟨Ax, x⟩r,
+when r ≥ 1
+⟨Arx, x⟩ ≤ ∥x∥2(1−r)⟨Ax, x⟩r,
+when 0 ≤ r ≤ 1
+for any x ∈ H .
+Lemma 2.3. [9](Buzano’s inequality) Let x, e, y ∈ H with ∥e∥ = 1, then
+|⟨x, e⟩⟨e, y⟩| ≤ 1
+2 (∥x∥∥y∥ + |⟨x, y⟩|).
+
+NUMERICAL RADIUS INEQUALITIES AND ESTIMATION OF ZEROS OF POLYNOMIALS
+3
+Now, we are in a position to present our results. First we develop the following upper
+bound for the numerical radius.
+Theorem 2.4. Let A ∈ B(H ). Then
+w2(A) ≤ 1
+4w2 (|A| + i|A∗|) + 1
+8
+��|A|2 + |A∗|2�� + 1
+4w (|A||A∗|) .
+Proof. Let x ∈ H with ∥x∥ = 1. Then we have
+|⟨Ax, x⟩|2
+≤
+⟨|A|x, x⟩⟨|A∗|x, x⟩ (by Lemma 2.1)
+≤
+1
+4 (⟨|A|x, x⟩ + ⟨|A∗|x, x⟩)2
+=
+1
+4
+�
+⟨|A|x, x⟩2 + ⟨|A∗|x, x⟩2 + 2⟨|A|x, x⟩⟨|A∗|x, x⟩
+�
+≤
+1
+4
+�
+|⟨|A|x, x⟩ + i⟨|A∗|x, x⟩|2 + ∥|A|x∥∥|A∗|x∥ + |⟨|A|x, |A∗|x⟩|
+�
+(by Lemma 2.3)
+≤
+1
+4
+�
+|⟨(|A| + i|A∗|)x, x⟩|2 + 1
+2∥|A|x∥2 + 1
+2∥|A∗|x∥2 + |⟨|A∗||A|x, x⟩|
+�
+≤
+1
+4w2 (|A| + i|A∗|) + 1
+8
+��|A|2 + |A∗|2�� + 1
+4w (|A||A∗|) .
+Taking supremum over all x ∈ H with ∥x∥ = 1, we get the desired inequality.
+□
+Clearly, we see that
+1
+4w2 (|A| + i|A∗|) + 1
+8
+��|A|2 + |A∗|2�� + 1
+4w (|A||A∗|)
+≤
+1
+4
+��|A|2 + |A∗|2�� + 1
+8
+��|A|2 + |A∗|2�� + 1
+4 ∥|A||A∗|∥
+=
+3
+8
+��|A|2 + |A∗|2�� + 1
+4
+��A2��
+≤
+3
+8
+��|A|2 + |A∗|2�� + 1
+8
+��|A|2 + |A∗|2��
+=
+1
+2
+��|A|2 + |A∗|2�� .
+Thus, we would like to remark that the upper bound obtained in Theorem 2.4 refines
+the second inequality in (1.4). Next result reads as follows.
+Theorem 2.5. Let X, Y ∈ B(H ), and 0 ≤ α ≤ 1, 0 ≤ β ≤ 1. Then for each x ∈ H
+with ∥x∥ = 1,
+|⟨Xx, x⟩⟨Y x, x⟩|
+≤
+1
+4
+��α|X|2 + (1 − α)|X∗|2 + β|Y |2 + (1 − β)|Y ∗|2�� + 1
+8
+��|X|2 + |Y ∗|2�� + 1
+4w(Y X).
+
+4
+S. JANA P. BHUNIA AND K. PAUL
+Proof. We have
+|⟨Xx, x⟩⟨Y x, x⟩|
+≤
+1
+4 {|⟨Xx, x⟩| + |⟨Y x, x⟩|}2
+=
+1
+4
+�
+|⟨Xx, x⟩|2 + |⟨Y x, x⟩|2 + 2|⟨Xx, x⟩||⟨Y x, x⟩|
+�
+≤
+1
+4
+�
+⟨|X|2αx, x⟩⟨|X∗|2(1−α)x, x⟩ + ⟨|Y |2βx, x⟩⟨|Y ∗|2(1−β)x, x⟩ + 2|⟨Xx, x⟩||⟨x, Y ∗x⟩|
+�
+(using Lemma 2.1)
+≤
+1
+4
+�
+⟨|X|2x, x⟩α⟨|X∗|2x, x⟩(1−α) + ⟨|Y |2x, x⟩β⟨|Y ∗|2x, x⟩(1−β) + ∥Xx∥∥Y ∗x∥ + |⟨Xx, Y ∗x⟩|
+�
+(using Lemma 2.2 and Lemma 2.3 )
+≤
+1
+4
+�
+α⟨|X|2x, x⟩ + (1 − α)⟨|X∗|2x, x⟩ + β⟨|Y |2x, x⟩ + (1 − β)⟨|Y ∗|2x, x⟩
+�
++1
+4
+�1
+2
+�
+⟨|X|2x, x⟩ + ⟨|Y ∗|2x, x⟩
+�
++ |⟨Y Xx, x⟩|
+�
+≤
+1
+4∥α|X|2 + (1 − α)|X∗|2 + β|Y |2 + (1 − β)|Y ∗|2∥ + 1
+8∥|X|2 + |Y ∗|2∥ + 1
+4w(Y X).
+□
+Applying the inequality in Theorem 2.5 we derive the following upper bound for the
+numerical radius.
+Corollary 2.6. If A ∈ B(H ), then
+w2(A) ≤ 1
+4
+��µ|A|2 + (2 − µ)|A∗|2�� + 1
+8
+��|A|2 + |A∗|2�� + 1
+4w(A2),
+for 0 ≤ µ ≤ 2.
+Proof. Putting X = Y = A in Theorem 2.5, and then taking supremum over all x ∈ H
+with ∥x∥ = 1, we get
+w2(A) ≤ 1
+4
+��(α + β)|A|2 + (2 − α − β)|A∗|2�� + 1
+8
+��|A|2 + |A∗|2�� + 1
+4w(A2),
+for 0 ≤ α, β ≤ 1. This implies the desired bound.
+□
+It follows from Corollary 2.6 that
+w2(A) ≤ 1
+4 min
+µ∈[0,2]
+��µ|A|2 + (2 − µ)|A∗|2�� + 1
+8
+��|A|2 + |A∗|2�� + 1
+4w(A2).
+(2.1)
+
+NUMERICAL RADIUS INEQUALITIES AND ESTIMATION OF ZEROS OF POLYNOMIALS
+5
+Remark 2.7. Clearly, We have
+min
+µ∈[0,2]
+1
+4
+��µ|A|2 + (2 − µ)|A∗|2�� + 1
+8
+��|A|2 + |A∗|2�� + 1
+4w(A2)
+≤
+1
+4∥|A|2 + |A∗|2∥ + 1
+8∥|A|2 + |A∗|2∥ + 1
+4w(A2) (by taking µ = 1)
+=
+3
+8∥|A|2 + |A∗|2∥ + 1
+4w(A2)
+≤
+3
+8∥|A|2 + |A∗|2∥ + 1
+4w2(A)
+≤
+3
+8∥|A|2 + |A∗|2∥ + 1
+8∥|A|2 + |A∗|2∥ (using the second inequality of (1.4))
+=
+1
+2∥|A|2 + |A∗|2∥.
+Thus, we would like to remark that inequality (2.1) is stronger than that in (1.4).
+We also note that the minimum value is not always attained for µ = 1. For example,
+consider the matrix A =
+
+
+
+0
+1
+0
+0
+0
+2
+0
+0
+0
+
+
+ . Then, min
+µ∈[0,2] ∥µ|A|2 + (2 − µ)|A∗|2∥ =
+32
+7 for
+µ = 8
+7, and we see that
+1
+4 min
+µ∈[0,2]
+��µ|A|2 + (2 − µ)|A∗|2�� + 1
+8
+��|A|2 + |A∗|2�� + 1
+4w(A2)
+=
+113
+56 ≈ 2.01785714
+<
+5
+2 = 1
+2∥|A|2 + |A∗|2∥.
+To prove our next result we need the following two lemmas. First one is a gen-
+eralization of the inequality in Lemma 2.1, and the second one is known as Bohr’s
+inequality.
+Lemma 2.8. ([19, Th. 5]) Let A, B ∈ B(H ) with |A|B = B∗|A|. Let f, g be two
+non-negative continuous functions on [0, ∞) such that f(t)g(t) = t for all t ≥ 0. Then
+|⟨ABx, y⟩| ≤ r(B)∥f(|A|)x∥∥g(|A∗|)y∥,
+for all x, y ∈ H .
+Lemma 2.9. ([24]) For i = 1, 2, · · · , n, let ai ≥ 0. Then
+� n
+�
+i=1
+ai
+�p
+≤ np−1
+n
+�
+i=1
+ap
+i ,
+for all p ≥ 1.
+By using the above lemmas we prove the following inequality involving numerical
+radius and spectral radius.
+
+6
+S. JANA P. BHUNIA AND K. PAUL
+Theorem 2.10. Let Ai, Bi ∈ B(H ) be such that |Ai|Bi = B∗
+i |Ai| for i = 1, 2, · · · , n.
+Then
+wp
+� n
+�
+i=1
+AiBi
+�
+≤ np−1
+√
+2
+w
+� n
+�
+i=1
+rp(Bi)
+�
+f 2p(|Ai|) + ig2p(|A∗
+i |)
+�
+�
+,
+for all p ≥ 1.
+Proof. Let x ∈ H with ∥x∥ = 1. Then we have
+�����⟨
+� n
+�
+i=1
+AiBi
+�
+x, x⟩
+�����
+p
+=
+�����
+n
+�
+i=1
+⟨AiBix, x⟩
+�����
+p
+≤
+� n
+�
+i=1
+|⟨AiBix, x|
+�p
+≤
+� n
+�
+i=1
+r(Bi)∥f(|Ai|)x∥∥g(|A∗
+i|)x∥
+�p
+(by Lemma 2.8)
+=
+� n
+�
+i=1
+r(Bi)⟨f 2(|Ai|)x, x⟩
+1
+2⟨g2(|A∗
+i |)x, x⟩
+1
+2
+�p
+≤
+� n
+�
+i=1
+r(Bi)⟨f 2(|Ai|)x, x⟩ + ⟨g2(|A∗
+i |)x, x⟩
+2
+�p
+≤
+np−1
+n
+�
+i=1
+rp(Bi)
+�⟨f 2(|Ai|)x, x⟩ + ⟨g2(|A∗
+i |)x, x⟩
+2
+�p
+(by Lemma 2.9)
+≤
+np−1
+2
+n
+�
+i=1
+rp(Bi)
+�
+⟨f 2(|Ai|)x, x⟩p + ⟨g2(|A∗
+i |)x, x⟩p�
+(by convexity of f(t) = tp)
+≤
+np−1
+2
+n
+�
+i=1
+rp(Bi)
+�
+⟨f 2p(|Ai|)x, x⟩ + ⟨g2p(|A∗
+i |)x, x⟩
+�
+(by Lemma 2.2)
+≤
+np−1
+√
+2
+�����
+n
+�
+i=1
+rp(Bi)
+�
+⟨f 2p(|Ai|)x, x⟩ + i⟨g2p(|A∗
+i |)x, x⟩
+�
+�����
+(as |a + b| ≤
+√
+2|a + ib| for all a, b ∈ R)
+≤
+np−1
+√
+2
+�����
+n
+�
+i=1
+rp(Bi)⟨
+�
+f 2p(|Ai|) + ig2p(|A∗
+i |)
+�
+x, x⟩
+�����
+≤
+np−1
+√
+2 w
+� n
+�
+i=1
+rp(Bi)
+�
+f 2p(|Ai|) + ig2p(|A∗
+i |)
+�
+�
+.
+
+NUMERICAL RADIUS INEQUALITIES AND ESTIMATION OF ZEROS OF POLYNOMIALS
+7
+Now, taking supremum over all x ∈ H , ∥x∥ = 1 we get,
+wp
+� n
+�
+i=1
+AiBi
+�
+≤ np−1
+√
+2 w
+� n
+�
+i=1
+rp(Bi)
+�
+f 2p(|Ai|) + ig2p(|A∗
+i |)
+�
+�
+.
+as desired.
+□
+Observe that the inequality in Theorem 2.10 indeed does not depend on the number
+n of summands in the case p = 1. In particular, considering p = n = 1, A1 = A,
+B1 = B, f(t) = g(t) =
+√
+t in Theorem 2.10, we get the following corollary.
+Corollary 2.11. Let A, B ∈ B(H ) be such that |A|B = B∗|A|. Then
+w(AB) ≤ 1
+√
+2r(B)w(|A| + i|A∗|).
+In particular, for B = I we have the following inequality (also obtained in [8]):
+w(A) ≤ 1
+√
+2w(|A| + i|A∗|).
+(2.2)
+Note that the bound (2.2) refines that in (1.4), see [8, Remark 2.16]. Again, considering
+Bi = I for i = 1, 2, · · · , n in Theorem 2.10 we have the following inequality for the sum
+of operators.
+Corollary 2.12. Let Ai ∈ B(H ) for i = 1, 2, · · · , n, and let f, g be two non-negative
+continuous functions on [0, ∞) such that f(t)g(t) = t for all t ≥ 0. Then
+wp
+� n
+�
+i=1
+Ai
+�
+≤ np−1
+√
+2 w
+� n
+�
+i=1
+�
+f 2p(|Ai|) + ig2p(|A∗
+i |)
+�
+�
+,
+for all p ≥ 1.
+In particular, for n = 1 and f(t) = g(t) =
+√
+t in Corollary 2.12, we get the following
+upper bound for the numerical radius.
+Corollary 2.13. If A ∈ B(H ), then
+wp(A) ≤ 1
+√
+2w(|A|p + i|A∗|p),
+for all p ≥ 1.
+It is easy to verify that
+1
+√
+2w(|A|p +i|A∗|p) ≤ ∥A∥p for all p ≥ 1. Therefore, we would
+like to remark that Corollary 2.13 improves the classical bound w(A) ≤ ∥A∥ for all
+p ≥ 1.
+At the end of this section, we give a sufficient condition for the equality of w(A) =
+1
+2∥A∗A + AA∗∥1/2. For this purpose first we note the following known lemma.
+
+8
+S. JANA P. BHUNIA AND K. PAUL
+Lemma 2.14. [17] Let A, B ∈ B(H ) be positive. Then, ∥A + B∥ = ∥A∥ + ∥B∥ if and
+only if ∥AB∥ = ∥A∥∥B∥.
+Theorem 2.15. Let A ∈ B(H ). Then ∥A∥4 = ∥ℜ2(A)ℑ2(A)∥ implies
+w2(A) = 1
+4∥A∗A + AA∗∥.
+Proof. We have
+∥A∥4
+=
+∥ℜ2(A)ℑ2(A)∥ ≤ ∥ℜ2(A)∥∥ℑ2(A)∥ = ∥ℜ(A)∥2∥ℑ(A)∥2
+≤
+1
+2
+�
+∥ℜ(A)∥4 + ∥ℑ(A)∥4�
+≤ max
+�
+∥ℜ(A)∥4, ∥ℑ(A)∥4�
+≤
+w4(A) ≤ ∥A∥4.
+This implies that
+∥ℜ2(A)ℑ2(A)∥ = ∥ℜ(A)∥2∥ℑ(A)∥2.
+(2.3)
+Also, we have
+1
+2
+�
+∥ℜ(A)∥4 + ∥ℑ(A)∥4�
+= max
+�
+∥ℜ(A)∥4, ∥ℑ(A)∥4�
+= w4(A).
+(2.4)
+This implies that
+∥ℜ(A)∥ = ∥ℑ(A)∥ = w(A).
+(2.5)
+Now, by using lemma 2.14, it follows from the identity (2.3) that
+1
+2∥ℜ2(A) + ℑ2(A)∥
+=
+1
+2
+�
+∥ℜ2(A)∥ + ∥ℑ2(A)∥
+�
+=
+1
+2
+�
+∥ℜ(A)∥2 + ∥ℑ(A)∥2�
+=
+∥ℜ(A)∥2 = w2(A) (using (2.5)).
+This completes the proof.
+□
+It should be mentioned here that the converse of Theorem 2.15 is not true, in gen-
+eral. For example, we consider A =
+
+
+
+0
+3
+0
+0
+0
+0
+0
+0
+1
+
+
+ . Then, w2(A) = 1
+4∥A∗A + AA∗∥ = 9
+4,
+however ∥A∥4 ̸= ∥ℜ2(A)ℑ2(A)∥.
+
+NUMERICAL RADIUS INEQUALITIES AND ESTIMATION OF ZEROS OF POLYNOMIALS
+9
+3. Estimation of zeros of polynomials
+Suppose p(z) = zn +anzn−1 +. . .+a2z +a1 is a complex monic polynomial of degree
+n ≥ 2 and a1 ̸= 0. Location of the zeros of p(z) have been obtained by applying
+numerical radius inequalities to Frobenius companion matrix of the polynomial p(z).
+The Frobenius companion matrix of the polynomial p(z) is given by
+Cp =
+
+
+
+
+
+
+
+
+−an
+−an−1
+....
+−a2
+−a1
+1
+0
+...
+0
+0
+0
+1
+...
+0
+0
+...
+...
+...
+...
+...
+0
+0
+....
+1
+0
+
+
+
+
+
+
+
+
+.
+The characteristic polynomial of Cp is the polynomial p(z). Thus, the zeros of p(z) are
+exactly the eigenvalues of Cp, see [12, p. 316]. The square of Cp is given by
+C2
+p =
+
+
+
+
+
+
+
+
+
+
+
+bn
+bn−1
+.....
+b3
+b2
+b1
+−an
+−an−1
+....
+−a3
+−a2
+−a1
+1
+0
+...
+0
+0
+0
+0
+1
+...
+0
+0
+0
+...
+...
+...
+...
+...
+...
+0
+0
+....
+1
+0
+0
+
+
+
+
+
+
+
+
+
+
+
+,
+where bj = anaj − aj−1 for j = 1, 2, . . . , n, with a0 = 0.
+Also,
+C3
+p =
+
+
+
+
+
+
+
+
+
+
+
+
+
+cn
+cn−1
+.....
+c4
+c3
+c2
+c1
+bn
+bn−1
+.....
+b4
+b3
+b2
+b1
+−an
+−an−1
+....
+−a4
+−a3
+−a2
+−a1
+1
+0
+...
+0
+0
+0
+0
+0
+1
+...
+0
+0
+0
+0
+...
+...
+...
+...
+...
+...
+...
+0
+0
+....
+1
+0
+0
+0
+
+
+
+
+
+
+
+
+
+
+
+
+
+,
+where bj = anaj − aj−1 and cj = −anbj + an−1aj − aj−2 for j = 1, 2, . . . , n, with
+a0 = a−1 = 0,
+
+10
+S. JANA P. BHUNIA AND K. PAUL
+and
+C4
+p =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+dn
+dn−1
+.....
+d5
+d4
+d3
+d2
+d1
+cn
+cn−1
+.....
+c5
+c4
+c3
+c2
+c1
+bn
+bn−1
+.....
+b5
+b4
+b3
+b2
+b1
+−an
+−an−1
+....
+−a5
+−a4
+−a3
+−a2
+−a1
+1
+0
+...
+0
+0
+0
+0
+0
+0
+1
+...
+0
+0
+0
+0
+0
+...
+...
+...
+...
+...
+...
+...
+...
+0
+0
+....
+1
+0
+0
+0
+0
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+,
+where bj = anaj − aj−1, cj = −anbj + an−1aj − aj−2, and dj = −ancj − an−1bj−1 +
+an−2aj − aj−3 for j = 1, 2, . . . , n, with a0 = a−1 = a−2 = 0.
+The exact value of ∥Cp∥ is well known (see in [18]), it is given by
+∥Cp∥ =
+�
+α + 1 +
+�
+(α + 1)2 − 4|a1|2
+2
+,
+(3.1)
+where α = �n
+j=1 |aj|2.
+An estimation of ∥C2
+p∥ obtained in [16] is as follows
+∥C2
+p∥ ≤
+�
+δ + 1 +
+�
+(δ − 1)2 + 4δ′
+2
+,
+(3.2)
+where δ = 1
+2
+�
+α + β +
+�
+(α − β)2 + 4|γ|2
+�
+and δ′ = 1
+2
+�
+α′ + β′ +
+�
+(α′ − β′)2 + 4|γ′|2
+�
+,
+α = �n
+j=1 |aj|2, β = �n
+j=1 |bj|2, α′ = �n
+j=3 |aj|2, β′ = �n
+j=3 |bj|2, γ = − �n
+j=1 ¯ajbj,
+γ′ = − �n
+j=3 ¯ajbj.
+We note that
+∥C2
+p∥
+1
+2 ≤
+
+
+�
+δ + 1 +
+�
+(δ − 1)2 + 4δ′
+2
+
+
+1/2
+≤
+�
+α + 1 +
+�
+(α + 1)2 − 4|a1|2
+2
+= ∥Cp∥.
+Motivated by the above estimation, here we will obtain an estimation of ∥C4
+p∥1/4.
+For this purpose first we note the following norm inequality for the sum of two positive
+operators.
+Lemma 3.1. [17] If A, B ∈ B(H ) are positive, then
+∥A + B∥ ≤ 1
+2
+�
+∥A∥ + ∥B∥ +
+�
+(∥A∥ − ∥B∥)2 + 4
+���A
+1
+2B
+1
+2
+���
+2
+�
+.
+
+NUMERICAL RADIUS INEQUALITIES AND ESTIMATION OF ZEROS OF POLYNOMIALS
+11
+Now, we are in a position to obtain an estimation of ∥C4
+p∥1/4. Let C4
+p = R + S + T,
+where
+R =
+
+
+
+
+
+
+
+
+dn
+dn−1
+.....
+d5
+d4
+d3
+d2
+d1
+cn
+cn−1
+.....
+c5
+c4
+c3
+c2
+c1
+0
+0
+.....
+0
+0
+0
+0
+0
+...
+...
+...
+...
+...
+...
+...
+...
+0
+0
+....
+0
+0
+0
+0
+0
+
+
+
+
+
+
+
+
+,
+S =
+
+
+
+
+
+
+
+
+
+
+
+
+
+0
+0
+.....
+0
+0
+0
+0
+0
+0
+0
+.....
+0
+0
+0
+0
+0
+bn
+bn−1
+.....
+b5
+b4
+b3
+b2
+b1
+−an
+−an−1
+....
+−a5
+−a4
+−a3
+−a2
+−a1
+0
+0
+...
+0
+0
+0
+0
+0
+...
+...
+...
+...
+...
+...
+...
+...
+0
+0
+....
+0
+0
+0
+0
+0
+
+
+
+
+
+
+
+
+
+
+
+
+
+and
+T =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+0
+0
+.....
+0
+0
+0
+0
+0
+0
+0
+.....
+0
+0
+0
+0
+0
+0
+0
+.....
+0
+0
+0
+0
+0
+0
+0
+....
+0
+0
+0
+0
+0
+1
+0
+...
+0
+0
+0
+0
+0
+0
+1
+...
+0
+0
+0
+0
+0
+...
+...
+...
+...
+...
+...
+...
+...
+0
+0
+....
+1
+0
+0
+0
+0
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+.
+Now,
+∥C4
+p∥2
+=
+∥R + S + T∥2
+=
+∥(R + S + T)∗(R + S + T)∥
+=
+∥R∗R + S∗S + T ∗T∥ (since R∗S = R∗T = S∗R = S∗T = T ∗R = T ∗S = 0)
+≤
+∥R∗R + S∗S∥ + ∥T ∗T∥
+≤
+1
+2
+�
+∥R∥2 + ∥S∥2 +
+�
+(∥R∥2 − ∥S∥2)2 + 4∥RS∗∥2
+�
++ 1 (using Lemma 3.1).
+By simple calculations, we have
+∥R∥2
+=
+∥R∗R∥ = ∥RR∗∥
+=
+1
+2
+�
+α1 + β1 +
+�
+(α1 − β1)2 + 4|γ1|2
+�
+= δ1,
+
+12
+S. JANA P. BHUNIA AND K. PAUL
+where α1 = �n
+j=1 |dj|2, β1 = �n
+j=1 |cj|2 , γ1 = �n
+j=1 dj ¯cj,
+∥S∥2
+=
+∥S∗S∥ = ∥SS∗∥
+=
+1
+2
+�
+α + β +
+�
+(α − β)2 + 4|γ|2
+�
+= δ,
+where α = �n
+j=1 |aj|2, β = �n
+j=1 |bj|2 , γ = − �n
+j=1 bj ¯aj,
+∥RS∗∥2
+=
+1
+2
+�
+|γ2|2 + |γ3|2 + |γ4|2 + |γ5|2 +
+�
+((|γ2|2 + |γ3|2) − (|γ4|2 + |γ5|2))2 + 4|γ2 ¯γ4 + γ3 ¯γ5|2
+�
+=
+δ2,
+where γ2 = �n
+j=1 dj ¯bj, γ3 = �n
+j=1 dj ¯aj, γ4 = �n
+j=1 cj ¯bj, γ5 = �n
+j=1 cj ¯aj.
+Therefore,
+∥C4
+p∥ ≤
+�
+1
+2
+�
+δ1 + δ +
+�
+(δ1 − δ)2 + 4δ2
+�
++ 1.
+(3.3)
+We observe that the estimation of ∥C4
+p∥1/4 in (3.3) is incomparable with the existing
+estimation of ∥C2
+p∥1/2 in (3.2). In the following theorem we derive an upper bound for
+the spectral radius of the Frobenius companion matrix Cp, by using the estimations in
+(3.2) and (3.3).
+Theorem 3.2. The following inequality holds:
+r(Cp) ≤
+�
+1
+4
+�
+δ + 1 +
+�
+(δ − 1)2 + 4δ′
+2
+�
++ 3
+4
+�1
+2
+�
+δ1 + δ +
+�
+(δ1 − δ)2 + 4δ2
+�
++ 1
+� 1
+2� 1
+4
+,
+where δ′ = 1
+2
+�
+α′ + β′ +
+�
+(α′ − β′)2 + 4|γ′|2
+�
+,
+δ = 1
+2
+�
+α + β +
+�
+(α − β)2 + 4|γ|2
+�
+,
+δ1 = 1
+2
+�
+α1 + β1 +
+�
+(α1 − β1)2 + 4|γ1|2
+�
+,
+δ2 = 1
+2
+�
+|γ2|2 + |γ3|2 + |γ4|2 + |γ5|2 +
+�
+((|γ2|2 + |γ3|2) − (|γ4|2 + |γ5|2))2 + 4|γ2 ¯γ4 + γ3 ¯γ5|2
+�
+,
+α′ = �n
+j=3 |aj|2, β′ = �n
+j=3 |bj|2, γ′ = − �n
+j=3 ¯ajbj,
+α = �n
+j=1 |aj|2, β = �n
+j=1 |bj|2 , γ = − �n
+j=1 bj ¯aj,
+α1 = �n
+j=1 |dj|2, β1 = �n
+j=1 |cj|2 , γ1 = �n
+j=1 dj ¯cj,
+γ2 = �n
+j=1 dj ¯bj, γ3 = �n
+j=1 dj ¯aj, γ4 = �n
+j=1 cj ¯bj, γ5 = �n
+j=1 cj ¯aj.
+Proof. Let A ∈ B(H ). Putting A = A2 in the inequality w2(A) ≤ 1
+4∥A∗A + AA∗∥ +
+1
+2w(A2) (see [1, Th. 2.4]), we get
+w2(A2)
+≤
+1
+4
+��|A2|2 + |(A∗)2|2�� + 1
+2w(A4).
+
+NUMERICAL RADIUS INEQUALITIES AND ESTIMATION OF ZEROS OF POLYNOMIALS
+13
+It follows that
+r2(A) = r(A2) ≤ w(A2) ≤
+�1
+4
+��|A2|2 + |(A∗)2|2�� + 1
+2w(A4)
+� 1
+2
+,
+i.e.,
+r(A) ≤
+�1
+4
+��|A2|2 + |(A∗)2|2�� + 1
+2w(A4)
+� 1
+4
+.
+(3.4)
+Now, it follows from (3.4) and the inequality ∥C∗
+pCp + CpC∗
+p∥ ≤ ∥Cp∥2 + ∥C2
+p∥ (see [6,
+Remark 3.9]) that
+r(Cp)
+≤
+�1
+4
+��|C2
+p|2 + |(C∗
+p)2|2�� + 1
+2w(C4
+p)
+� 1
+4
+≤
+�1
+4(∥C2
+p∥2 + ∥C4
+p∥) + 1
+2∥C4
+p∥
+� 1
+4
+≤
+�1
+4
+��C2
+p
+��2 + 3
+4
+��C4
+p
+��
+� 1
+4
+.
+Therefore, the required inequality follows by using the estimations in (3.2) and (3.3).
+□
+By using the fact |λj(Cp)| ≤ r(Cp), where λj(Cp) is the j-th eigenvalue of Cp, we
+infer the following estimation for the zeros of the polynomial p(z).
+Theorem 3.3. If z is any zero of p(z), then
+|z| ≤
+�
+1
+4
+�
+δ + 1 +
+�
+(δ − 1)2 + 4δ′
+2
+�
++ 3
+4
+�1
+2
+�
+δ1 + δ +
+�
+(δ1 − δ)2 + 4δ2
+�
++ 1
+� 1
+2� 1
+4
+,
+where δ, δ1, δ2 and δ′ are same as in Theorem 3.2.
+Applying the spectral mapping theorem, we conclude that if z is any zero of p(z) then
+|z| ≤ ∥C4
+p∥
+1
+4. Thus, by using the inequality (3.3) we achieve another new estimation
+for the zeros of p(z).
+Theorem 3.4. If z is any zero of p(z), then
+|z| ≤
+�1
+2
+�
+δ1 + δ +
+�
+(δ1 − δ)2 + 4δ2
+�
++ 1
+� 1
+8
+,
+where δ, δ1 and δ2 are given in Theorem 3.2.
+Again, putting A = A2 in the inequality w(A) ≤ 1
+2
+�
+∥A∥ + ∥A2∥
+1
+2
+�
+(see [16, Th. 1]),
+and proceeding as (3.4), we get
+r(A)
+≤
+�1
+2∥A2∥ + 1
+2∥A4∥
+1
+2
+� 1
+2
+.
+(3.5)
+
+14
+S. JANA P. BHUNIA AND K. PAUL
+Proceeding similarly as in Theorem 3.2 we obtain the following estimation by using
+the inequalities in (3.5), (3.2) and (3.3).
+Theorem 3.5. If z is any zero of p(z), then
+|z| ≤
+
+
+
+1
+2
+�
+δ + 1 +
+�
+(δ − 1)2 + 4δ′
+2
++ 1
+2
+�1
+2
+�
+δ1 + δ +
+�
+(δ1 − δ)2 + 4δ2
+�
++ 1
+� 1
+4
+
+
+
+1
+2
+,
+where δ, δ1, δ1 and δ′ are given in Theorem 3.2.
+Finally, we compare the bounds obtained here for the zeros of p(z) with the existing
+ones. First we note some well known existing bounds. Let z be any zero of p(z). Then
+Linden [20] obtained that
+|z| ≤ |an|
+n
++
+�
+n − 1
+n
+�
+n − 1 +
+n
+�
+j=1
+|aj|2 − |an|2
+n
+�� 1
+2
+.
+Montel [11, Th. 3] obtained that
+|z| ≤ max {1, |a1| + · · · + |an|} .
+Cauchy [12] obtained that
+|z| ≤ 1 + max {|a1|, · · · , |an|} .
+Kittaneh [15] proved that
+|z| ≤ 1
+2
+
+
+|an| + 1 +
+�
+�
+�
+�
+�(|an| − 1)2 + 4
+�
+�
+�
+�
+n−1
+�
+j=1
+|aj|2
+
+
+ .
+Fujii and Kubo [10] obtained that
+|z| ≤ cos
+π
+n + 1 + 1
+2
+
+|an| +
+�
+�
+�
+�
+n
+�
+j=1
+|aj|2
+
+ .
+Bhunia and Paul [4, Th. 2.6] proved that
+|z|2 ≤ cos2
+π
+n + 1 + |an−1| + 1
+4
+�
+|an| + √α
+�2 + 1
+2
+�
+α − |an|2 + 1
+2
+√α,
+where α = �n
+j=1 |aj|2.
+We consider a polynomial p(z) = z3 + z2 + 1
+2z + 1. Different upper bounds for the
+modulus of the zeros of this polynomial, mentioned above, are as shown in the following
+table.
+
+NUMERICAL RADIUS INEQUALITIES AND ESTIMATION OF ZEROS OF POLYNOMIALS
+15
+Linden [20]
+1.9492
+Montel[11]
+2.5
+Cauchy[12]
+2
+Kittaneh[15]
+2.0547
+Fujii and Kubo[10]
+1.9571
+Bhunia and Paul[4]
+1.96761
+However, Theorem 3.3 gives |z| ≤ 1.38047091798, Theorem 3.4 gives |z| ≤ 1.3798438819
+and Theorem 3.5 gives |z| ≤ 1.381095966, which are better than the above mentioned
+bounds.
+Statements & Declarations:
+Funding. The authors declare that no funds, grants, or other support were received
+during the preparation of this manuscript.
+Competing interests. The authors have no relevant financial or non-financial inter-
+ests to disclose.
+Data availability statements.
+Data sharing not applicable to this article as no
+datasets were generated or analysed during the current study.
+Author Contributions. All authors have contributed equally in the preparation of
+the manuscript.
+References
+1. A. Abu-Omar and F. Kittaneh, Upper and lower bounds for numerical radius with an application
+to involution operators, Rocky Mountain J. Math. 45 (2015), no. 4, 1055–1064.
+2. P. Bhunia, S.S. Dragomir, M.S. Moslehian and K. Paul, Lectures on Numerical Radius
+Inequalities, Infosys Science Foundation Series in Mathematical Sciences, Springer, (2022).
+https://doi.org/10.1007/978-3-031-13670-2
+3. P. Bhunia, S. Jana, M.S. Moslehian and K. Paul, Improved inequalities for the numerical radius
+via Cartesian decomposition, Funct. Anal. Appl. (2022), to appear.
+4. P. Bhunia and K. Paul, Annular bounds for the zeros of a polynomial from companion matrices,
+Adv. Oper. Theory 7 (2022), no. 1, Paper No. 8, 19 pp.
+5. P. Bhunia and K. Paul, Proper improvement of well-known numerical radius inequalities and their
+applications. Results Math. 76 (2021), no. 4, Paper No. 177, 12 pp.
+6. P. Bhunia and K. Paul, Furtherance of numerical radius inequality of Hilbert space operators,
+Arch. Math. (Basel) 117 (2021), no. 5, 537–546.
+7. P. Bhunia, S. Bag and K. Paul, Bounds for zeros of a polynomial using numerical radius of Hilbert
+space operators, Ann. Funct. Anal. 12 (2021), no. 2, Paper No. 21, 14 pp.
+8. P. Bhunia and K. Paul, New upper bounds for the numerical radius of Hilbert space operators,
+Bull. Sci. Math. 167 ( 2021), Paper No. 102959, 11 pp.
+9. M.L. Buzano, Generalizzatione della diseguaglianza di Cauchy-Schwarz, Rend. Sem. Mat. Univ.
+e Politech. Torino 31(1971/73) (1974) 405–409.
+
+16
+S. JANA P. BHUNIA AND K. PAUL
+10. M. Fujii and F. Kubo, Buzano’s inequality and bounds for roots of algebraic equations, Proc.
+Amer. Math. Soc. 117 (1993), no. 2, 359–361.
+11. M. Fujii and F. Kubo, Operator norms as bounds for roots of algebraic equations, Proc. Japan
+Acad. Sci. 49 (1973), 805–808.
+12. R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge, 1985.
+13. T. Kato, Notes on some inequalities for linear operators, Math. Ann. 125 (1952), 208–212.
+14. F. Kittaneh, Numerical radius inequalities for Hilbert space operators, Studia Math. 168 (2005),
+no. 1, 73–80.
+15. F. Kittaneh, Bounds for the zeros of polynomials from matrix inequalities, Arch. Math. (Basel)
+81 (2003), no. 5, 601–608.
+16. F. Kittaneh, A numerical radius inequality and an estimate for the numerical radius of the Frobe-
+nius companion matrix, Studia Math. 158 (2003), no. 1, 11–17.
+17. F. Kittaneh, Norm inequalities for sums of positive operators, J. Operator Theory 48 (2002),
+95–103.
+18. F. Kittaneh, Singular values of companion matrices and bounds on zeros of polynomials, SIAM
+J. Matrix Anal. Appl. 16 (1995), 333–340.
+19. F. Kittaneh, Notes on some inequalities for Hilbert space operators, Publ. Res. Inst. Math. Sci.
+24 (1988), no. 2, 283–293.
+20. H. Linden, Bounds for zeros of polynomials using traces and determinants, Seminarberichte Fachb.
+Math. FeU Hagen. 69 (2000), 127–146.
+21. C.A. McCarthy, Cp , Israel J. Math. 5 (1967), 249–271.
+22. S. Sahoo, N.C. Rout and M. Sababheh, Some extended numerical radius inequalities, Linear
+Multilinear Algebra 69 (2021), no. 5, 907–920.
+23. Y. Seo, Numerical radius inequalities related to the geometric means of negative power, Oper.
+Matrices 13 (2019), no. 2, 489–493.
+24. M.P. Vasi´c and D.J. Keˆcki´c, Some inequalities for complex numbers, Math. Balkanica 1 (1971),
+282–286.
+1 Department of Mathematics, Mahisadal Girls’ College, Purba Medinipur 721628,
+West Bengal, India
+Email address: janasuva8@gmail.com
+2 Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal,
+India
+Email address: pintubhunia5206@gmail.com
+Email address: kalloldada@gmail.com; kallol.paul@jadavpuruniversity.in
+
diff --git a/PNE1T4oBgHgl3EQfaQRP/content/tmp_files/load_file.txt b/PNE1T4oBgHgl3EQfaQRP/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..31007300adee94928098a0e8f9a49a11c030634e
--- /dev/null
+++ b/PNE1T4oBgHgl3EQfaQRP/content/tmp_files/load_file.txt
@@ -0,0 +1,718 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf,len=717
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='03159v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='FA]  9 Jan 2023  NUMERICAL RADIUS INEQUALITIES AND ESTIMATION OF  ZEROS OF POLYNOMIALS  SUVENDU JANA1, PINTU BHUNIA2 and KALLOL PAUL2  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Let A be a bounded linear operator defined on a complex Hilbert space  and let |A| = (A∗A)1/2 be the positive square root of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Among other refinements of  the well known numerical radius inequality w2(A) ≤ 1  2∥A∗A + AA∗∥, we show that  w2(A)  ≤  1  4w2 (|A| + i|A∗|) + 1  8  ��|A|2 + |A∗|2�� + 1  4w (|A||A∗|)  ≤  1  2∥A∗A + AA∗∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Also, we develop inequalities involving numerical radius and spectral radius for the  sum of the product operators, from which we derive the following inequalities  wp(A) ≤  1  √  2w(|A|p + i|A∗|p) ≤ ∥A∥p  for all p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Further, we derive new bounds for the zeros of complex polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Introduction  Let H be a complex Hilbert space with usual inner product ⟨·, ·⟩ and the correspond-  ing norm ∥ · ∥ induced by the inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Let B(H ) denote the C∗-algebra of  all bounded linear operators on H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  For A ∈ B(H ), |A| = (A∗A)1/2 is the posi-  tive square root of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' The numerical range of A, denoted as W(A), is defined by  W(A) = {⟨Ax, x⟩ : x ∈ H , ∥x∥ = 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Let ∥A∥, r(A) and w(A) denote the operator  norm, the spectral radius and the numerical radius of A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Recall that  w(A) = sup {|⟨Ax, x⟩| : x ∈ H , ∥x∥ = 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' The numerical radius w(·) defines a norm  on B(H ), (is equivalent to the operator norm ∥·∥) is satisfying the following inequality  1  2∥A∥ ≤ w(A) ≤ ∥A∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='1)  The first inequality becomes equality if A2 = 0 and the second one turns into equality  if A is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Similar as the operator norm, numerical radius also satisfies the power  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Primary 47A12, 26C10 Secondary 47A30, 30C15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Numerical radius, Operator norm, Frobenius companion matrix, Zeros of  a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Pintu Bhunia would like to thank UGC, Govt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' of India for the financial support in the form of  SRF under the mentorship of Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Kallol Paul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  1  2  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' JANA P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' BHUNIA AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' PAUL  inequality:  w(An) ≤ wn(A) for every n = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='.  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='2)  It is well known that for A ∈ B(H ),  r(A) ≤ w(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='3)  The inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='3) is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' In fact, if A is normal, then r(A) = w(A) = ∥A∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' For  A, B ∈ B(H ), we have r(AB) = r(BA) and r(An) = rn(A) for every positive integer  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Over the years many eminent mathematicians have studied various refinements of  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='1) and obtained various bounds for the zeros of a complex polynomial, we refer  the readers to [2, 3, 5, 7, 15, 18, 22, 23] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' In [14], Kittaneh  improved the inequalities in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='1) to prove that  1  4∥A∗A + AA∗∥ ≤ w2(A) ≤ 1  2∥A∗A + AA∗∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='4)  In this article, we develop new refinements of the second inequality in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  We  obtain inequalities involving numerical radius and spectral radius of the sum of the  product operators, from which we achieve a nice refinement of the classical inequality  w(A) ≤ ∥A∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' As application of the numerical radius inequalities, we give new bounds  for the zeros of a complex monic polynomial which improve on the existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Numerical radius inequalities  We begin the section with the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' [13](Generalized Cauchy-Schwarz inequality) If A ∈ B(H ) and 0 ≤ α ≤  1, then  |⟨Ax, y⟩|2 ≤ ⟨|A|2αx, x⟩⟨|A∗|2(1−α)y, y⟩  for all x, y ∈ H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' [21](Holder-McCarthy inequality) Let A ∈ B(H ) be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Then the  following inequalities hold:  ⟨Arx, x⟩ ≥ ∥x∥2(1−r)⟨Ax, x⟩r,  when r ≥ 1  ⟨Arx, x⟩ ≤ ∥x∥2(1−r)⟨Ax, x⟩r,  when 0 ≤ r ≤ 1  for any x ∈ H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' [9](Buzano’s inequality) Let x, e, y ∈ H with ∥e∥ = 1, then  |⟨x, e⟩⟨e, y⟩| ≤ 1  2 (∥x∥∥y∥ + |⟨x, y⟩|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  NUMERICAL RADIUS INEQUALITIES AND ESTIMATION OF ZEROS OF POLYNOMIALS  3  Now, we are in a position to present our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' First we develop the following upper  bound for the numerical radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Let A ∈ B(H ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Then  w2(A) ≤ 1  4w2 (|A| + i|A∗|) + 1  8  ��|A|2 + |A∗|2�� + 1  4w (|A||A∗|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Let x ∈ H with ∥x∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Then we have  |⟨Ax, x⟩|2  ≤  ⟨|A|x, x⟩⟨|A∗|x, x⟩ (by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='1)  ≤  1  4 (⟨|A|x, x⟩ + ⟨|A∗|x, x⟩)2  =  1  4  �  ⟨|A|x, x⟩2 + ⟨|A∗|x, x⟩2 + 2⟨|A|x, x⟩⟨|A∗|x, x⟩  �  ≤  1  4  �  |⟨|A|x, x⟩ + i⟨|A∗|x, x⟩|2 + ∥|A|x∥∥|A∗|x∥ + |⟨|A|x, |A∗|x⟩|  �  (by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='3)  ≤  1  4  �  |⟨(|A| + i|A∗|)x, x⟩|2 + 1  2∥|A|x∥2 + 1  2∥|A∗|x∥2 + |⟨|A∗||A|x, x⟩|  �  ≤  1  4w2 (|A| + i|A∗|) + 1  8  ��|A|2 + |A∗|2�� + 1  4w (|A||A∗|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Taking supremum over all x ∈ H with ∥x∥ = 1, we get the desired inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  □  Clearly, we see that  1  4w2 (|A| + i|A∗|) + 1  8  ��|A|2 + |A∗|2�� + 1  4w (|A||A∗|)  ≤  1  4  ��|A|2 + |A∗|2�� + 1  8  ��|A|2 + |A∗|2�� + 1  4 ∥|A||A∗|∥  =  3  8  ��|A|2 + |A∗|2�� + 1  4  ��A2��  ≤  3  8  ��|A|2 + |A∗|2�� + 1  8  ��|A|2 + |A∗|2��  =  1  2  ��|A|2 + |A∗|2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Thus, we would like to remark that the upper bound obtained in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='4 refines  the second inequality in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Next result reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Let X, Y ∈ B(H ), and 0 ≤ α ≤ 1, 0 ≤ β ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Then for each x ∈ H  with ∥x∥ = 1,  |⟨Xx, x⟩⟨Y x, x⟩|  ≤  1  4  ��α|X|2 + (1 − α)|X∗|2 + β|Y |2 + (1 − β)|Y ∗|2�� + 1  8  ��|X|2 + |Y ∗|2�� + 1  4w(Y X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  4  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' JANA P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' BHUNIA AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' PAUL  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' We have  |⟨Xx, x⟩⟨Y x, x⟩|  ≤  1  4 {|⟨Xx, x⟩| + |⟨Y x, x⟩|}2  =  1  4  �  |⟨Xx, x⟩|2 + |⟨Y x, x⟩|2 + 2|⟨Xx, x⟩||⟨Y x, x⟩|  �  ≤  1  4  �  ⟨|X|2αx, x⟩⟨|X∗|2(1−α)x, x⟩ + ⟨|Y |2βx, x⟩⟨|Y ∗|2(1−β)x, x⟩ + 2|⟨Xx, x⟩||⟨x, Y ∗x⟩|  �  (using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='1)  ≤  1  4  �  ⟨|X|2x, x⟩α⟨|X∗|2x, x⟩(1−α) + ⟨|Y |2x, x⟩β⟨|Y ∗|2x, x⟩(1−β) + ∥Xx∥∥Y ∗x∥ + |⟨Xx, Y ∗x⟩|  �  (using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='2 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='3 )  ≤  1  4  �  α⟨|X|2x, x⟩ + (1 − α)⟨|X∗|2x, x⟩ + β⟨|Y |2x, x⟩ + (1 − β)⟨|Y ∗|2x, x⟩  �  +1  4  �1  2  �  ⟨|X|2x, x⟩ + ⟨|Y ∗|2x, x⟩  �  + |⟨Y Xx, x⟩|  �  ≤  1  4∥α|X|2 + (1 − α)|X∗|2 + β|Y |2 + (1 − β)|Y ∗|2∥ + 1  8∥|X|2 + |Y ∗|2∥ + 1  4w(Y X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  □  Applying the inequality in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='5 we derive the following upper bound for the  numerical radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' If A ∈ B(H ), then  w2(A) ≤ 1  4  ��µ|A|2 + (2 − µ)|A∗|2�� + 1  8  ��|A|2 + |A∗|2�� + 1  4w(A2),  for 0 ≤ µ ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Putting X = Y = A in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='5, and then taking supremum over all x ∈ H  with ∥x∥ = 1, we get  w2(A) ≤ 1  4  ��(α + β)|A|2 + (2 − α − β)|A∗|2�� + 1  8  ��|A|2 + |A∗|2�� + 1  4w(A2),  for 0 ≤ α, β ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' This implies the desired bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  □  It follows from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='6 that  w2(A) ≤ 1  4 min  µ∈[0,2]  ��µ|A|2 + (2 − µ)|A∗|2�� + 1  8  ��|A|2 + |A∗|2�� + 1  4w(A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='1)  NUMERICAL RADIUS INEQUALITIES AND ESTIMATION OF ZEROS OF POLYNOMIALS  5  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Clearly, We have  min  µ∈[0,2]  1  4  ��µ|A|2 + (2 − µ)|A∗|2�� + 1  8  ��|A|2 + |A∗|2�� + 1  4w(A2)  ≤  1  4∥|A|2 + |A∗|2∥ + 1  8∥|A|2 + |A∗|2∥ + 1  4w(A2) (by taking µ = 1)  =  3  8∥|A|2 + |A∗|2∥ + 1  4w(A2)  ≤  3  8∥|A|2 + |A∗|2∥ + 1  4w2(A)  ≤  3  8∥|A|2 + |A∗|2∥ + 1  8∥|A|2 + |A∗|2∥ (using the second inequality of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='4))  =  1  2∥|A|2 + |A∗|2∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Thus, we would like to remark that inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='1) is stronger than that in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  We also note that the minimum value is not always attained for µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' For example,  consider the matrix A =  \uf8eb  \uf8ec  \uf8ed  0  1  0  0  0  2  0  0  0  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Then, min  µ∈[0,2] ∥µ|A|2 + (2 − µ)|A∗|2∥ =  32  7 for  µ = 8  7, and we see that  1  4 min  µ∈[0,2]  ��µ|A|2 + (2 − µ)|A∗|2�� + 1  8  ��|A|2 + |A∗|2�� + 1  4w(A2)  =  113  56 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='01785714  <  5  2 = 1  2∥|A|2 + |A∗|2∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  To prove our next result we need the following two lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' First one is a gen-  eralization of the inequality in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='1, and the second one is known as Bohr’s  inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' ([19, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' 5]) Let A, B ∈ B(H ) with |A|B = B∗|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Let f, g be two  non-negative continuous functions on [0, ∞) such that f(t)g(t) = t for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Then  |⟨ABx, y⟩| ≤ r(B)∥f(|A|)x∥∥g(|A∗|)y∥,  for all x, y ∈ H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' ([24]) For i = 1, 2, · · · , n, let ai ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Then  � n  �  i=1  ai  �p  ≤ np−1  n  �  i=1  ap  i ,  for all p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  By using the above lemmas we prove the following inequality involving numerical  radius and spectral radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  6  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' JANA P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' BHUNIA AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' PAUL  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Let Ai, Bi ∈ B(H ) be such that |Ai|Bi = B∗  i |Ai| for i = 1, 2, · · · , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Then  wp  � n  �  i=1  AiBi  �  ≤ np−1  √  2  w  � n  �  i=1  rp(Bi)  �  f 2p(|Ai|) + ig2p(|A∗  i |)  �  �  ,  for all p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Let x ∈ H with ∥x∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Then we have  �����⟨  � n  �  i=1  AiBi  �  x, x⟩  �����  p  =  �����  n  �  i=1  ⟨AiBix, x⟩  �����  p  ≤  � n  �  i=1  |⟨AiBix, x|  �p  ≤  � n  �  i=1  r(Bi)∥f(|Ai|)x∥∥g(|A∗  i|)x∥  �p  (by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='8)  =  � n  �  i=1  r(Bi)⟨f 2(|Ai|)x, x⟩  1  2⟨g2(|A∗  i |)x, x⟩  1  2  �p  ≤  � n  �  i=1  r(Bi)⟨f 2(|Ai|)x, x⟩ + ⟨g2(|A∗  i |)x, x⟩  2  �p  ≤  np−1  n  �  i=1  rp(Bi)  �⟨f 2(|Ai|)x, x⟩ + ⟨g2(|A∗  i |)x, x⟩  2  �p  (by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='9)  ≤  np−1  2  n  �  i=1  rp(Bi)  �  ⟨f 2(|Ai|)x, x⟩p + ⟨g2(|A∗  i |)x, x⟩p�  (by convexity of f(t) = tp)  ≤  np−1  2  n  �  i=1  rp(Bi)  �  ⟨f 2p(|Ai|)x, x⟩ + ⟨g2p(|A∗  i |)x, x⟩  �  (by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='2)  ≤  np−1  √  2  �����  n  �  i=1  rp(Bi)  �  ⟨f 2p(|Ai|)x, x⟩ + i⟨g2p(|A∗  i |)x, x⟩  �  �����  (as |a + b| ≤  √  2|a + ib| for all a, b ∈ R)  ≤  np−1  √  2  �����  n  �  i=1  rp(Bi)⟨  �  f 2p(|Ai|) + ig2p(|A∗  i |)  �  x, x⟩  �����  ≤  np−1  √  2 w  � n  �  i=1  rp(Bi)  �  f 2p(|Ai|) + ig2p(|A∗  i |)  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  NUMERICAL RADIUS INEQUALITIES AND ESTIMATION OF ZEROS OF POLYNOMIALS  7  Now, taking supremum over all x ∈ H , ∥x∥ = 1 we get,  wp  � n  �  i=1  AiBi  �  ≤ np−1  √  2 w  � n  �  i=1  rp(Bi)  �  f 2p(|Ai|) + ig2p(|A∗  i |)  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  □  Observe that the inequality in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='10 indeed does not depend on the number  n of summands in the case p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' In particular, considering p = n = 1, A1 = A,  B1 = B, f(t) = g(t) =  √  t in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='10, we get the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Let A, B ∈ B(H ) be such that |A|B = B∗|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Then  w(AB) ≤ 1  √  2r(B)w(|A| + i|A∗|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  In particular, for B = I we have the following inequality (also obtained in [8]):  w(A) ≤ 1  √  2w(|A| + i|A∗|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='2)  Note that the bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='2) refines that in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='4), see [8, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Again, considering  Bi = I for i = 1, 2, · · · , n in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='10 we have the following inequality for the sum  of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Let Ai ∈ B(H ) for i = 1, 2, · · · , n, and let f, g be two non-negative  continuous functions on [0, ∞) such that f(t)g(t) = t for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Then  wp  � n  �  i=1  Ai  �  ≤ np−1  √  2 w  � n  �  i=1  �  f 2p(|Ai|) + ig2p(|A∗  i |)  �  �  ,  for all p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  In particular, for n = 1 and f(t) = g(t) =  √  t in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='12, we get the following  upper bound for the numerical radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' If A ∈ B(H ), then  wp(A) ≤ 1  √  2w(|A|p + i|A∗|p),  for all p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  It is easy to verify that  1  √  2w(|A|p +i|A∗|p) ≤ ∥A∥p for all p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Therefore, we would  like to remark that Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='13 improves the classical bound w(A) ≤ ∥A∥ for all  p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  At the end of this section, we give a sufficient condition for the equality of w(A) =  1  2∥A∗A + AA∗∥1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' For this purpose first we note the following known lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  8  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' JANA P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' BHUNIA AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' PAUL  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' [17] Let A, B ∈ B(H ) be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Then, ∥A + B∥ = ∥A∥ + ∥B∥ if and  only if ∥AB∥ = ∥A∥∥B∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Let A ∈ B(H ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Then ∥A∥4 = ∥ℜ2(A)ℑ2(A)∥ implies  w2(A) = 1  4∥A∗A + AA∗∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' We have  ∥A∥4  =  ∥ℜ2(A)ℑ2(A)∥ ≤ ∥ℜ2(A)∥∥ℑ2(A)∥ = ∥ℜ(A)∥2∥ℑ(A)∥2  ≤  1  2  �  ∥ℜ(A)∥4 + ∥ℑ(A)∥4�  ≤ max  �  ∥ℜ(A)∥4, ∥ℑ(A)∥4�  ≤  w4(A) ≤ ∥A∥4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  This implies that  ∥ℜ2(A)ℑ2(A)∥ = ∥ℜ(A)∥2∥ℑ(A)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='3)  Also, we have  1  2  �  ∥ℜ(A)∥4 + ∥ℑ(A)∥4�  = max  �  ∥ℜ(A)∥4, ∥ℑ(A)∥4�  = w4(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='4)  This implies that  ∥ℜ(A)∥ = ∥ℑ(A)∥ = w(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='5)  Now, by using lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='14, it follows from the identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='3) that  1  2∥ℜ2(A) + ℑ2(A)∥  =  1  2  �  ∥ℜ2(A)∥ + ∥ℑ2(A)∥  �  =  1  2  �  ∥ℜ(A)∥2 + ∥ℑ(A)∥2�  =  ∥ℜ(A)∥2 = w2(A) (using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  □  It should be mentioned here that the converse of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='15 is not true, in gen-  eral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' For example, we consider A =  \uf8eb  \uf8ec  \uf8ed  0  3  0  0  0  0  0  0  1  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Then, w2(A) = 1  4∥A∗A + AA∗∥ = 9  4,  however ∥A∥4 ̸= ∥ℜ2(A)ℑ2(A)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  NUMERICAL RADIUS INEQUALITIES AND ESTIMATION OF ZEROS OF POLYNOMIALS  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Estimation of zeros of polynomials  Suppose p(z) = zn +anzn−1 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='+a2z +a1 is a complex monic polynomial of degree  n ≥ 2 and a1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Location of the zeros of p(z) have been obtained by applying  numerical radius inequalities to Frobenius companion matrix of the polynomial p(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  The Frobenius companion matrix of the polynomial p(z) is given by  Cp =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  −an  −an−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='.  −a2  −a1  1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  0  0  0  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='.  1  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  The characteristic polynomial of Cp is the polynomial p(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Thus, the zeros of p(z) are  exactly the eigenvalues of Cp, see [12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' 316].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' The square of Cp is given by  C2  p =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  bn  bn−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  b3  b2  b1  −an  −an−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='.  −a3  −a2  −a1  1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  0  0  0  0  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='.  1  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  ,  where bj = anaj − aj−1 for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' , n, with a0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Also,  C3  p =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  cn  cn−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  c4  c3  c2  c1  bn  bn−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  b4  b3  b2  b1  −an  −an−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='.  −a4  −a3  −a2  −a1  1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  0  0  0  0  0  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  0  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='.  1  0  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  ,  where bj = anaj − aj−1 and cj = −anbj + an−1aj − aj−2 for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' , n, with  a0 = a−1 = 0,  10  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' JANA P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' BHUNIA AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' PAUL  and  C4  p =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  dn  dn−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  d5  d4  d3  d2  d1  cn  cn−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  c5  c4  c3  c2  c1  bn  bn−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  b5  b4  b3  b2  b1  −an  −an−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='.  −a5  −a4  −a3  −a2  −a1  1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  0  0  0  0  0  0  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
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+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
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+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='.  1  0  0  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  ,  where bj = anaj − aj−1, cj = −anbj + an−1aj − aj−2, and dj = −ancj − an−1bj−1 +  an−2aj − aj−3 for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' , n, with a0 = a−1 = a−2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  The exact value of ∥Cp∥ is well known (see in [18]), it is given by  ∥Cp∥ =  �  α + 1 +  �  (α + 1)2 − 4|a1|2  2  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='1)  where α = �n  j=1 |aj|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  An estimation of ∥C2  p∥ obtained in [16] is as follows  ∥C2  p∥ ≤  �  δ + 1 +  �  (δ − 1)2 + 4δ′  2  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='2)  where δ = 1  2  �  α + β +  �  (α − β)2 + 4|γ|2  �  and δ′ = 1  2  �  α′ + β′ +  �  (α′ − β′)2 + 4|γ′|2  �  ,  α = �n  j=1 |aj|2, β = �n  j=1 |bj|2, α′ = �n  j=3 |aj|2, β′ = �n  j=3 |bj|2, γ = − �n  j=1 ¯ajbj,  γ′ = − �n  j=3 ¯ajbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  We note that  ∥C2  p∥  1  2 ≤  \uf8eb  \uf8ed  �  δ + 1 +  �  (δ − 1)2 + 4δ′  2  \uf8f6  \uf8f8  1/2  ≤  �  α + 1 +  �  (α + 1)2 − 4|a1|2  2  = ∥Cp∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Motivated by the above estimation, here we will obtain an estimation of ∥C4  p∥1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  For this purpose first we note the following norm inequality for the sum of two positive  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' [17] If A, B ∈ B(H ) are positive, then  ∥A + B∥ ≤ 1  2  �  ∥A∥ + ∥B∥ +  �  (∥A∥ − ∥B∥)2 + 4  ���A  1  2B  1  2  ���  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  NUMERICAL RADIUS INEQUALITIES AND ESTIMATION OF ZEROS OF POLYNOMIALS  11  Now, we are in a position to obtain an estimation of ∥C4  p∥1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Let C4  p = R + S + T,  where  R =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  dn  dn−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  d5  d4  d3  d2  d1  cn  cn−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  c5  c4  c3  c2  c1  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
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+page_content='  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
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+page_content='.  0  0  0  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  ,  S =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
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+page_content='  0  0  0  0  0  bn  bn−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
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+page_content='  b5  b4  b3  b2  b1  −an  −an−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
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+page_content='.  −a5  −a4  −a3  −a2  −a1  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
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+page_content='.  0  0  0  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  and  T =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
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+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='.  1  0  0  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Now,  ∥C4  p∥2  =  ∥R + S + T∥2  =  ∥(R + S + T)∗(R + S + T)∥  =  ∥R∗R + S∗S + T ∗T∥ (since R∗S = R∗T = S∗R = S∗T = T ∗R = T ∗S = 0)  ≤  ∥R∗R + S∗S∥ + ∥T ∗T∥  ≤  1  2  �  ∥R∥2 + ∥S∥2 +  �  (∥R∥2 − ∥S∥2)2 + 4∥RS∗∥2  �  + 1 (using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  By simple calculations, we have  ∥R∥2  =  ∥R∗R∥ = ∥RR∗∥  =  1  2  �  α1 + β1 +  �  (α1 − β1)2 + 4|γ1|2  �  = δ1,  12  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' JANA P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' BHUNIA AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' PAUL  where α1 = �n  j=1 |dj|2, β1 = �n  j=1 |cj|2 , γ1 = �n  j=1 dj ¯cj,  ∥S∥2  =  ∥S∗S∥ = ∥SS∗∥  =  1  2  �  α + β +  �  (α − β)2 + 4|γ|2  �  = δ,  where α = �n  j=1 |aj|2, β = �n  j=1 |bj|2 , γ = − �n  j=1 bj ¯aj,  ∥RS∗∥2  =  1  2  �  |γ2|2 + |γ3|2 + |γ4|2 + |γ5|2 +  �  ((|γ2|2 + |γ3|2) − (|γ4|2 + |γ5|2))2 + 4|γ2 ¯γ4 + γ3 ¯γ5|2  �  =  δ2,  where γ2 = �n  j=1 dj ¯bj, γ3 = �n  j=1 dj ¯aj, γ4 = �n  j=1 cj ¯bj, γ5 = �n  j=1 cj ¯aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Therefore,  ∥C4  p∥ ≤  �  1  2  �  δ1 + δ +  �  (δ1 − δ)2 + 4δ2  �  + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='3)  We observe that the estimation of ∥C4  p∥1/4 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='3) is incomparable with the existing  estimation of ∥C2  p∥1/2 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' In the following theorem we derive an upper bound for  the spectral radius of the Frobenius companion matrix Cp, by using the estimations in  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' The following inequality holds:  r(Cp) ≤  �  1  4  �  δ + 1 +  �  (δ − 1)2 + 4δ′  2  �  + 3  4  �1  2  �  δ1 + δ +  �  (δ1 − δ)2 + 4δ2  �  + 1  � 1  2� 1  4  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  where δ′ = 1  2  �  α′ + β′ +  �  (α′ − β′)2 + 4|γ′|2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  δ = 1  2  �  α + β +  �  (α − β)2 + 4|γ|2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  δ1 = 1  2  �  α1 + β1 +  �  (α1 − β1)2 + 4|γ1|2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  δ2 = 1  2  �  |γ2|2 + |γ3|2 + |γ4|2 + |γ5|2 +  �  ((|γ2|2 + |γ3|2) − (|γ4|2 + |γ5|2))2 + 4|γ2 ¯γ4 + γ3 ¯γ5|2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  α′ = �n  j=3 |aj|2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' β′ = �n  j=3 |bj|2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' γ′ = − �n  j=3 ¯ajbj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  α = �n  j=1 |aj|2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' β = �n  j=1 |bj|2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' γ = − �n  j=1 bj ¯aj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  α1 = �n  j=1 |dj|2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' β1 = �n  j=1 |cj|2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' γ1 = �n  j=1 dj ¯cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  γ2 = �n  j=1 dj ¯bj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' γ3 = �n  j=1 dj ¯aj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' γ4 = �n  j=1 cj ¯bj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' γ5 = �n  j=1 cj ¯aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Let A ∈ B(H ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Putting A = A2 in the inequality w2(A) ≤ 1  4∥A∗A + AA∗∥ +  1  2w(A2) (see [1, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='4]), we get  w2(A2)  ≤  1  4  ��|A2|2 + |(A∗)2|2�� + 1  2w(A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  NUMERICAL RADIUS INEQUALITIES AND ESTIMATION OF ZEROS OF POLYNOMIALS  13  It follows that  r2(A) = r(A2) ≤ w(A2) ≤  �1  4  ��|A2|2 + |(A∗)2|2�� + 1  2w(A4)  � 1  2  ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=',  r(A) ≤  �1  4  ��|A2|2 + |(A∗)2|2�� + 1  2w(A4)  � 1  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='4)  Now, it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='4) and the inequality ∥C∗  pCp + CpC∗  p∥ ≤ ∥Cp∥2 + ∥C2  p∥ (see [6,  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='9]) that  r(Cp)  ≤  �1  4  ��|C2  p|2 + |(C∗  p)2|2�� + 1  2w(C4  p)  � 1  4  ≤  �1  4(∥C2  p∥2 + ∥C4  p∥) + 1  2∥C4  p∥  � 1  4  ≤  �1  4  ��C2  p  ��2 + 3  4  ��C4  p  ��  � 1  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Therefore, the required inequality follows by using the estimations in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  □  By using the fact |λj(Cp)| ≤ r(Cp), where λj(Cp) is the j-th eigenvalue of Cp, we  infer the following estimation for the zeros of the polynomial p(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' If z is any zero of p(z), then  |z| ≤  �  1  4  �  δ + 1 +  �  (δ − 1)2 + 4δ′  2  �  + 3  4  �1  2  �  δ1 + δ +  �  (δ1 − δ)2 + 4δ2  �  + 1  � 1  2� 1  4  ,  where δ, δ1, δ2 and δ′ are same as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Applying the spectral mapping theorem, we conclude that if z is any zero of p(z) then  |z| ≤ ∥C4  p∥  1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Thus, by using the inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='3) we achieve another new estimation  for the zeros of p(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' If z is any zero of p(z), then  |z| ≤  �1  2  �  δ1 + δ +  �  (δ1 − δ)2 + 4δ2  �  + 1  � 1  8  ,  where δ, δ1 and δ2 are given in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Again, putting A = A2 in the inequality w(A) ≤ 1  2  �  ∥A∥ + ∥A2∥  1  2  �  (see [16, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' 1]),  and proceeding as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='4), we get  r(A)  ≤  �1  2∥A2∥ + 1  2∥A4∥  1  2  � 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='5)  14  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' JANA P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' BHUNIA AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' PAUL  Proceeding similarly as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='2 we obtain the following estimation by using  the inequalities in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='5), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' If z is any zero of p(z), then  |z| ≤  \uf8f1  \uf8f2  \uf8f3  1  2  �  δ + 1 +  �  (δ − 1)2 + 4δ′  2  + 1  2  �1  2  �  δ1 + δ +  �  (δ1 − δ)2 + 4δ2  �  + 1  � 1  4  \uf8fc  \uf8fd  \uf8fe  1  2  ,  where δ, δ1, δ1 and δ′ are given in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Finally, we compare the bounds obtained here for the zeros of p(z) with the existing  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' First we note some well known existing bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Let z be any zero of p(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Then  Linden [20] obtained that  |z| ≤ |an|  n  +  �  n − 1  n  �  n − 1 +  n  �  j=1  |aj|2 − |an|2  n  �� 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Montel [11, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' 3] obtained that  |z| ≤ max {1, |a1| + · · · + |an|} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Cauchy [12] obtained that  |z| ≤ 1 + max {|a1|, · · · , |an|} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Kittaneh [15] proved that  |z| ≤ 1  2  \uf8eb  \uf8ec  \uf8ed|an| + 1 +  �  �  �  �  �(|an| − 1)2 + 4  �  �  �  �  n−1  �  j=1  |aj|2  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Fujii and Kubo [10] obtained that  |z| ≤ cos  π  n + 1 + 1  2  \uf8eb  \uf8ed|an| +  �  �  �  �  n  �  j=1  |aj|2  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Bhunia and Paul [4, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='6] proved that  |z|2 ≤ cos2  π  n + 1 + |an−1| + 1  4  �  |an| + √α  �2 + 1  2  �  α − |an|2 + 1  2  √α,  where α = �n  j=1 |aj|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  We consider a polynomial p(z) = z3 + z2 + 1  2z + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Different upper bounds for the  modulus of the zeros of this polynomial, mentioned above, are as shown in the following  table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  NUMERICAL RADIUS INEQUALITIES AND ESTIMATION OF ZEROS OF POLYNOMIALS  15  Linden [20]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='9492  Montel[11]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='5  Cauchy[12]  2  Kittaneh[15]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='0547  Fujii and Kubo[10]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='9571  Bhunia and Paul[4]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='96761  However, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='3 gives |z| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='38047091798, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='4 gives |z| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='3798438819  and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='5 gives |z| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='381095966, which are better than the above mentioned  bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Statements & Declarations:  Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' The authors declare that no funds, grants, or other support were received  during the preparation of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' The authors have no relevant financial or non-financial inter-  ests to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Data availability statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Data sharing not applicable to this article as no  datasets were generated or analysed during the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  Author Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' All authors have contributed equally in the preparation of  the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
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+page_content=' Paul, Improved inequalities for the numerical radius  via Cartesian decomposition, Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' (2022), to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Bhunia and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Paul, Annular bounds for the zeros of a polynomial from companion matrices,  Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Theory 7 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' 1, Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' 8, 19 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Bhunia and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Paul, Proper improvement of well-known numerical radius inequalities and their  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Results Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' 76 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' 4, Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' 177, 12 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Bhunia and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Paul, Furtherance of numerical radius inequality of Hilbert space operators,  Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' (Basel) 117 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' 5, 537–546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Bhunia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Bag and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Paul, Bounds for zeros of a polynomial using numerical radius of Hilbert  space operators, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' 12 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' 2, Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' 21, 14 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Bhunia and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Paul, New upper bounds for the numerical radius of Hilbert space operators,  Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' 167 ( 2021), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' 102959, 11 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Buzano, Generalizzatione della diseguaglianza di Cauchy-Schwarz, Rend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Sem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  e Politech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' Torino 31(1971/73) (1974) 405–409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='  16  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' JANA P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content=' BHUNIA AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
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+page_content='  1 Department of Mathematics, Mahisadal Girls’ College, Purba Medinipur 721628,  West Bengal, India  Email address: janasuva8@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='com  2 Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal,  India  Email address: pintubhunia5206@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
+page_content='com  Email address: kalloldada@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE1T4oBgHgl3EQfaQRP/content/2301.03159v1.pdf'}
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+Towards Real-Time Panoptic Narrative Grounding by an End-to-End
+Grounding Network
+Haowei Wang1*, Jiayi Ji1*, Yiyi Zhou1, 2, Yongjian Wu4, Xiaoshuai Sun1, 2, 3†
+1Media Analytics and Computing Lab, Department of Artificial Intelligence,
+School of Informatics, Xiamen University, 361005, China.
+2Institute of Artificial Intelligence, Xiamen University, China.
+3Fujian Engineering Research Center of Trusted Artificial Intelligence Analysis and Application, Xiamen University, China.
+4Tencent Youtu Lab, Shanghai, China
+wanghaowei@stu.xmu.edu.cn, jjyxmu@gmail.com, zhouyiyi@xmu.edu.cn,
+littlekenwu@tencent.com, xssun@xmu.edu.cn
+Abstract
+Panoptic Narrative Grounding (PNG) is an emerging cross-
+modal grounding task, which locates the target regions of
+an image corresponding to the text description. Existing ap-
+proaches for PNG are mainly based on a two-stage paradigm,
+which is computationally expensive. In this paper, we propose
+a one-stage network for real-time PNG, termed End-to-End
+Panoptic Narrative Grounding network (EPNG), which di-
+rectly generates masks for referents. Specifically, we propose
+two innovative designs, i.e., Locality-Perceptive Attention
+(LPA) and a bidirectional Semantic Alignment Loss (SAL),
+to properly handle the many-to-many relationship between
+textual expressions and visual objects. LPA embeds the local
+spatial priors into attention modeling, i.e., a pixel may belong
+to multiple masks at different scales, thereby improving seg-
+mentation. To help understand the complex semantic relation-
+ships, SAL proposes a bidirectional contrastive objective to
+regularize the semantic consistency inter modalities. Exten-
+sive experiments on the PNG benchmark dataset demonstrate
+the effectiveness and efficiency of our method. Compared to
+the single-stage baseline, our method achieves a significant
+improvement of up to 9.4% accuracy. More importantly, our
+EPNG is 10 times faster than the two-stage model. Mean-
+while, the generalization ability of EPNG is also validated by
+zero-shot experiments on other grounding tasks. The source
+codes and trained models for all our experiments are publicly
+available at https://github.com/Mr-Neko/EPNG.git.
+Introduction
+Panoptic Narrative Grounding (Gonz´alez et al. 2021) is a
+new challenging task that locates the target instances of an
+image corresponding to the text description via binary pixel
+masks. Its main challenges not only lie in the joint under-
+standing of multi-modal information but also in many-to-
+many language-vision alignment, i.e., grounding all related
+instances or amorphous regions mentioned in the text de-
+scription. This property also makes it different from a sim-
+ilar grounding task called Referring Expression Segmenta-
+tion (RES) (Hu, Rohrbach, and Darrell 2016; Yu et al. 2018;
+*These authors contributed equally.
+†The corresponding author.
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+Panoptic 
+Segmentation
+Feature 
+Interactive
+Matching 
+Score
+Mask 
+Allocation
+Text
+Visual Feature Extraction
+Text
+Multi-modal 
+Communicator
+Dense
+Prediction
+Noun
+Visual
+100ms
+7ms
+11ms
+Total: 107ms
+Test Environment: 
+Nvidia RTX 3090, without the time of data loading
+PNG
+(two-stage)
+EPNG
+(one-stage)
+PNG
+(two-stage)
+EPNG
+(one-stage)
+(a)
+(b)
+Textual Feature Extraction
+Multimodal 
+Encoding
+Total: 11ms
+Figure 1: Comparison of pipeline and inference speed be-
+tween the proposed EPNG and two-stage PNG. (a) EPNG
+jointly processes visual and text information to generate re-
+ferred masks in a one-stage fashion, while PNG relies on
+mask proposals. (b) Our single-stage EPNG is 10x faster
+than the two-stage approach, enabling real-time deployment.
+Ye et al. 2019; Shi et al. 2018), which segments only one
+instance per expression.
+Gonzalez et al. (Gonz´alez et al. 2021) first explore this
+task and propose a preliminary solution in a two-stage fash-
+ion, as illustrated in Fig. 1 (a). First, the pre-trained panoptic
+segmentation models like PFPN (Kirillov et al. 2019a) are
+used to provide a set of candidate masks of the given image.
+Secondly, these masks are further transformed into convo-
+lution features and then ranked by cross-modal matching.
+Overall, with the help of panoramic segmentation models,
+this two-stage solution defines PNG as a mask-text match-
+ing problem, greatly reducing the difficulty of prediction.
+However, this solution still suffers from two limitations.
+On the one hand, such a two-stage approach requires of-
+fline feature extraction, storage, and alignment, which is
+inevitably time-consuming. This limitation poses a huge
+obstacle to real-time applications, e.g., text-to-image re-
+trieval, and video matting. On the other hand, the pre-trained
+arXiv:2301.03160v1  [cs.CV]  9 Jan 2023
+
+panoramic segmentation model requires massive mask an-
+notations, which place a greater burden on the already ex-
+pensive expenditure of PNG. More importantly, the perfor-
+mance of these panoptic segmentation models also limits the
+upper bound of PNG models.
+To solve the above problems, a natural way is to design an
+efficient single-stage network for end-to-end training from
+scratch. However, this solution also encounters two chal-
+lenges that are critical for PNG. First, in PNG, each pixel can
+be subordinated to different masks, which is greatly differ-
+ent from panoptic segmentation (Kirillov et al. 2019a). This
+property makes the model need to capture visual seman-
+tics from macro- to micro-views. However, existing meth-
+ods only focus on global modeling and overlook local in-
+formation, resulting in limited performance. Second, PNG
+involves more complicated relationships than other ground-
+ing or segmentation tasks (Liu, Wang, and Yang 2017; Luo
+and Shakhnarovich 2017; Yu et al. 2017). In each example,
+multiple nouns of an expression may correspond to the same
+mask, or one noun may refer to multiple masks. This case
+further increases the difficulty of vision-language alignment.
+In this paper, we propose a novel End-to-End Panoptic
+Narrative Grounding network (EPNG) for real-time panop-
+tic narrative grounding, as shown in Fig. 1. Specifically,
+EPNG adopts a visual encoder to extract the features of the
+given image, based on which a decoder is deployed to pre-
+dict masks for different noun phrases.
+To enhance local semantic modeling, we introduce
+Locality-Perceptive Attention (LPA) to enhance grid fea-
+tures via neighborhood interactions based on their spatial
+priors. In LPA, different attention heads are allowed to per-
+ceive visual information in different receptive fields, thus
+achieving multi-scale modeling. To ensure the semantic con-
+sistency of many-to-many relationships in PNG, we design
+a new bidirectional Semantic Alignment Loss (SAL), which
+uses one modality as an anchor to eliminate the deviation
+of similar semantic tokens of the other modality. With these
+innovative designs, EPNG is superior in cross-modal reason-
+ing while keeping real-time inference.
+Conclusively, the contributions of our work are as below:
+• We propose a real-time End-to-End Panoptic Narrative
+Grounding network (EPNG), which greatly reduces com-
+putation overhead via unifying cross-modal alignment
+and mask prediction in one forward structure.
+• We propose two novel designs, namely Locality-
+Perceptive Attention (LPA) and bidirectional Semantic
+Alignment Loss (SAL). LPA enhances visual features
+at different scales to understand complex cross-modal
+relationships. SAL regularizes the semantic consistency
+problem by performing contrastive learning between pix-
+els and noun phrases.
+• On the benchmark dataset, EPNG is on par with or even
+better than existing two-stage methods, while its infer-
+ence is 10 times faster. In addition, it requires no addi-
+tional mask annotations for pre-training.
+Related Work
+Panoptic Segmentation
+Panoptic segmentation aims to entirely understand scenes
+containing things and stuff. Following the benchmark pro-
+posed by (Kirillov et al. 2019b), the earlier methods
+treated it as the combination of things masks and stuff
+masks (de Geus, Meletis, and Dubbelman 2018), e.g.,
+PFPN (Kirillov et al. 2019a), Panoptic-DeepLab (Cheng
+et al. 2020a), and UPSNet (Xiong et al. 2019). Recently,
+things and stuff are expected to be treated uniformly (Car-
+ion et al. 2020; Wang et al. 2021; Cheng, Schwing, and Kir-
+illov 2021). To eliminate the difference between things and
+stuff, part of those like PFCN (Li et al. 2021), K-Net (Zhang
+et al. 2021), and Panoptic SegFormer (Li et al. 2022) try to
+use the kernel to represent things and stuff uniformly and
+generate masks by the convolution on feature maps, which
+obtain significant performance. Benefiting from those meth-
+ods, our model utilizes word features as reliable kernels to
+get corresponding masks through the convolution on multi-
+modal features.
+Referring Expression Segmentation
+Recently, multi-modal applications have received a lot of at-
+tention and made significant progress (Ji et al. 2022b, 2021;
+Ma et al. 2022b,a; Zhou et al. 2019). Among them, as a
+prevalent task in multi-modal communities, Referring Ex-
+pression Segmentation (RES) (Hu, Rohrbach, and Darrell
+2016; Yu et al. 2018; Ye et al. 2019; Shi et al. 2018) is to seg-
+ment a referent based on the understanding of a related short
+phrase. In sequential order, previous models (Li et al. 2018;
+Liu et al. 2017; Margffoy-Tuay et al. 2018) obtain a set of
+proposals by a general method of segmentation and pick up
+a better one that is described by the given short phrase. With
+the strength of leveraging visual information, however, the
+upper bound of those methods is seriously restricted by the
+performance of the segmentation models. After that, a batch
+of methods is developed for refining segmentation masks
+by a single-stage network (Ye et al. 2019; Liu et al. 2019;
+Zhou et al. 2021), which brings higher rates of false positive
+segmentation. In summary, RES is an incomplete task with
+the neglect of stuff and many-to-many relationships between
+natural language and images. Additionally, whether things
+and stuff or the many-to-many relationships should be con-
+sidered in Panoptic Narrative Grounding.
+Panoptic Narrative Grounding
+The existing method (Gonz´alez et al. 2021) handles it with
+a two-stage paradigm, which first obtains a lot of candi-
+date panoptic masks by a pre-trained panoptic segmenta-
+tion model. With those candidates, a scoring module is used
+to assign plural masks to referred phrases. This paradigm
+achieves impressive performance, nevertheless, the expen-
+sive computation cost and space cost on the stage of seg-
+ment becomes the barrier to real-time. Because of the rea-
+sons above, we propose an End-to-End Panoptic Narrative
+Grounding network (EPNG) to generate the corresponding
+mask directly from the noun phrases.
+
+In the center of the image 
+there are two elephants. At 
+the bottom there is grass. In 
+the background we can see 
+hills and sky.
+BERT
+…
+FPN
+FFN
+Linear
+⊛
+×S
+𝑫
+Relative pos
+Dense Prediction
+𝑭𝑽
+Visual Feature
+𝑇2
+∙ 𝐼2
+𝑇2
+∙ 𝐼3
+𝑇1
+∙ 𝐼1
+𝑇2
+∙ 𝐼1
+…
+𝑇1
+∙ 𝐼4
+…
+𝑇2
+∙ 𝐼4
+𝑇𝑛
+∙ 𝐼1
+𝑇𝑛
+∙ 𝐼2
+𝑇𝑛
+∙ 𝐼3 …
+𝑇𝑛
+∙ 𝐼4
+…
+…
+…
+…
+𝑇1
+∙ 𝐼𝑚
+𝑇2
+∙ 𝐼𝑚
+𝑇𝑛
+∙ 𝐼𝑚
+…
+𝐼1
+𝐼2 𝐼3 … 𝐼4 𝐼𝑚
+𝑇1
+𝑇2
+𝑇𝑛
+…
+𝑇1
+∙ 𝐼2
+𝑇1
+∙ 𝐼3
+𝑴
+Mask
+𝑮
+Ground-Truth
+𝑻
+𝑰
+𝑭𝑵
+Noun Feature
+Text Encoder
+Visual Encoder
+Multi-modal 
+Communicator
+𝑪
+Contrastive Matrix
+Q
+K
+V
+V
+K
+Q
+BCE Loss
+Segmentation 
+Loss
+Dice Loss
+Bidirectional 
+Semantic 
+Alignment 
+Loss
+Phrase to 
+Pixel
+𝒍𝒗
+Pixel to 
+Phrase
+𝒍𝒕
+LPA
+Cross Attention
+Text Input
+Image Input
+Figure 2: The framework of the proposed EPNG. The solid lines denote the pipeline of EPNG, while the dotted lines represent
+the loss computation during training. During the pipeline, a Multi-modal Encoding module is used to extract the features.
+Then a Multi-modal Communicator fuses multi-modal features with a Cross Attention module and the proposed LPA. Finally,
+traditional segmentation loss and the proposed SAL are set to improve the quality of segmentation and align the multi-modal
+information.
+End-to-End Panoptic Narrative Grounding
+Network
+In this section, we give a detailed description of our EPNG,
+of which the framework is illustrated in Fig. 2. The input
+images and descriptions are first processed by the visual and
+text encoders, respectively. A multi-modal fusion module is
+further deployed for image-text interaction, based on which
+a dense prediction head is used to predict masks.
+Problem Definition
+Unlike the existing two-stage PNG, the proposed one-stage
+PNG is free of mask proposals, which generates the mask
+directly based on the expressions and images. We formulate
+it as a cross-modal dense prediction task.
+Specifically, given an image I and the corresponding text
+T, the goal of PNG is to find the nouns N = {nℓ}L
+ℓ=0 that
+each pixel i belongs to, where nℓ is the ℓ-th noun and L de-
+notes the number of the noun phrases. Then the probability
+of the obtained mask M ∈ {0, 1} is formulated as:
+p (M) =
+�
+i∈I
+L
+�
+ℓ=0
+p (i|I, T, nℓ) .
+(1)
+Multi-modal Encoding
+Visual Encoder
+Given an image I ∈ RH×W ×3, we first
+adopt a visual backbone (Lin et al. 2017) to extract the
+multi-scale visual features, e.g., Fv1 ∈ R
+H
+8 × W
+8 ×C1, Fv2 ∈
+R
+H
+16 × W
+16 ×C2, and Fv3 ∈ R
+H
+32 × W
+32 ×C3 . Then we obtain the
+final visual feature Fv ∈ R
+H
+16 × W
+16 ×C by:
+Fv = concat [Down (Fv1) ; Fv2; Up (Fv3)] .
+(2)
+where Up (·) denotes 2× upsampling, Down (·) denotes 2×
+downsampling and concat [·] denotes feature concatenation.
+Text Encoder
+Given a sentence T, we follow (Gonz´alez
+et al. 2021) to adopt a pre-trained BERT (Kenton and
+Toutanova 2019) to extract the word embeddings FT =
+{vt}|T |
+t=0, where vt denotes the embedding of t-th word. After
+that, we filter out the noun phrases according to the annota-
+tions given by (Gonz´alez et al. 2021) and then obtain the
+phrase features by average-pooling the word embeddings in
+each phrase. These features are then projected by a linear
+layer, making their feature dimension consistent with the vi-
+sual features. As a result, the phrase embedding is denoted
+as FN = {fnℓ}L
+nℓ=0 ∈ RL×C, where nℓ represents the ℓ-th
+noun phrases, and L is the number of phrases.
+
+Multi-modal Communicator
+Based on the visual feature Fv and the textual feature FN,
+Multi-Modal Communicator is designed for cross-modal in-
+teraction and fusion. It consists of S serial identical layer,
+and each layer is composed of two modules called Locality-
+Perceptive Attention (LPA) and Cross Attention (CA).
+Locality-Perceptive
+Attention
+Similar
+to
+self-
+attention (Vaswani et al. 2017), LPA aims to improve
+the input features via modeling their inter-relationships.
+As argued in
+(Cheng et al. 2020b), local information is
+important for the visual segmentation tasks. Then EPNG,
+going a step further, presents multi-scale local modeling.
+Each pixel in an image may belong to different masks at the
+same time. For example, a pixel in cloth may also belong
+to a person. However, the standard self-attention treats all
+tokens in the feature map equally. To this end, we reinforce
+the role of neighborhood information of each pixel when in
+attention modeling, following (Wu, Wu, and Huang 2021;
+Ji et al. 2022a).
+Specifically, in the features Fi, the 2D spatial coordinates
+of the m-th and n-th vectors are denoted as (xm, ym) and
+(xn, yn), where the superscript i indicates that the feature
+map is the output of the layer i. Then we calculate the Eu-
+clidean Distance between these two coordinates:
+Dm,n =
+�
+(xm − xn)2 + (ym − yn)2,
+(3)
+where D ∈ R(H×W )×(H×W ), and we truncate the values
+in D with an upper bound 2 to explicitly inject the local
+receptive information. Afterward, for an attention head j in
+LPA, we transform distance matrix into a coefficient matrix
+Rj ∈ R(H×W )×(H×W ), obtained by:
+Rj = WjD.
+(4)
+The obtained matrix Rj is used to re-weight the attention,
+which is given by:
+Aj = Softmax
+�(FiWj
+Q)(FiWj
+K)T
+√dk
+⊗ Rj
+�
+,
+(5)
+where the projections Wj
+Q ∈ Rd× d
+h and Wj
+K ∈ Rd× d
+h are
+weight matrices, and dk is a scaling factor. The subscript j
+represents the j-th head, and the number of heads h is set
+to 8. ⊗ represents an element-wise product. In this way, we
+naturally embed local information into attention modeling.
+Next, we sum the features using the attention weights to
+obtain the results for head j, and aggregate all the results:
+Headj = Aj(FiWj
+V ),
+(6)
+LPA(Fi, Fi, Fi) = concat(Head1, · · · , Headh)WO,
+(7)
+where Wj
+V ∈ Rd× d
+h , and WO ∈ Rd×d are projection ma-
+trices.
+Fi′ = LN
+�
+LPA
+�
+Fi, Fi, Fi�
++ Fi�
+,
+(8)
+where LN (·) means the layer normalization (Ba, Kiros, and
+Hinton 2016), and shortcut connection (He et al. 2016) is
+applied after the LPA module.
+The difference between LPA and Multi-Head Attention
+(MHA). As shown in Eq. 5, the definition of LPA is based
+on Multi-head Attention (Vaswani et al. 2017), but it still has
+an obvious difference in principle. MHA treats all tokens in
+the feature map equally and excels at capturing long-range
+dependencies. However, local information is inevitably ig-
+nored in this process. We introduce the local prior of each
+grid, which is obtained from the distance matrix to attention
+modeling. Note that, these local priors can be dynamically
+adjusted by the coefficient matrix Rj in Eq. 4.
+Cross-modal Attention
+Following (Yu et al. 2019), we
+use Cross-modal Attention for modality interactions:
+Fi+1 = FFN
+�
+LN
+�
+MHA
+�
+Fi′, FN, FN
+�
++ Fi′��
+,
+(9)
+where FFN (·) denotes feed-forward network, and FN is the
+noun features.
+Dense Prediction
+Given the fused feature after multi-modal communicator,
+F ∈ R
+H
+16 × W
+16 ×C, we upsample it to a tensor shape of
+H
+4 × W
+4 × C. Afterward, we apply each noun phrase feature
+as a kernel to convolve F. The final masks M are obtained
+by:
+M = Up (Sigmoid (FN ∗ F)) ,
+(10)
+where ∗ represents convolution operation, and Sigmoid
+transforms the results to (0, 1). After upsampling, we set a
+threshold to force M ∈ {0, 1}.
+Training loss
+Since panoptic narrative grounding is a seg-
+mentation task, we try different seg. losses in existing tasks,
+including BCE loss and Dice loss (Milletari, Navab, and Ah-
+madi 2016). For BCE loss, the loss function is formulated as
+LBCE =
+�
+ˆ
+yi∈M
+−(yi · log ( ˆyi)+(1 − yi) · log (1 − ˆyi)) ,
+(11)
+where ˆyi is the prediction of the i-th pixel and yi is the
+ground-truth. Dice loss is defined by
+LDice = 1 − 2|M � G|
+|M| + |G|,
+(12)
+where M is the generated mask and G is the ground truth,
+the value of which all belongs to {0, 1}. Considering these
+loss functions are designed for single-modal tasks (He et al.
+2017), we propose a new loss called bidirectional Semantic
+Alignment Loss (SAL) for regularizing the semantic consis-
+tency between modalities.
+Bidirectional Semantic Alignment Loss
+As mentioned
+above, PNG has complex many-to-many relationships, i.e.,
+a mask may belong to several noun phrases or vice versa.
+However, the above segmentation loss only considers the
+one-to-one interaction between the phrases and mask, while
+ignoring the semantic connection between them. Inspired
+by (Kamath et al. 2021), we design a SAL to guarantee se-
+mantic consistency, which encourages the multi-modal fea-
+tures with the same semantics to be similar.
+
+Method
+Average Recall
+Inference Time
+Params
+Training Data
+All
+Thing
+Stuff
+Single
+Plural
+Stage-1
+Stage-2
+All
+Stage-1
+Stage-2
+All
+Stage-1
+Stage-2
+All
+PNG (Gonz´alez et al. 2021)
+55.4
+56.2
+54.3
+56.2
+48.8
+100ms
+7ms
+107ms
+21.0M
+240.3M
+261.3M
+1.3M
+0.8M
+2.1M
+Baseline (ours)
+40.3
+34.5
+50.5
+42.3
+31.4
+-
+-
+9.5ms
+-
+-
+76.5M
+-
+-
+0.8M
+EPNG (ours)
+49.7
+45.6
+55.5
+50.2
+45.1
+-
+-
+11ms
+-
+-
+76.5M
+-
+-
+0.8M
+EPNG∗ (ours)
+58.0
+54.8
+62.4
+58.6
+52.1
+-
+-
+11ms
+-
+-
+76.5M
+-
+-
+2.1M
+Table 1: Comparison of the EPNG and the previous two-stage method. Baseline means the same design as EPNG except for
+LPA and SAL. EPNG∗ is trained with the same data as PNG.
+Communicator
+Average Recall
+All
+Thing
+Stuff
+Single
+Plural
+w/o PE
+42.8
+36.5
+52.1
+44.2
+32.5
+PE (Vaswani et al. 2017)
+46.7
+43.0
+52.9
+47.6
+42.4
+SPE (Liu et al. 2021)
+46.1
+41.8
+52.2
+46.6
+42.2
+DPE (Zhu et al. 2020)
+45.9
+41.5
+52.0
+46.5
+40.3
+RPE (Dosovitskiy et al. 2020)
+45.3
+42.1
+52.2
+46.9
+41.5
+LPA
+49.7
+45.6
+55.5
+50.2
+45.1
+Table 2: Ablation study of the LPA module, where “w/o
+PE” denotes no position embedding, and “SPE”, “DPE”,
+and “RPE” mean different relative position embedding from
+other methods.
+Loss
+Average Recall
+All
+Thing
+Stuff
+Single
+Plural
+BCE + Dice
+43.7
+38.5
+49.6
+43.7
+37.6
+BCE + Dice + SAL
+49.7
+45.6
+55.5
+50.2
+45.1
+Table 3: Ablation study of the SAL.
+Specifically, we first adopt noun phrases as anchors to im-
+prove the semantic consistency within visual features. For
+i-th noun phrase Fi
+N ∈ RC and the ground-truth Gi ∈
+{0, 1}H×W , the collection of pixels with the class of “1”
+is considered as the positive set, while the one of “0” is
+gathered as the negative set. By increasing the similarity
+within the positive set, multi-modal information is forced to
+be aligned. Considering all nouns together, the loss is intro-
+duced as follows:
+lv = 1
+L
+L
+�
+i=0
+1
+|G+|
+�
+j∈G+
+−log
+�
+exp
+�
+Fi
+n · Fj/τ
+�
+�
+k∈G exp (Fin · Fk/τ)
+�
+, (13)
+where G+ is the positive set, denotes the class of “1” in the
+ground truth, and G is the ground-truth. τ is a temperature
+coefficient.
+Next, We adopt pixel features as anchors to improve the
+semantic consistency within noun features. Similarly, the
+loss is introduced as follows:
+lt = 1
+|G|
+|G|
+�
+i=0
+1
+|T +|
+�
+j∈T +
+−log
+�
+exp
+�
+Fi·Fj
+n/τ
+�
+�
+k∈T exp (Fi·Fkn/τ)
+�
+,
+(14)
+where T + is the positive gather of the noun set, and T is
+the whole set, where ”1” denotes the pixel belonging to this
+noun phrase. We combine Eq. 13 and Eq. 14 as the SAL loss
+function.
+During the training, we use the summation of Dice loss,
+BCE loss, and SAL:
+L = λ1LBCE + λ2LDice + λ3LSAL,
+(15)
+where λ1, λ2 and λ3 are the hyper-parameters.
+Dataset
+Type
+mIoU
+p@0.3
+p@0.4
+p@0.5
+RefCOCO
+testA
+random
+9.2
+2.5
+0.8
+0.1
+zero-shot
+28.0
+42.5
+26.5
+12.4
+testB
+random
+9.8
+3.3
+1.3
+0.3
+zero-shot
+17.7
+22.4
+13.4
+7.8
+RefCOCO+
+testA
+random
+9.3
+2.6
+0.7
+0.2
+zero-shot
+27.7
+41.1
+25.6
+11.9
+testB
+random
+10.5
+3.8
+1.4
+0.3
+zero-shot
+20.6
+27.4
+16.3
+9.4
+RefCOCOg
+test
+random
+9.9
+4.0
+1.5
+0.6
+zero-shot
+27.4
+40.1
+27.3
+16.3
+Table 4: Zero-shot results of EPNG on RES. EPNG is not
+trained with RES data. We average the IoU of every case as
+the mIoU.
+Experiment
+Datasets
+We train and compare our model with the existing method
+on the Panoptic Narrative Grounding dataset (Gonz´alez et al.
+2021). It is consist of images and the corresponding text. Un-
+like the brief phrase in other datasets such as RefCOCO (Yu
+et al. 2016), the texts of PNG are long and are a narrative
+of all items in the complete image and their relationships.
+It often has hundreds of words and more complex semantic
+information. The noun-level segmentations are provided for
+each text. It encompasses both the thing and the stuff, simi-
+lar to panoptic segmentation. The difference is that the thing
+will include both the singular and the plural according to the
+semantics, which also brings more difficulty for vision-text
+alignment. The dataset includes a total of 133,103 training
+images and 8,380 test images with 875,073 and 56,531 seg-
+mentation annotations, respectively.
+Implementation Details
+Experimental Settings
+In this paper, we follow (Gonz´alez
+et al. 2021) to use the ResNet-101 as our visual back-
+bone, which is pre-trained on the ImageNet (Krizhevsky,
+Sutskever, and Hinton 2012). BERT is used as the text
+backbone. During the training process, all the backbones
+are frozen except for the last two layers of ResNet-101.
+For parallel training, we increase the input image resolu-
+tion to 640 × 640, so the shapes of the last three layers are
+20×20×256, 40×40×256, and 80×80×256, respectively.
+Moreover, the dimension of text features is 768. The num-
+ber of attention heads is 8 and the hidden dimension is 2048.
+Besides, the number of Layers S is 3. In terms of hyper-
+parameters, we use λ1 = 2, λ2 = 2 and λ3 = 1 to balance
+the final loss. We set the initial learning rate η = 1e−5 which
+
+Narrative
+Image
+Baseline
+Ground Truth
+In this image they are two buffaloes
+stand-ing on an open area. On the
+background we can see a monuments.
+On the top there is a sky with full
+clouds. In the center of the image we
+can see trees and plants. On the
+bottom we can see a grass on the
+ground.
+This image is a black and white
+image. This image is taken indoors.
+In the back-ground there is a wall. In
+the middle of the image there is a
+monitor on the table. At the bottom of
+the image there is a table with a
+keyboard and a mouse on it.
+This picture shows a woman standing
+and flying a kite with string and we
+see water and a cloudy sky and and
+women wore a black jacket and we
+see grass on the ground
+His picture shows a bus on the road
+and we see couple of trees and a
+electric pole and a cloudy sky
+EPNG
+Figure 3: Visualization of EPNG. We mark the same color between the nouns in the narrative and the referred pixels.
+Input Image
+Baseline
+EPNG
+Head#1
+Head#2
+Head#3
+Head#4
+Head#5
+Head#6
+Head#7
+Head#8
+Figure 4: Comparison of the attention of the sampled point for the LPA and MHA from the top layer. The red point represents
+the sampled point.
+is half decayed by every 5 epochs, and fix η = 5e−7 after
+10 epochs. The batch size is 32. We train it on 4 RTX3090
+GPUs, which cost 20 hours in total. The optimizer is Adam.
+Metrics
+Following (Gonz´alez et al. 2021), we use Average
+Recall (Gonz´alez et al. 2021) as our metric. Specifically, we
+calculate Intersection over Union (IoU) between masks and
+the ground truth. We use the integral of the IoU curve as the
+final metric. Additionally, we simultaneously analyze this
+measure for the thing, stuff, single, and plural.
+Quantitative Analysis
+Comparison with the state-of-the-arts.
+We first evaluate
+the overall performance of the model using the Average Re-
+call metric, as shown in Tab. 1. In Tab. 1, we also introduce a
+
+WORWORWORKPLACEBenc
+WORKPLACEReferring
+Image
+Prediction
+Ground Truth
+the man playing 
+tennis
+a lady wearing jeans
+and a pink and gray
+north face jacket
+front left chicken
+man
+Figure 5: The visualization of zero-shot setting for RES.
+baseline model for comparison, which is the same as EPNG
+except for LPA and SAL. The performance of the one-stage
+baseline is much inferior to the two-stage PNG. It is because
+single-stage models are trained end-to-end from scratch, suf-
+fering from greater training difficulties. After adding LPA
+and SAL, the proposed EPNG can bring up to 23.3% gain
+on all the masks, 32.2% on the thing, and 9.9% on the stuff,
+respectively. This fully demonstrates the effectiveness of the
+proposed method. In terms of the inference speed, we set the
+batch size to 12, and calculate the average inference time on
+each image. Our EPNG has a significant benefit, being 10×
+faster than the two-stage model and using only 38% of its pa-
+rameters, allowing for model deployment on edge devices.
+To make a fair comparison with PNG, we adopt FPN with
+a ResNet-101 backbone pre-trained with Panoptic Feature
+Pyramid Network (Kirillov et al. 2019a) on MS COCO. Our
+EPNG achieves better performance (i.e., 2.6 points of per-
+formance gain) at 10× inference speed, which better demon-
+strates the contribution of our EPNG.
+Ablation Study
+To verify the contribution of our proposed
+LPA and SAL, we conduct ablation experiments on the two
+modules, respectively. In Tab. 2, we compare LPA with other
+position embeddings. As can be seen that the method that
+does not use any location and distance information performs
+the worst, demonstrating the importance of location infor-
+mation for this task. Meanwhile, our LPA achieves the best
+performance than other relative position embeddings, which
+are widely used to capture location information (Dosovitskiy
+et al. 2020; Liu et al. 2021; Zhu et al. 2020).
+To verify the efficiency of SAL, we perform two exper-
+iments with and without it, of which results are given in
+Tab. 3. It can be seen that SAL brings a significant improve-
+ment of 6.0 points to EPNG, which fully validates our mo-
+tivation that the improvement of semantic consistency can
+well improve the segmentation accuracy in PNG.
+Zero-Shot Study for RES
+Meanwhile, we validate our
+model’s generalization by conducting zero-shot experiments
+on the datasets of the RES task, e.g., RefCOCO (Yu et al.
+2016), RefCOCO+ (Yu et al. 2016), and RefCOCOg (Mao
+et al. 2016; Nagaraja, Morariu, and Davis 2016). These
+datasets are built based on MS COCO (Lin et al. 2014) and
+each image has a phrase that refers to an object in the im-
+age. We use the feature of the whole phrase as the text fea-
+ture. The results are shown in Tab. 4. By comparing with the
+fully randomized model, we can find that the zero-shot per-
+formance of EPNG is significantly improved. Even on the
+more complex dataset i.e., RefCOCO+, our performance is
+close to some early supervised REC models, like (Liu et al.
+2017), mIoU of which is 30.48 and 29.5 on the testA and
+testB of RefCOCO+, respectively.
+Qualitative Analysis
+Visualization
+As shown in Fig. 3, we present some typical
+grounding results from EPNG compared to the ground truth.
+Compared to the baseline, our method can generate more
+accurate masks, especially for the edge parts. This further
+proves the effectiveness of our method.
+Attention Visualization
+To figure out the role of LPA in
+our proposed method, we visualize its attention weights dur-
+ing inference in Fig. 4. Compared with the baseline, LPA
+presents diverse local attention patterns with miscellaneous
+scopes, which brings a powerful capability to model local
+semantic relationships. This also validates the motivation of
+LPA, and our proposed method does allow the model to fo-
+cus locally and improve the accuracy of attention.
+Zero-Shot for RES
+Additionally, Fig. 5 shows some qual-
+itative results of our zero-shot study on RES. Our method
+can achieve very accurate segmentation, which fully demon-
+strates its transferability. For example, in the second case,
+our model identifies the segmentation of “man”, which is
+wrong even in the ground truth. Due to the ability of the
+finest-grained and complex semantic understanding, EPNG
+can handle more general scenarios with greater potential.
+Conclusion
+In this paper, we proposed an End-to-End Panoptic Narrative
+Grounding network (EPNG) for real-time inference. To bet-
+ter handle the many-to-many relationships between pixels
+and phrases, two innovative designs are proposed, namely
+Local-Sensitive Attention (LPA) and bidirectional Semantic
+Alignment Loss (SAL), respectively. Extensive experiments
+show that our proposed EPNG achieves significant perfor-
+mance gains compared to the baseline. More importantly,
+compared to the two-stage model, EPNG achieves compet-
+itive performance with faster inference (10x) and fewer pa-
+rameters. Furthermore, we conducted zero-shot experiments
+on RES and achieved surprising performance. These results
+demonstrate the excellent generalization of our model and
+also provide a reference for a subsequent unified vision-
+language segmentation framework.
+
+DavidDDaCCCPIACaDavidDaAcknowledgments
+This work was supported by the National Science Fund for
+Distinguished Young Scholars (No. 62025603), the National
+Natural Science Foundation of China (No. U21B2037, No.
+U22B2051, No. 62176222, No. 62176223, No. 62176226,
+No.
+62072386,
+No.
+62072387,
+No.
+62072389,
+No.
+62002305 and No. 62272401), Guangdong Basic and Ap-
+plied Basic Research Foundation (No. 2019B1515120049),
+and the Natural Science Foundation of Fujian Province of
+China (No. 2021J01002, No. 2022J06001).
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf,len=1256
+page_content='Towards Real-Time Panoptic Narrative Grounding by an End-to-End  Grounding Network  Haowei Wang1*, Jiayi Ji1*, Yiyi Zhou1, 2, Yongjian Wu4, Xiaoshuai Sun1, 2, 3†  1Media Analytics and Computing Lab, Department of Artificial Intelligence,  School of Informatics, Xiamen University, 361005, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  2Institute of Artificial Intelligence, Xiamen University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  3Fujian Engineering Research Center of Trusted Artificial Intelligence Analysis and Application, Xiamen University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  4Tencent Youtu Lab, Shanghai, China  wanghaowei@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='xmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='cn, jjyxmu@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='com, zhouyiyi@xmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='cn,  littlekenwu@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='com, xssun@xmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='cn  Abstract  Panoptic Narrative Grounding (PNG) is an emerging cross-  modal grounding task, which locates the target regions of  an image corresponding to the text description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Existing ap-  proaches for PNG are mainly based on a two-stage paradigm,  which is computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' In this paper, we propose  a one-stage network for real-time PNG, termed End-to-End  Panoptic Narrative Grounding network (EPNG), which di-  rectly generates masks for referents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Specifically, we propose  two innovative designs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=', Locality-Perceptive Attention  (LPA) and a bidirectional Semantic Alignment Loss (SAL),  to properly handle the many-to-many relationship between  textual expressions and visual objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' LPA embeds the local  spatial priors into attention modeling, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=', a pixel may belong  to multiple masks at different scales, thereby improving seg-  mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' To help understand the complex semantic relation-  ships, SAL proposes a bidirectional contrastive objective to  regularize the semantic consistency inter modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Exten-  sive experiments on the PNG benchmark dataset demonstrate  the effectiveness and efficiency of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Compared to  the single-stage baseline, our method achieves a significant  improvement of up to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='4% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' More importantly, our  EPNG is 10 times faster than the two-stage model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Mean-  while, the generalization ability of EPNG is also validated by  zero-shot experiments on other grounding tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' The source  codes and trained models for all our experiments are publicly  available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='com/Mr-Neko/EPNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='git.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Introduction  Panoptic Narrative Grounding (Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2021) is a  new challenging task that locates the target instances of an  image corresponding to the text description via binary pixel  masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Its main challenges not only lie in the joint under-  standing of multi-modal information but also in many-to-  many language-vision alignment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=', grounding all related  instances or amorphous regions mentioned in the text de-  scription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' This property also makes it different from a sim-  ilar grounding task called Referring Expression Segmenta-  tion (RES) (Hu, Rohrbach, and Darrell 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  †The corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Copyright © 2023, Association for the Advancement of Artificial  Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Panoptic  Segmentation  Feature  Interactive  Matching  Score  Mask  Allocation  Text  Visual Feature Extraction  Text  Multi-modal  Communicator  Dense  Prediction  Noun  Visual  100ms  7ms  11ms  Total: 107ms  Test Environment:  Nvidia RTX 3090,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' without the time of data loading  PNG  (two-stage)  EPNG  (one-stage)  PNG  (two-stage)  EPNG  (one-stage)  (a)  (b)  Textual Feature Extraction  Multimodal  Encoding  Total: 11ms  Figure 1: Comparison of pipeline and inference speed be-  tween the proposed EPNG and two-stage PNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' (a) EPNG  jointly processes visual and text information to generate re-  ferred masks in a one-stage fashion, while PNG relies on  mask proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' (b) Our single-stage EPNG is 10x faster  than the two-stage approach, enabling real-time deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2018), which segments only one  instance per expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Gonzalez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' (Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2021) first explore this  task and propose a preliminary solution in a two-stage fash-  ion, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' First, the pre-trained panoptic  segmentation models like PFPN (Kirillov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2019a) are  used to provide a set of candidate masks of the given image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Secondly, these masks are further transformed into convo-  lution features and then ranked by cross-modal matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Overall, with the help of panoramic segmentation models,  this two-stage solution defines PNG as a mask-text match-  ing problem, greatly reducing the difficulty of prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  However, this solution still suffers from two limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  On the one hand, such a two-stage approach requires of-  fline feature extraction, storage, and alignment, which is  inevitably time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' This limitation poses a huge  obstacle to real-time applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=', text-to-image re-  trieval, and video matting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' On the other hand, the pre-trained  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='03160v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='CV]  9 Jan 2023  panoramic segmentation model requires massive mask an-  notations, which place a greater burden on the already ex-  pensive expenditure of PNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' More importantly, the perfor-  mance of these panoptic segmentation models also limits the  upper bound of PNG models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  To solve the above problems, a natural way is to design an  efficient single-stage network for end-to-end training from  scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' However, this solution also encounters two chal-  lenges that are critical for PNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' First, in PNG, each pixel can  be subordinated to different masks, which is greatly differ-  ent from panoptic segmentation (Kirillov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' This  property makes the model need to capture visual seman-  tics from macro- to micro-views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' However, existing meth-  ods only focus on global modeling and overlook local in-  formation, resulting in limited performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Second, PNG  involves more complicated relationships than other ground-  ing or segmentation tasks (Liu, Wang, and Yang 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Luo  and Shakhnarovich 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' In each example,  multiple nouns of an expression may correspond to the same  mask, or one noun may refer to multiple masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' This case  further increases the difficulty of vision-language alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  In this paper, we propose a novel End-to-End Panoptic  Narrative Grounding network (EPNG) for real-time panop-  tic narrative grounding, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Specifically,  EPNG adopts a visual encoder to extract the features of the  given image, based on which a decoder is deployed to pre-  dict masks for different noun phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  To enhance local semantic modeling, we introduce  Locality-Perceptive Attention (LPA) to enhance grid fea-  tures via neighborhood interactions based on their spatial  priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' In LPA, different attention heads are allowed to per-  ceive visual information in different receptive fields, thus  achieving multi-scale modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' To ensure the semantic con-  sistency of many-to-many relationships in PNG, we design  a new bidirectional Semantic Alignment Loss (SAL), which  uses one modality as an anchor to eliminate the deviation  of similar semantic tokens of the other modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' With these  innovative designs, EPNG is superior in cross-modal reason-  ing while keeping real-time inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Conclusively, the contributions of our work are as below:  We propose a real-time End-to-End Panoptic Narrative  Grounding network (EPNG), which greatly reduces com-  putation overhead via unifying cross-modal alignment  and mask prediction in one forward structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  We propose two novel designs, namely Locality-  Perceptive Attention (LPA) and bidirectional Semantic  Alignment Loss (SAL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' LPA enhances visual features  at different scales to understand complex cross-modal  relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' SAL regularizes the semantic consistency  problem by performing contrastive learning between pix-  els and noun phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  On the benchmark dataset, EPNG is on par with or even  better than existing two-stage methods, while its infer-  ence is 10 times faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' In addition, it requires no addi-  tional mask annotations for pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Related Work  Panoptic Segmentation  Panoptic segmentation aims to entirely understand scenes  containing things and stuff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Following the benchmark pro-  posed by (Kirillov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2019b), the earlier methods  treated it as the combination of things masks and stuff  masks (de Geus, Meletis, and Dubbelman 2018), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=',  PFPN (Kirillov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2019a), Panoptic-DeepLab (Cheng  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2020a), and UPSNet (Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Recently,  things and stuff are expected to be treated uniformly (Car-  ion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Cheng, Schwing, and Kir-  illov 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' To eliminate the difference between things and  stuff, part of those like PFCN (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2021), K-Net (Zhang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2021), and Panoptic SegFormer (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2022) try to  use the kernel to represent things and stuff uniformly and  generate masks by the convolution on feature maps, which  obtain significant performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Benefiting from those meth-  ods, our model utilizes word features as reliable kernels to  get corresponding masks through the convolution on multi-  modal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Referring Expression Segmentation  Recently, multi-modal applications have received a lot of at-  tention and made significant progress (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2022b, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2022b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Among them, as a  prevalent task in multi-modal communities, Referring Ex-  pression Segmentation (RES) (Hu, Rohrbach, and Darrell  2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2018) is to seg-  ment a referent based on the understanding of a related short  phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' In sequential order, previous models (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Margffoy-Tuay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2018) obtain a set of  proposals by a general method of segmentation and pick up  a better one that is described by the given short phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' With  the strength of leveraging visual information, however, the  upper bound of those methods is seriously restricted by the  performance of the segmentation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' After that, a batch  of methods is developed for refining segmentation masks  by a single-stage network (Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2021), which brings higher rates of false positive  segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' In summary, RES is an incomplete task with  the neglect of stuff and many-to-many relationships between  natural language and images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Additionally, whether things  and stuff or the many-to-many relationships should be con-  sidered in Panoptic Narrative Grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Panoptic Narrative Grounding  The existing method (Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2021) handles it with  a two-stage paradigm, which first obtains a lot of candi-  date panoptic masks by a pre-trained panoptic segmenta-  tion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' With those candidates, a scoring module is used  to assign plural masks to referred phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' This paradigm  achieves impressive performance, nevertheless, the expen-  sive computation cost and space cost on the stage of seg-  ment becomes the barrier to real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Because of the rea-  sons above, we propose an End-to-End Panoptic Narrative  Grounding network (EPNG) to generate the corresponding  mask directly from the noun phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  In the center of the image  there are two elephants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' At  the bottom there is grass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' In  the background we can see  hills and sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='Linear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='Relative pos  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Dense Prediction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='𝑭𝑽  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Visual Feature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='…  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='𝐼3 …  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='…  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='…  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='…  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='…  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='𝑇𝑛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='𝐼𝑚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='…  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='𝐼2 𝐼3 … 𝐼4 𝐼𝑚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='𝑇1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='…  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='Mask  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='𝑮  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Ground-Truth  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='𝑻  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='𝑭𝑵  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Noun Feature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Text Encoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Visual Encoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Multi-modal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Communicator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='𝑪  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Contrastive Matrix  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Q  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='V  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='V  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Q  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='BCE Loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Segmentation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Dice Loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Bidirectional  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Semantic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Alignment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Phrase to  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Pixel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='𝒍𝒗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Pixel to  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Phrase  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='𝒍𝒕  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='LPA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Cross Attention  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Text Input  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Image Input  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='Figure 2: The framework of the proposed EPNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' The solid lines denote the pipeline of EPNG, while the dotted lines represent  the loss computation during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' During the pipeline, a Multi-modal Encoding module is used to extract the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Then a Multi-modal Communicator fuses multi-modal features with a Cross Attention module and the proposed LPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Finally,  traditional segmentation loss and the proposed SAL are set to improve the quality of segmentation and align the multi-modal  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  End-to-End Panoptic Narrative Grounding  Network  In this section, we give a detailed description of our EPNG,  of which the framework is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' The input  images and descriptions are first processed by the visual and  text encoders, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' A multi-modal fusion module is  further deployed for image-text interaction, based on which  a dense prediction head is used to predict masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Problem Definition  Unlike the existing two-stage PNG, the proposed one-stage  PNG is free of mask proposals, which generates the mask  directly based on the expressions and images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' We formulate  it as a cross-modal dense prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Specifically, given an image I and the corresponding text  T, the goal of PNG is to find the nouns N = {nℓ}L  ℓ=0 that  each pixel i belongs to, where nℓ is the ℓ-th noun and L de-  notes the number of the noun phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Then the probability  of the obtained mask M ∈ {0, 1} is formulated as:  p (M) =  �  i∈I  L  �  ℓ=0  p (i|I, T, nℓ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  (1)  Multi-modal Encoding  Visual Encoder  Given an image I ∈ RH×W ×3, we first  adopt a visual backbone (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2017) to extract the  multi-scale visual features, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=', Fv1 ∈ R  H  8 × W  8 ×C1, Fv2 ∈  R  H  16 × W  16 ×C2, and Fv3 ∈ R  H  32 × W  32 ×C3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Then we obtain the  final visual feature Fv ∈ R  H  16 × W  16 ×C by:  Fv = concat [Down (Fv1) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Fv2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Up (Fv3)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  (2)  where Up (·) denotes 2× upsampling, Down (·) denotes 2×  downsampling and concat [·] denotes feature concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Text Encoder  Given a sentence T, we follow (Gonz´alez  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2021) to adopt a pre-trained BERT (Kenton and  Toutanova 2019) to extract the word embeddings FT =  {vt}|T |  t=0, where vt denotes the embedding of t-th word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' After  that, we filter out the noun phrases according to the annota-  tions given by (Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2021) and then obtain the  phrase features by average-pooling the word embeddings in  each phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' These features are then projected by a linear  layer, making their feature dimension consistent with the vi-  sual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' As a result, the phrase embedding is denoted  as FN = {fnℓ}L  nℓ=0 ∈ RL×C, where nℓ represents the ℓ-th  noun phrases, and L is the number of phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Multi-modal Communicator  Based on the visual feature Fv and the textual feature FN,  Multi-Modal Communicator is designed for cross-modal in-  teraction and fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' It consists of S serial identical layer,  and each layer is composed of two modules called Locality-  Perceptive Attention (LPA) and Cross Attention (CA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Locality-Perceptive  Attention  Similar  to  self-  attention (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2017), LPA aims to improve  the input features via modeling their inter-relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  As argued in  (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2020b), local information is  important for the visual segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Then EPNG,  going a step further, presents multi-scale local modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Each pixel in an image may belong to different masks at the  same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' For example, a pixel in cloth may also belong  to a person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' However, the standard self-attention treats all  tokens in the feature map equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' To this end, we reinforce  the role of neighborhood information of each pixel when in  attention modeling, following (Wu, Wu, and Huang 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Specifically, in the features Fi, the 2D spatial coordinates  of the m-th and n-th vectors are denoted as (xm, ym) and  (xn, yn), where the superscript i indicates that the feature  map is the output of the layer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Then we calculate the Eu-  clidean Distance between these two coordinates:  Dm,n =  �  (xm − xn)2 + (ym − yn)2,  (3)  where D ∈ R(H×W )×(H×W ), and we truncate the values  in D with an upper bound 2 to explicitly inject the local  receptive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Afterward, for an attention head j in  LPA, we transform distance matrix into a coefficient matrix  Rj ∈ R(H×W )×(H×W ), obtained by:  Rj = WjD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  (4)  The obtained matrix Rj is used to re-weight the attention,  which is given by:  Aj = Softmax  �(FiWj  Q)(FiWj  K)T  √dk  ⊗ Rj  �  ,  (5)  where the projections Wj  Q ∈ Rd× d  h and Wj  K ∈ Rd× d  h are  weight matrices, and dk is a scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' The subscript j  represents the j-th head, and the number of heads h is set  to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' ⊗ represents an element-wise product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' In this way, we  naturally embed local information into attention modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Next, we sum the features using the attention weights to  obtain the results for head j, and aggregate all the results:  Headj = Aj(FiWj  V ),  (6)  LPA(Fi, Fi, Fi) = concat(Head1, · · · , Headh)WO,  (7)  where Wj  V ∈ Rd× d  h , and WO ∈ Rd×d are projection ma-  trices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Fi′ = LN  �  LPA  �  Fi, Fi, Fi�  + Fi�  ,  (8)  where LN (·) means the layer normalization (Ba, Kiros, and  Hinton 2016), and shortcut connection (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2016) is  applied after the LPA module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  The difference between LPA and Multi-Head Attention  (MHA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' As shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 5, the definition of LPA is based  on Multi-head Attention (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2017), but it still has  an obvious difference in principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' MHA treats all tokens in  the feature map equally and excels at capturing long-range  dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' However, local information is inevitably ig-  nored in this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' We introduce the local prior of each  grid, which is obtained from the distance matrix to attention  modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Note that, these local priors can be dynamically  adjusted by the coefficient matrix Rj in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Cross-modal Attention  Following (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2019), we  use Cross-modal Attention for modality interactions:  Fi+1 = FFN  �  LN  �  MHA  �  Fi′, FN, FN  �  + Fi′��  ,  (9)  where FFN (·) denotes feed-forward network, and FN is the  noun features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Dense Prediction  Given the fused feature after multi-modal communicator,  F ∈ R  H  16 × W  16 ×C, we upsample it to a tensor shape of  H  4 × W  4 × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Afterward, we apply each noun phrase feature  as a kernel to convolve F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' The final masks M are obtained  by:  M = Up (Sigmoid (FN ∗ F)) ,  (10)  where ∗ represents convolution operation, and Sigmoid  transforms the results to (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' After upsampling, we set a  threshold to force M ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Training loss  Since panoptic narrative grounding is a seg-  mentation task, we try different seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' losses in existing tasks,  including BCE loss and Dice loss (Milletari, Navab, and Ah-  madi 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' For BCE loss, the loss function is formulated as  LBCE =  �  ˆ  yi∈M  −(yi · log ( ˆyi)+(1 − yi) · log (1 − ˆyi)) ,  (11)  where ˆyi is the prediction of the i-th pixel and yi is the  ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Dice loss is defined by  LDice = 1 − 2|M � G|  |M| + |G|,  (12)  where M is the generated mask and G is the ground truth,  the value of which all belongs to {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Considering these  loss functions are designed for single-modal tasks (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  2017), we propose a new loss called bidirectional Semantic  Alignment Loss (SAL) for regularizing the semantic consis-  tency between modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Bidirectional Semantic Alignment Loss  As mentioned  above, PNG has complex many-to-many relationships, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=',  a mask may belong to several noun phrases or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  However, the above segmentation loss only considers the  one-to-one interaction between the phrases and mask, while  ignoring the semantic connection between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Inspired  by (Kamath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2021), we design a SAL to guarantee se-  mantic consistency, which encourages the multi-modal fea-  tures with the same semantics to be similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Method  Average Recall  Inference Time  Params  Training Data  All  Thing  Stuff  Single  Plural  Stage-1  Stage-2  All  Stage-1  Stage-2  All  Stage-1  Stage-2  All  PNG (Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2021)  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='4  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='2  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='3  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='2  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='8  100ms  7ms  107ms  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='0M  240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='3M  261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='3M  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='3M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='8M  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='1M  Baseline (ours)  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='3  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='5  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='5  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='3  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='5ms 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='8M  EPNG (ours)  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='7  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='1 11ms 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='8M  EPNG∗ (ours)  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='0  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='8  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='4  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='6  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='1 11ms 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='5M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='1M  Table 1: Comparison of the EPNG and the previous two-stage method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Baseline means the same design as EPNG except for  LPA and SAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' EPNG∗ is trained with the same data as PNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Communicator  Average Recall  All  Thing  Stuff  Single  Plural  w/o PE  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='8  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='1  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='2  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='5  PE (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2017)  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='7  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='6  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='4  SPE (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2021)  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='1  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='8  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='2  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='6  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='2  DPE (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2020)  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='9  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='5  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='0  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='5  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='3  RPE (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2020)  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='3  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='1  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='2  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='9  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='5  LPA  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='7  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='6  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='5  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='2  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='1  Table 2: Ablation study of the LPA module, where “w/o  PE” denotes no position embedding, and “SPE”, “DPE”,  and “RPE” mean different relative position embedding from  other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Loss  Average Recall  All  Thing  Stuff  Single  Plural  BCE + Dice  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='7  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='5  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='6  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='7  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='6  BCE + Dice + SAL  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='7  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='6  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='5  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='2  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='1  Table 3: Ablation study of the SAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Specifically, we first adopt noun phrases as anchors to im-  prove the semantic consistency within visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' For  i-th noun phrase Fi  N ∈ RC and the ground-truth Gi ∈  {0, 1}H×W , the collection of pixels with the class of “1”  is considered as the positive set, while the one of “0” is  gathered as the negative set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' By increasing the similarity  within the positive set, multi-modal information is forced to  be aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Considering all nouns together, the loss is intro-  duced as follows:  lv = 1  L  L  �  i=0  1  |G+|  �  j∈G+  −log  �  exp  �  Fi  n · Fj/τ  �  �  k∈G exp (Fin · Fk/τ)  �  , (13)  where G+ is the positive set, denotes the class of “1” in the  ground truth, and G is the ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' τ is a temperature  coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Next, We adopt pixel features as anchors to improve the  semantic consistency within noun features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Similarly, the  loss is introduced as follows:  lt = 1  |G|  |G|  �  i=0  1  |T +|  �  j∈T +  −log  �  exp  �  Fi·Fj  n/τ  �  �  k∈T exp (Fi·Fkn/τ)  �  ,  (14)  where T + is the positive gather of the noun set, and T is  the whole set, where ”1” denotes the pixel belonging to this  noun phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' We combine Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 13 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 14 as the SAL loss  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  During the training, we use the summation of Dice loss,  BCE loss, and SAL:  L = λ1LBCE + λ2LDice + λ3LSAL,  (15)  where λ1, λ2 and λ3 are the hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Dataset  Type  mIoU  p@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='3  p@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='4  p@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='5  RefCOCO  testA  random  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='1  zero-shot  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='0  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='5  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='5  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='4  testB  random  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='3  zero-shot  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='7  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='4  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='4  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='8  RefCOCO+  testA  random  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='2  zero-shot  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='7  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='1  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='6  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='9  testB  random  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='3  zero-shot  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='6  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='4  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='3  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='4  RefCOCOg  test  random  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='6  zero-shot  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='4  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='1  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='3  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='3  Table 4: Zero-shot results of EPNG on RES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' EPNG is not  trained with RES data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' We average the IoU of every case as  the mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Experiment  Datasets  We train and compare our model with the existing method  on the Panoptic Narrative Grounding dataset (Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' It is consist of images and the corresponding text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Un-  like the brief phrase in other datasets such as RefCOCO (Yu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2016), the texts of PNG are long and are a narrative  of all items in the complete image and their relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  It often has hundreds of words and more complex semantic  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' The noun-level segmentations are provided for  each text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' It encompasses both the thing and the stuff, simi-  lar to panoptic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' The difference is that the thing  will include both the singular and the plural according to the  semantics, which also brings more difficulty for vision-text  alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' The dataset includes a total of 133,103 training  images and 8,380 test images with 875,073 and 56,531 seg-  mentation annotations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Implementation Details  Experimental Settings  In this paper, we follow (Gonz´alez  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2021) to use the ResNet-101 as our visual back-  bone, which is pre-trained on the ImageNet (Krizhevsky,  Sutskever, and Hinton 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' BERT is used as the text  backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' During the training process, all the backbones  are frozen except for the last two layers of ResNet-101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  For parallel training, we increase the input image resolu-  tion to 640 × 640, so the shapes of the last three layers are  20×20×256, 40×40×256, and 80×80×256, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Moreover, the dimension of text features is 768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' The num-  ber of attention heads is 8 and the hidden dimension is 2048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Besides, the number of Layers S is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' In terms of hyper-  parameters, we use λ1 = 2, λ2 = 2 and λ3 = 1 to balance  the final loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' We set the initial learning rate η = 1e−5 which  Narrative  Image  Baseline  Ground Truth  In this image they are two buffaloes  stand-ing on an open area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' On the  background we can see a monuments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  On the top there is a sky with full  clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' In the center of the image we  can see trees and plants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' On the  bottom we can see a grass on the  ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  This image is a black and white  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' This image is taken indoors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  In the back-ground there is a wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' In  the middle of the image there is a  monitor on the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' At the bottom of  the image there is a table with a  keyboard and a mouse on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  This picture shows a woman standing  and flying a kite with string and we  see water and a cloudy sky and and  women wore a black jacket and we  see grass on the ground  His picture shows a bus on the road  and we see couple of trees and a  electric pole and a cloudy sky  EPNG  Figure 3: Visualization of EPNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' We mark the same color between the nouns in the narrative and the referred pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Input Image  Baseline  EPNG  Head#1  Head#2  Head#3  Head#4  Head#5  Head#6  Head#7  Head#8  Figure 4: Comparison of the attention of the sampled point for the LPA and MHA from the top layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' The red point represents  the sampled point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  is half decayed by every 5 epochs, and fix η = 5e−7 after  10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' The batch size is 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' We train it on 4 RTX3090  GPUs, which cost 20 hours in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' The optimizer is Adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Metrics  Following (Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2021), we use Average  Recall (Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2021) as our metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Specifically, we  calculate Intersection over Union (IoU) between masks and  the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' We use the integral of the IoU curve as the  final metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Additionally, we simultaneously analyze this  measure for the thing, stuff, single, and plural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Quantitative Analysis  Comparison with the state-of-the-arts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  We first evaluate  the overall performance of the model using the Average Re-  call metric, as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 1, we also introduce a  WORWORWORKPLACEBenc  WORKPLACEReferring  Image  Prediction  Ground Truth  the man playing  tennis  a lady wearing jeans  and a pink and gray  north face jacket  front left chicken  man  Figure 5: The visualization of zero-shot setting for RES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  baseline model for comparison, which is the same as EPNG  except for LPA and SAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' The performance of the one-stage  baseline is much inferior to the two-stage PNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' It is because  single-stage models are trained end-to-end from scratch, suf-  fering from greater training difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' After adding LPA  and SAL, the proposed EPNG can bring up to 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='3% gain  on all the masks, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='2% on the thing, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='9% on the stuff,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' This fully demonstrates the effectiveness of the  proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' In terms of the inference speed, we set the  batch size to 12, and calculate the average inference time on  each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Our EPNG has a significant benefit, being 10×  faster than the two-stage model and using only 38% of its pa-  rameters, allowing for model deployment on edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  To make a fair comparison with PNG, we adopt FPN with  a ResNet-101 backbone pre-trained with Panoptic Feature  Pyramid Network (Kirillov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2019a) on MS COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Our  EPNG achieves better performance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=', 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='6 points of per-  formance gain) at 10× inference speed, which better demon-  strates the contribution of our EPNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Ablation Study  To verify the contribution of our proposed  LPA and SAL, we conduct ablation experiments on the two  modules, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2, we compare LPA with other  position embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' As can be seen that the method that  does not use any location and distance information performs  the worst, demonstrating the importance of location infor-  mation for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Meanwhile, our LPA achieves the best  performance than other relative position embeddings, which  are widely used to capture location information (Dosovitskiy  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  To verify the efficiency of SAL, we perform two exper-  iments with and without it, of which results are given in  Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' It can be seen that SAL brings a significant improve-  ment of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='0 points to EPNG, which fully validates our mo-  tivation that the improvement of semantic consistency can  well improve the segmentation accuracy in PNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Zero-Shot Study for RES  Meanwhile, we validate our  model’s generalization by conducting zero-shot experiments  on the datasets of the RES task, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=', RefCOCO (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  2016), RefCOCO+ (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2016), and RefCOCOg (Mao  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Nagaraja, Morariu, and Davis 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' These  datasets are built based on MS COCO (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 2014) and  each image has a phrase that refers to an object in the im-  age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' We use the feature of the whole phrase as the text fea-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' The results are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' By comparing with the  fully randomized model, we can find that the zero-shot per-  formance of EPNG is significantly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Even on the  more complex dataset i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=', RefCOCO+, our performance is  close to some early supervised REC models, like (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  2017), mIoU of which is 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='48 and 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='5 on the testA and  testB of RefCOCO+, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Qualitative Analysis  Visualization  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 3, we present some typical  grounding results from EPNG compared to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Compared to the baseline, our method can generate more  accurate masks, especially for the edge parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' This further  proves the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Attention Visualization  To figure out the role of LPA in  our proposed method, we visualize its attention weights dur-  ing inference in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Compared with the baseline, LPA  presents diverse local attention patterns with miscellaneous  scopes, which brings a powerful capability to model local  semantic relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' This also validates the motivation of  LPA, and our proposed method does allow the model to fo-  cus locally and improve the accuracy of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Zero-Shot for RES  Additionally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 5 shows some qual-  itative results of our zero-shot study on RES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Our method  can achieve very accurate segmentation, which fully demon-  strates its transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' For example, in the second case,  our model identifies the segmentation of “man”, which is  wrong even in the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Due to the ability of the  finest-grained and complex semantic understanding, EPNG  can handle more general scenarios with greater potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Conclusion  In this paper, we proposed an End-to-End Panoptic Narrative  Grounding network (EPNG) for real-time inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' To bet-  ter handle the many-to-many relationships between pixels  and phrases, two innovative designs are proposed, namely  Local-Sensitive Attention (LPA) and bidirectional Semantic  Alignment Loss (SAL), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Extensive experiments  show that our proposed EPNG achieves significant perfor-  mance gains compared to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' More importantly,  compared to the two-stage model, EPNG achieves compet-  itive performance with faster inference (10x) and fewer pa-  rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Furthermore, we conducted zero-shot experiments  on RES and achieved surprising performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' These results  demonstrate the excellent generalization of our model and  also provide a reference for a subsequent unified vision-  language segmentation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  DavidDDaCCCPIACaDavidDaAcknowledgments  This work was supported by the National Science Fund for  Distinguished Young Scholars (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 62025603), the National  Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' U21B2037, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  U22B2051, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 62176222, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 62176223, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' 62176226,  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  62072386,  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  62072387,  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content=' 2019B1515120049),  and the Natural Science Foundation of Fujian Province of  China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content=' Weissenborn,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content=' Zhai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content=' Unterthiner, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content=' Dehghani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='  Heigold, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content=' Gelly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content=' An Image is Worth 16x16  Words: Transformers for Image Recognition at Scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='  Shao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content=' and Ji, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content=' Multi-Branch Distance-Sensitive  Self-Attention Network for Image Captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' IEEE Trans-  actions on Multimedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content=' Improving image captioning by leverag-  ing intra-and inter-layer global representation in transformer  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='  Knowing what to learn: a metric-oriented focal mechanism  for image captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' IEEE Transactions on Image Process-  ing, 31: 4321–4335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Kamath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Singh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' LeCun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' Synnaeve, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='  and Carion, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='  Kenton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content=' Panoptic SegFormer: Delv-  ing deeper into panoptic segmentation with transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content=' Knowing what it is: Semantic-enhanced Dual  Attention Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' IEEE Transactions on Multimedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content=' and Ji, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content='  X-CLIP: End-to-End Multi-grained Contrastive Learning  for Video-Text Retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content=' In Proceedings of the 30th ACM  International Conference on Multimedia, 638–647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+page_content=' In Proceedings of the IEEE  conference on computer vision and pattern recognition, 11–  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
+page_content='  Margffoy-Tuay, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf'}
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+In submission 
+Russia-Ukraine war: Modeling and Clustering the Sentiments Trends of 
+Various Countries 
+Hamed Vahdat-Nejad*, Mohammad Ghasem Akbari**, Fatemeh Salmani*, Faezeh Azizi*,  
+Hamid-Reza Nili-Sani** 
+* PerLab, Faculty of Electrical and Computer Engineering, University of Birjand, Birjand, Iran 
+**Department of Statistics, University of Birjand, Birjand, Iran 
+Email: vahdatnejad@birjand.ac.ir, g_z_akbari@birjand.ac.ir, salmani_fatemeh98@birjand.ac.ir, 
+faezeh.azizi1995@birjand.ac.ir, hnilisani@birjand.ac.ir  
+Abstract 
+With Twitter’s growth and popularity, a huge number of views are shared by users on various 
+topics, making this platform a valuable information source on various political, social, and 
+economic issues. This paper investigates English tweets on the Russia-Ukraine war to analyze 
+trends reflecting users’ opinions and sentiments regarding the conflict. The tweets’ positive and 
+negative sentiments are analyzed using a BERT-based model, and the time series associated with 
+the frequency of positive and negative tweets for various countries is calculated. Then, we propose 
+a method based on the neighborhood average for modeling and clustering the time series of 
+countries. The clustering results provide valuable insight into public opinion regarding this 
+conflict. Among other things, we can mention the similar thoughts of users from the United States, 
+Canada, the United Kingdom, and most Western European countries versus the shared views of 
+Eastern European, Scandinavian, Asian, and South American nations toward the conflict. 
+Keywords: Social network mining, Russia-Ukraine war, Sentiment analysis, Clustering 
+1. Introduction 
+
+ 
+Social networks are currently one of the most significant platforms for population information 
+extraction. Users of social networks share innumerable posts in many domains, including social, 
+economic, and political ones, that reflect their perspectives and sentiments regarding global events 
+and incidents (Crannell, Clark, Jones, James, & Moore, 2016). As a social network, Twitter is a 
+key source of users’ sentiments and perspectives. Users’ views can be analyzed using natural 
+language processing and data retrieval. 
+Natural language processing (NLP) is “focused on the design and analysis of computational 
+algorithms and representations for processing natural human language” (Eisenstein, 2018). It aims 
+to provide “new computational capabilities around human language” (Eisenstein, 2018). Indeed, 
+we can extract information from a mass of texts using NLP techniques. 
+Sentiment analysis is a promising technique used in natural language processing to extract 
+emotional states and subjective information from a text (Zad, Heidari, Jones, & Uzuner, 2021). It 
+allows for the extraction of sentiments from users’ tweets regarding a variety of events and issues. 
+The outbreak of COVID-19, for instance, was a globally important event about which many users 
+have expressed their opinions. Alongside this, there have been sentiment analyses of tweets related 
+to COVID-19 (Vahdat-Nejad et al., 2022), the economy during COVID-19 (Salmani, Vahdat-
+Nejad, & Hajiabadi, 2021), and education during the epidemic of COVID-19 (Jamalian, Vahdat-
+Nejad, & Hajiabadi, 2022), among others. Other events, including wars and conflicts, have also 
+prompted users to post and analyze tweets. For instance, the sentiment analysis of Turkish and 
+English tweets about Syrian refugees has revealed a predominantly positive tone in Turkish tweets 
+but a negative tone in English tweets (Öztürk & Ayvaz, 2018).  
+Similarly, an analysis of opinions regarding recent events in Afghanistan has revealed that tweets 
+containing negative terms such as terrorist, attack, destroy, and violence have dominated 
+
+ 
+(Aggarwal, Khan, & Kakkar, 2022). Furthermore, the ratio of negative to positive tweets was 
+identical across all eight investigated countries, indicating that these nations shared similar 
+sentiments regarding the recent events in Afghanistan (Aggarwal et al., 2022). Likewise, the 
+sentiment analysis of tweets about the Syrian chemical war reveals that users negatively view the 
+Syrian chemical attack (Bashir et al., 2021).  
+The Russia-Ukraine war has probably been the most prominent event of 2022. Recently, the main 
+topics discussed on the Chinese Weibo social network regarding this conflict have been extracted 
+(Chen et al., 2022). To our best knowledge, no study has analyzed tweets regarding this war to 
+investigate and categorize the attitude of various countries. To this end, this research collects and 
+analyzes tweets on this conflict to cluster various countries according to their population’s 
+attitudes. For this purpose, 140,000 tweets posted in the first month of the war (March 2022) were 
+collected using the keyword Ukraine. In order to analyze the sentiments of tweets, the language-
+based RoBERTa (Liu et al., 2019) model is employed, which has superior accuracy and 
+performance compared to similar models (Briskilal & Subalalitha, 2022). Following this, the time 
+series of the frequency of positive and negative sentiments are calculated and modeled for several 
+countries that have a sufficient number of English tweets. Next, the time series models of the 
+countries are clustered based on the neighborhood average method. Results indicate that countries 
+such as the United States, England, Canada, and those of Western Europe are in one cluster, while 
+Eastern Europe, Scandinavian, South American, and Asian countries are in another. In addition, 
+Ukraine was placed alone in one cluster, indicating a divergent trend in public opinion between 
+this country and the rest of the world in the first month of the conflict. The main contributions of 
+the paper are as follows: 
+
+ 
+• To the best of our knowledge, it is the first work investigating and clustering the attitudes 
+of nations on Twitter regarding the Ukraine war. 
+• It proposes a new method for modeling and clustering the time series of countries’ 
+sentiments. 
+The article’s remaining sections are as follows: Section 2 reviews previous research. Section 3 
+describes the proposed methodology. Implementation and results are described in Section 4. 
+Lastly, the fifth section summarizes the conclusions and limitations. 
+2- Related work 
+As a result of their growth and popularity, social networks have become an important and valuable 
+information source on users’ attitudes in various fields. Techniques such as big data and natural 
+language processing are required to analyze this unstructured textual information of social network 
+users. Twitter is one of the most popular social networks on which users share their thoughts and 
+feelings about current events and issues (Mathur, Kubde, & Vaidya, 2020). Numerous analyses of 
+user opinions have been conducted in a variety of fields, including politics (Ansari, Aziz, Siddiqui, 
+Mehra, & Singh, 2020), stock return forecasting (Sul, Dennis, & Yuan, 2017), sustainable energy 
+(Corbett & Savarimuthu, 2022), and tourism (Abbasi-Moud, Vahdat-Nejad, & Mansoor, 2019; 
+Asani, Vahdatnejad, Hosseinabadi, & Sadri, 2020). For instance, with the spread of COVID-19 
+over the past few years, researchers have analyzed the sentiments of COVID-19-related tweets 
+(Salmani et al., 2021; Vahdat-Nejad et al., 2022) and extracted the primary topics discussed (Azizi, 
+Vahdat-Nejad, Hajiabadi, & Khosravi, 2021).  
+The incidence of wars and conflicts is among the occasions that prompt users to post their opinions 
+on social networks. In this regard, users’ views in various countries regarding recent events in 
+
+ 
+Afghanistan were analyzed from August to November 2021 (Aggarwal et al., 2022). For this 
+purpose, the eight nations with the most tweets were investigated. One of the findings of this study 
+was the high number of tweets containing negative terms such as terrorist, attack, destruction, and 
+violence, as well as the fact that all eight investigated countries shared a similar perspective 
+regarding the recent events in Afghanistan (Aggarwal et al., 2022). Likewise, a sentiment 
+analysis (positive, neutral, negative, very positive, and very negative) was conducted on English 
+and Turkish tweets related to the Syrian civil war in April 2017 (Öztürk & Ayvaz, 2018). The R-
+Sentiment package was used to analyze the sentiments of tweets written in English. Due to the 
+substantial influx of Syrian refugees into Turkey, Turkish tweets were also analyzed. In the 
+absence of a model for sentiment analysis of tweets written in Turkish, a sentiment lexicon of 
+Turkish vocabulary was presented. Accordingly, there was a large number of positive tweets in 
+Turkish and a large number of neutral, negative, and extremely negative tweets in English. 
+Additionally, English tweets were more political and focused more on the legality of immigrants, 
+whereas the Turkish-language tweets focused more on the war’s details, as the conflict in 
+proximity to the Turkish border was of interest to the Turkish community (Öztürk & Ayvaz, 2018). 
+Similar sentiment analyses have been conducted for the ten countries with the highest volume of 
+tweets regarding Syria’s chemical warfare (Bashir et al., 2021). Overall, a greater proportion of 
+tweets were negative, indicating that users had a negative view regarding the Syrian chemical 
+attack (Bashir et al., 2021). 
+The Russia-Ukraine war is one of the most important events of 2022, about which countless tweets 
+have been published. Recently, Chinese Weibo texts have been investigated to extract the main 
+topics discussed by Chinese users about this conflict(Chen et al., 2022). We aim to continue that 
+research by modeling the sentiment trends of various countries and clustering them. To the best of 
+
+ 
+our knowledge, no study has been conducted to investigate and classify the sentiments trends of 
+users from different countries regarding this war. 
+3- Proposed Method 
+Numerous tweets have been posted regarding the Russia-Ukraine war, which is one of the most 
+significant events in 2022. To the best of our knowledge, no research has been conducted to 
+analyze the sentiments of tweets related to this war and to examine and classify the sentiments of 
+users from various countries. This research aims to collect and analyze tweets pertaining to the 
+Ukrainian dispute in order to classify the sentiment trends of users from varying countries over 
+time. 
+3-1- Tweet collection and processing 
+In March 2022, the war between Ukraine and Russia was the prominent topic of tweets. First, 
+tweets related to the war in Ukraine are collected and separated by location. Afterward, positive 
+and negative tweets are identified using sentiment analysis. The final step concerns clustering the 
+time series of tweet sentiments. 
+We collected 140,000 English tweets associated with the Russia-Ukraine war using the keyword 
+Ukraine during March 2022. The geotag is then applied to separate tweets from different countries. 
+Thus, tweets with an unknown location are deleted. 
+The process of determining whether a tweet is positive, negative, or neutral is known as tweet 
+sentiment analysis. A large share of sentiment analysis research is conducted on social network 
+posts, such as those on Twitter, since these posts contain users’ views and feelings. We utilize 
+sentiment analysis to determine the trends of public sentiment regarding this war over time. To 
+calculate the sentiment scores of tweets, we employed the promising language-based RoBERTa 
+
+ 
+(Liu et al., 2019) model. Recommended by Google, RoBERTa (Liu et al., 2019) is the optimized 
+version of the language-based BERT (Devlin, Chang, Lee, & Toutanova, 2019) model. Identifying 
+positive and negative tweets gives us insight into the direction of public opinion regarding the 
+Ukraine conflict. The number of positive and negative tweets for each country (for which there are 
+sufficient tweets) was then calculated weekly to yield a time series for each country. The analysis 
+and clustering of time series pertaining to various nations unveiled intriguing information 
+regarding public opinion. 
+3-2- Clustering countries' sentiments trends 
+Various methods have been presented for modeling and clustering users' opinions in social 
+networks (Asadolahi, Akbari, Hesamian, & Arefi, 2021; G. Hesamian & M. Akbari, 2021; 
+Hesamian & Akbari, 2018; G. Hesamian & M. G. Akbari, 2021; Hesamian & Akbari, 2022). This 
+research draws on the neighborhood average method to cluster countries. Because the number of 
+tweets in different countries varies, with the majority of these differences being substantial, the 
+weekly data for each country is first normalized. Subsequently, the weekly frequency of the 
+number of positive and negative tweets are computed for each country. Therefore, four positive 
+and four negative frequencies corresponding to the first four weeks of the war’s onset are 
+calculated for each country and modeled as a vector with eight features. The neighborhood average 
+method is then used to cluster the coefficients obtained from the proposed model for the nations. 
+The distance between countries is calculated using the Euclidean meter, and the countries with a 
+smaller distance are put in one cluster. The proposed model, which uses the support vector method 
+(SVM), is elaborated on below. 
+First, we consider the following model for time data {𝑍𝑡:𝑡 =  1,2,…, 𝑇   (T = 8)}: 
+
+ 
+𝑍𝑡 = ∑ 𝑊𝑗𝑍𝑡−𝑗 + 𝑊0 = 𝑊
+⃗⃗⃗ 𝑇𝑍𝑡
+⃗⃗⃗ + 𝑊0               𝑍 𝑡 = ( 𝑍𝑡−1, … , 𝑍𝑡−𝑝)
+𝑝
+𝑗=1
+ 
+To estimate the vector 𝑊
+⃗⃗⃗  and 𝑊0, we use the SVM as per figure1 and following equations: 
+ 
+ 
+𝑍𝑡 =  ∑
+𝑊𝑗𝑍𝑡−𝑗
+𝑝
+𝑗=1
++ 𝑊0  ± 𝑒  
+ 
+𝑍𝑡 =  ∑
+𝑊𝑗𝑍𝑡−𝑗
+𝑝
+𝑗=1
++ 𝑊0  ± 𝑒  
+ 
+𝜀𝑡 
+Figure 1. Using SVM for estimation 
+min𝑊
+⃗⃗⃗ ,𝑊0,𝜀⃗ ,𝜀⃗ ∗ 𝐽(𝑊,
+⃗⃗⃗⃗⃗ 𝑊0, 𝜀 , 𝜀 ∗) = 
+1
+2 𝑊
+⃗⃗⃗  𝑇𝑊
+⃗⃗⃗ + 𝑐 ∑
+(𝜀𝑡 + 𝜀𝑡
+∗)             (1)
+𝑇
+𝑡=𝑝+1
+         
+S.T 
+𝑍𝑡 − 𝑊
+⃗⃗⃗ 𝑇𝑍𝑡
+⃗⃗⃗ − 𝑊0 ≤ 𝑒 + 𝜀𝑡                     𝑡 = 𝑝 + 1 , … , 𝑇       
+ 𝑊
+⃗⃗⃗ 𝑇𝑍𝑡
+⃗⃗⃗ + 𝑊0 − 𝑍𝑡 ≤ 𝑒 + 𝜀𝑡                           
+∗
+𝑡 = 𝑝 + 1 , … , 𝑇 
+𝜀𝑡 , 𝜀𝑡
+∗  ≥ 0 
+Support vectors
+ 
+𝜀𝑡 
+
+ 
+Lagrange’s equation for the above optimization problem with coefficients 𝛼𝑡 , 𝛼𝑡
+∗, 𝜇𝑡 , 𝜇𝑡
+∗ is as 
+follows:  
+𝐿(𝑊,
+⃗⃗⃗⃗⃗ 𝑊0, 𝜀  , 𝜀 ∗, 𝛼  , 𝛼 ∗𝜇 , 𝜇 ∗)
+=  𝐽(𝑊,
+⃗⃗⃗⃗⃗ 𝑊0, 𝜀 , 𝜀 ∗)
+− ∑ 𝛼𝑡 (𝑒 + 𝜀  − 𝑍𝑡 + 𝑊
+⃗⃗⃗ 𝑇𝑍𝑡
+⃗⃗⃗ + 𝑊0)
+𝑇
+𝑡=𝑝+1
+− ∑ 𝛼𝑡
+∗ (𝑒 + 𝜀 ∗ + 𝑍𝑡 − 𝑊
+⃗⃗⃗ 𝑇𝑍𝑡
+⃗⃗⃗ − 𝑊0) − ∑  (𝜇𝑡𝜀𝑡 + 𝜇𝑡
+∗𝜀𝑡
+∗) 
+𝑇
+𝑡=𝑝+1
+𝑇
+𝑡=𝑝+1
+      (2) 
+Where 𝐿 is used in Eq (2) to reach the optimal value 𝐽 in Eq (1), i.e., 
+max(𝛼 , 𝛼 ∗, 𝜇 , 𝜇 ∗)      min(𝑊,
+⃗⃗⃗⃗ 𝑊0, 𝜀 , 𝜀 
+∗)      𝐿(𝑊,
+⃗⃗⃗⃗ 𝑊0, 𝜀  , 𝜀 
+∗, 𝛼⃗  , 𝛼⃗ 
+∗𝜇 , 𝜇⃗ 
+∗)        
+We have: 
+𝜕𝐿
+𝜕𝑊
+⃗⃗⃗ = 0  → 𝑊
+⃗⃗⃗ = ∑ (𝛼𝑡 − 𝛼𝑡
+∗)
+𝑇
+𝑡=𝑝+1
+𝑍𝑡 
+𝜕𝐿
+𝜕𝑊0
+= 0  → ∑ (𝛼𝑡 − 𝛼𝑡
+∗)
+𝑇
+𝑡=𝑝+1
+= 0                                                         (3) 
+𝜕𝐿
+𝜕𝜀𝑡
+= 0  →   𝑐 − 𝛼𝑡−𝜇𝑡 = 0           𝜕𝐿
+𝜕𝜀𝑡
+∗ = 0   → 𝑐 − 𝛼𝑡
+∗ − 𝜇𝑡
+∗ = 0 
+By embedding Eq (3) in (2), the dual (1) is yielded as follows: 
+
+ 
+m𝑎𝑥𝛼⃗⃗ ,𝛼⃗⃗ ∗ 𝐽𝐷(𝛼, 𝛼∗)
+= − 1
+2 𝑊
+⃗⃗⃗  𝑇𝑊
+⃗⃗⃗ 
+− 𝑒 ∑ (𝛼𝑡 + 𝛼𝑡
+∗)
+𝑇
+𝑡=𝑝+1
++ ∑ 𝑍𝑡(𝛼𝑡 − 𝛼𝑡
+∗)
+𝑇
+𝑡=𝑝+1
+= ∑
+∑ (𝛼𝑡 − 𝛼𝑡
+∗)(𝛼𝑘 − 𝛼𝑘
+∗)
+𝑇
+𝑘=𝑝+1
+ 
+𝑇
+𝑡=𝑝+1
+  𝑍𝑡
+⃗⃗⃗ 
+ 𝑇𝑍𝑘
+⃗⃗⃗⃗ 
+− 𝑒   ∑ (𝛼𝑡 + 𝛼𝑡
+∗) + 
+𝑇
+𝑡=𝑝+1
+∑ 𝑍𝑡(𝛼𝑡 − 𝛼𝑡
+∗)
+𝑇
+𝑡=𝑝+1
+                   
+S.T 
+∑ (𝛼𝑡 − 𝛼𝑡
+∗)
+𝑇
+𝑡=𝑝+1
+= 0          0 ≤ 𝛼𝑡, 𝛼𝑡
+∗  ≤   𝑐 
+The KKT method is used to obtain 𝑊0 as follows: 
+𝛼𝑡 (𝑒 + 𝜀𝑡 − 𝑍𝑡 + 𝑊
+⃗⃗⃗⃗  𝑇𝑍𝑡
+⃗⃗⃗⃗ + 𝑊0) = 0 
+𝛼𝑡
+∗ (𝑒 + 𝜀𝑡
+∗ + 𝑍𝑡 − 𝑊
+⃗⃗⃗⃗  𝑇𝑍𝑡
+⃗⃗⃗⃗ − 𝑊0) = 0 
+As a result, with a few simple calculations, we have: 
+𝑊0 = 𝑍𝑡 − 𝑊
+⃗⃗⃗⃗  𝑇𝑍𝑡
+⃗⃗⃗⃗ − 𝑒                   𝛼𝑡𝜖[0, 𝑐]           𝜀𝑡 = 0 
+𝑊0 = 𝑍𝑡 − 𝑊
+⃗⃗⃗⃗  𝑇𝑍𝑡
+⃗⃗⃗⃗ + 𝑒                   𝛼𝑡
+∗𝜖[0, 𝑐]           𝜀𝑡
+∗ = 0     
+
+ 
+Therefore: 
+𝑊0
+̂ = 1
+|𝑆| ∑ (𝑍𝑡 − 𝑊
+⃗⃗⃗⃗  𝑇𝑍𝑡
+⃗⃗⃗⃗ − 𝑆𝑖𝑔𝑛(𝛼𝑡 − 𝛼𝑡
+∗)𝑒)
+𝑡𝜖𝑆
+ 
+𝑆 = {𝑡:  0 < 𝛼𝑡 − 𝛼𝑡
+∗ < 𝑐} 
+If so, we will have: 
+𝑍𝑡̂ = ∑ (𝛼̂𝑘 − 𝛼̂𝑘
+∗)
+𝑇
+𝑘=𝑝+1
+𝑍𝑘
+⃗⃗⃗⃗⃗  𝑇𝑍𝑡
+⃗⃗⃗⃗ + Ŵ0 
+Now, if we put 𝑍𝑡 = 𝑊
+⃗⃗⃗  𝑇𝜑(𝑍 𝑡) + 𝑊0 where φ: 𝑅𝑝 → 𝑅𝑛, we will similarly have:  
+𝑍𝑡̂ = ∑ (𝛼𝑘 − 𝛼𝑘
+∗)
+𝑇
+𝑘=𝑝+1
+ 𝜑 𝑇 (𝑍⃗⃗ 𝑘)𝜑 (𝑍⃗⃗ 𝑡) = ∑ (𝛼𝑘 − 𝛼𝑘
+∗)
+𝑇
+𝑘=𝑝+1
+ 𝐾 (𝑍⃗⃗ 𝑘,𝑍⃗⃗ 𝑡)                (4) 
+in which k is a kernel function. Instead of model (4), we consider the following model:  
+𝑍𝑡 = ∑ 𝑊𝑗 𝐾ℎ(𝑍 𝑗, 𝑍 𝑡) + 𝑊0               
+𝑇
+𝑗=𝑝+1
+ 
+This model has many characteristics, including:  
+• The target equation is derived from the SVM method.  
+• Instead of using errors 𝜀𝑡 , 𝜀𝑡
+∗, the loss function 𝜌 is used to estimate coefficients. 
+• The kernel function K h is used in dual equations in the model. 
+• There are fewer time twists when programming to obtain the values of the coefficients of 
+the above equation. 
+
+ 
+• The smoothing constant can be calculated using the trial-and-error method or the 
+generalized Wasserman cross-validation criterion. 
+• The optimal value of the smoothing constant h in the loss function makes it possible to 
+consider many outlier data and alterations in linear or non-linear 𝑍𝑡 modes. 
+All 𝑍𝑝, … , 𝑍𝑝+1 observations via function ∑
+𝑊𝑗 𝐾ℎ(𝑍 𝑗, 𝑍 𝑡)
+𝑇
+𝑗=𝑝+1
+, which is the weighted sum of the 
+neighborhood to the center 𝑍𝑡 and radius h, are used to estimate 𝑍𝑡. Besides, h is the smoothing 
+constant obtained by the GCV criterion. Moreover, we use the following objective equation to 
+obtain 𝑊0 and 𝑊
+⃗⃗⃗ : 
+(𝑊̂ , 𝑊0) = min𝑊
+⃗⃗⃗ ,𝑊0
+𝑊
+⃗⃗⃗  𝑇𝑊
+2
++ 𝑐 ∑ 𝜌(𝑍𝑡 − ∑ 𝑊𝑗 𝑘 (𝑍 𝑗, 𝑍 𝑡)
+𝑇
+𝑗=𝑝+1
+)
+𝑇
+𝑡=𝑝+1
+ 
+, where 𝜌 is the loss function. 
+The mean least squared error (𝜌𝐿𝑆), which measures the distance between predicted and actual 
+values, is one of the most well-known and widely used loss functions in the analysis and modeling 
+of time-dependent data. Occasionally, the data is arranged such that the predicted values tend 
+towards the outlier data and are so-called “crooked”. In this case, the loss function mentioned 
+above leads to problems when estimating parameters and predicting the response variable. Huber’s 
+loss function (𝜌𝐻) is utilized to solve such a problem. In most modeling problems involving real-
+world data, we must determine whether predictions are certain or uncertain. Knowledge of the 
+range of variations for predicted values is crucial for solving real-world issues. Using the quantile 
+loss function (𝜌𝑄) has the property of providing an interval for the response variable rather than a 
+specific value as a prediction. The forms of the aforementioned functions are summarized in Table 
+1. 
+
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+4-Evaluation 
+ The dataset contains 140,000 tweets related to the Russia-Ukraine war, collected during the first 
+month of the war (March 2022) using the keyword Ukraine. The Python programming language 
+was utilized for all implementations. Geotags were used to determine the tweets posted from each 
+country, and sentiment analysis was performed on each country’s tweets. At this stage, retweets 
+were deleted to avoid duplication. Figure 2 depicts the frequency of tweets with positive and 
+negative sentiments during the initial four weeks of the war per country. In all countries, the 
+number of negative tweets exceeds the number of positive tweets, indicating that users have a 
+negative view of the conflict in Ukraine. In addition, the proportion of countries that shared 
+sufficient tweets about the conflict is larger in Europe, with European states accounting for 50 
+percent of the countries. Asian nations came in second place and made up nearly 30% of the 
+nations. Switzerland, Singapore, Portugal, Ukraine, Spain, Italy, Austria, and Turkey have a higher 
+negative to positive ratio of tweets, indicating a more negative attitude towards the events of the 
+Table 1. Loss functions 
+loss function
+ 
+methods
+ 
+𝝆𝑳𝑺(𝒆) = 𝒆𝟐 
+Least-Squares
+ 
+𝝆𝑯(𝒆) = {
+𝟏
+𝟐 𝒆𝟐                  𝒇𝒐𝒓 |𝒆| ≤ 𝒌 
+𝒌|𝒆| − 𝟏
+𝟐 𝒌𝟐    𝒇𝒐𝒓 |𝒆| > 𝒌 
+ 
+Huber
+ 
+𝝆𝑸(𝒆) = 𝒆(𝑸 − 𝑰(𝒆<𝟎)),
+𝟎 ≤ 𝑸 ≤ 𝟏 
+Quantile
+ 
+
+ 
+Russia-Ukraine war. In contrast, the positive to negative tweet ratio is higher in Belgium, 
+Denmark, China, Argentina, the Philippines, and Sweden. This ratio indicates that the citizens’ 
+opinions of these nations were less negative, in the first month of the conflict. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+0
+500
+1000
+1500
+2000
+1
+2
+3
+4
+Number of Tweets
+Week
+USA
+Positive
+Negative
+0
+100
+200
+300
+400
+1
+2
+3
+4
+Number of Tweets
+Week
+Germany
+Positive
+Negative
+0
+5
+10
+15
+20
+1
+2
+3
+4
+Number of Tweets
+Week
+UAE
+Positive
+Negative
+0
+2
+4
+6
+8
+10
+12
+1
+2
+3
+4
+Number of Tweets
+Week
+Argentina
+Positive
+Negative
+0
+10
+20
+30
+40
+50
+60
+1
+2
+3
+4
+Number of Tweets
+Week
+Turkey
+Positive
+Negative
+0
+5
+10
+15
+20
+25
+1
+2
+3
+4
+Number of Tweets
+Week
+Denmark
+Positive
+Negative
+0
+50
+100
+150
+200
+1
+2
+3
+4
+Number of Tweets
+Week
+Canada
+Positive
+Negative
+0
+10
+20
+30
+40
+50
+60
+1
+2
+3
+4
+Number of Tweets
+Week
+Ireland
+Positive
+Negative
+0
+20
+40
+60
+80
+1
+2
+3
+4
+Number of Tweets
+Week
+Belgium
+Positive
+Negative
+0
+50
+100
+150
+200
+1
+2
+3
+4
+Number of Tweets
+Week
+India
+Positive
+Negative
+0
+20
+40
+60
+80
+100
+120
+1
+2
+3
+4
+Number of Tweets
+Week
+France
+Positive
+Negative
+0
+5
+10
+15
+20
+25
+1
+2
+3
+4
+Number of Tweets
+Week
+Brazil
+Positive
+Negative
+0
+10
+20
+30
+40
+50
+1
+2
+3
+4
+Number of Tweets
+Week
+Australia
+Positive
+Negative
+0
+5
+10
+15
+20
+25
+1
+2
+3
+4
+Number of Tweets
+Week
+Czechia
+Positive
+Negative
+0
+50
+100
+150
+1
+2
+3
+4
+Number of Tweets
+Week
+Poland
+Positive
+Negative
+0
+20
+40
+60
+80
+1
+2
+3
+4
+Number of Tweets
+Week
+Spain
+Positive
+Negative
+
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 2. Frequency of positive and negative tweets in the first four months of the war 
+0
+20
+40
+60
+80
+1
+2
+3
+4
+Number of Tweets
+Week
+Switzerland
+Positive
+Negative
+0
+5
+10
+15
+1
+2
+3
+4
+Number of Tweets
+Week
+Indonesia
+Positive
+Negative
+0
+50
+100
+150
+200
+250
+300
+1
+2
+3
+4
+Number of Tweets
+Week
+Ukraine
+Positive
+Negative
+0
+100
+200
+300
+400
+1
+2
+3
+4
+Number of Tweets
+Week
+UK
+Positive
+Negative
+0
+5
+10
+15
+20
+1
+2
+3
+4
+Number of Tweets
+Week
+Finland
+Positive
+Negative
+0
+5
+10
+15
+20
+25
+30
+1
+2
+3
+4
+Number of Tweets
+Week
+Estonia
+Positive
+Negative
+0
+5
+10
+15
+1
+2
+3
+4
+Number of Tweets
+Week
+China
+Positive
+Negative
+0
+10
+20
+30
+40
+50
+60
+1
+2
+3
+4
+Number of Tweets
+Week
+Portugal
+Positive
+Negative
+0
+10
+20
+30
+40
+1
+2
+3
+4
+Number of Tweets
+Week
+Italy
+Positive
+Negative
+0
+5
+10
+15
+20
+25
+1
+2
+3
+4
+Number of Tweets
+Week
+South Africa
+Positive
+Negative
+0
+5
+10
+15
+1
+2
+3
+4
+Number of Tweets
+Week
+Mexico
+Positive
+Negative
+0
+2
+4
+6
+8
+10
+12
+1
+2
+3
+4
+Number of Tweets
+Week
+Russia
+Positive
+Negative
+0
+5
+10
+15
+20
+25
+1
+2
+3
+4
+Number of Tweets
+Week
+Philippines
+Positive
+Negative
+0
+50
+100
+150
+200
+1
+2
+3
+4
+Number of Tweets
+Week
+Netherland
+Positive
+Negative
+0
+20
+40
+60
+80
+1
+2
+3
+4
+Number of Tweets
+Week
+Japan
+Positive
+Negative
+0
+5
+10
+15
+20
+25
+1
+2
+3
+4
+Number of Tweets
+Week
+Singapore
+Positive
+Negative
+0
+5
+10
+15
+1
+2
+3
+4
+Number of Tweets
+Week
+Sweden
+Positive
+Negative
+0
+20
+40
+60
+80
+100
+1
+2
+3
+4
+Number of Tweets
+Week
+Austria
+Positive
+Negative
+
+ 
+1-4- Clustering time series of countries 
+The proposed clustering model is applied to the 34 countries with the highest number of tweets. 
+These 34 nations are ultimately classified into five clusters. Figure 3 illustrates the clustering of 
+the countries. The greatest number of countries are located in clusters 1 and 5. Cluster 1 consists 
+of the United States, Canada, England, India, and the majority of Western European nations. 
+Except for India, all other states have fully backed Ukraine during the war. Cluster 2 only contains 
+Ukraine. Given that this nation has been attacked, it is understandable that their attitude regarding 
+this war differs from other nations.  
+Although the Japanese government sanctioned Russia, the model obtained for Japan is surprisingly 
+unlike any other country. Cluster 4 is composed of Australia, Italy, and Spain. Ultimately, Cluster 
+5 comprises the United Arab Emirates, Estonia, Russia, Singapore, Finland, Portugal, Brazil, 
+Argentina, Mexico, China, Denmark, Belgium, the Czech Republic, Poland, South Africa, 
+Switzerland, the Philippines, and Sweden, all of which held relatively similar opinions  with Russia. 
+Most of these nations are in Asia, southern and central Africa, eastern Europe, and Scandinavia. 
+When interpreting the clustering, it is important to consider the following two points, which also 
+reflect the limitations of the work: 
+• The clustering of countries was performed according to the users’ tweets in the first month of 
+the war. Naturally, the type of clustering may change as the war continued and other events 
+occurred. 
+• The dynamic trend of the time series of positive and negative tweets was clustered. As a result, 
+the time parameter and the importance of the topic over time have been somehow involved.  
+ 
+
+ 
+Figure 3. Clustering of 34 countries based on neighborhood average values 
+14
+34
+30
+23
+25
+22
+28
+21
+18
+19
+27
+17
+31
+20
+29
+26
+33
+32
+24
+16
+12
+10
+8
+9
+13
+11
+5
+3
+4
+7
+15
+2
+6
+1
+39.15
+59.43
+79.72
+100.00
+Observations
+Similarity
+Dendrogram
+Average Linkage; Euclidean Distance
+ 
+8
+7
+6
+5
+4
+3
+2
+1
+1.5
+1.0
+0.5
+0.0
+-0.5
+-1.0
+-1.5
+Index
+UKRAINE
+ 
+8
+7
+6
+5
+4
+3
+2
+1
+1.5
+1.0
+0.5
+0.0
+-0.5
+-1.0
+-1.5
+Index
+Data
+NETHERLAND
+USA
+INDIA
+GERMANY
+AUSTRIA
+FRANCE
+CANADA
+TURKEY
+IRELAND
+UK
+Variable
+ 
+8
+7
+6
+5
+4
+3
+2
+1
+1.5
+1.0
+0.5
+0.0
+-0.5
+-1.0
+-1.5
+Index
+Data
+AUSTRALIA
+SPAIN
+ITALY
+Variable
+ 
+8
+7
+6
+5
+4
+3
+2
+1
+1.5
+1.0
+0.5
+0.0
+-0.5
+-1.0
+Index
+JAPAN
+ 
+I 
+Country
+ 
+I
+ 
+Country
+ 
+1 
+USA
+ 
+18 
+DENMARK
+ 
+2 
+GERMANY
+ 
+19 
+BELGIUM
+ 
+3 
+TURCKY
+ 
+20 
+BRAZIL
+ 
+4 
+CANADA
+ 
+21 
+CZECHIA
+ 
+5 
+IRELAND
+ 
+22 
+POLAND
+ 
+6 
+INDIA
+ 
+23 
+INDONESIA
+ 
+7 
+FRANCE
+ 
+24 
+ESTONIA
+ 
+8 
+AUSTRALIA
+ 
+25 
+SWITZERLAND
+ 
+9 
+UKRAIN
+ 
+26 
+FINLAND
+ 
+10 
+SPAIN
+ 
+27 
+CHINA
+ 
+11 
+UK
+ 
+28 
+SOUTH AFRICA
+ 
+12 
+ITALY
+ 
+29 
+PORTUGAL
+ 
+13 
+NETHERLAND
+ 
+30 
+PHILIPPINES
+ 
+14 
+JAPAN
+ 
+31 
+MEXICO
+ 
+15 
+AUSTRIA
+ 
+32 
+RUSSIA
+ 
+16 
+UAE
+ 
+33 
+SINGAPORE
+ 
+17 
+ARGENTINA
+ 
+34 
+SWEDEN
+ 
+8
+7
+6
+5
+4
+3
+2
+1
+3
+2
+1
+0
+-1
+-2
+Index
+Data
+CHINA
+DENMARK
+BELGIUM
+CZECHIA
+POLAND
+SOUTH AFRICA
+Switzerland
+PHILIPPINES
+SWEDEN
+UAE
+ESTONIA
+RUSSIA
+SINGAPORE
+FINLAND
+PORTUGAL
+BRAZIL
+Argentina
+MEXICO
+Variable
+ 
+Cluster 1 
+Cluster 2 
+Cluster 3 
+Cluster 4 
+Cluster 5 
+
+ 
+ 
+5.  Conclusion 
+This article examines user sentiments regarding the Russia-Ukraine conflict during the first month 
+of the conflict. For this purpose, 140,000 related English tweets were collected through the 
+keyword Ukraine in March 2022. Then, the location of each tweet was determined based on its 
+geotag. RoBERTa, a language-based model with superior performance to similar models, was used 
+to analyze the sentiments of tweets. Afterward, the weekly time series of frequencies of positive 
+and negative tweets from countries with a sufficient number of tweets were analyzed. The analysis 
+of these time series yielded significant insight into users’ perspectives regarding the Russia-
+Ukraine conflict. Because the number of negative tweets was greater than the number of positive 
+tweets in all countries, it is safe to conclude that most users have a negative view of this war.  
+Furthermore, the trend of positive and negative tweets of the countries was clustered, which 
+accounted for the topic’s importance over time. The findings of this study allow us to draw 
+attention to the similarity of views held by users in the United States, Canada, and Western Europe 
+during the first month of the war, as well as the similarity of opinions held by users based 
+in Eastern Europe, South America, Asia, and Scandinavia. In addition, users’ views in Ukraine 
+and Japan were distinct and unlike those of any other nation. 
+One of the limitations of this article is that it does not consider tweets in languages other than 
+English, as users in many countries do not speak English fluently and publish tweets in other 
+languages. Additionally, we have only used Twitter for polling purposes, and other social networks 
+were omitted. Lastly, only tweets containing the keyword Ukraine were collected; therefore, all 
+relevant tweets might not have been captured. 
+In future work, the coefficients derived from the proposed model for each country can be clustered 
+based on various methods, and the results can be compared. Besides, the effects of this war on 
+
+ 
+international relations, as well as economic, political, and other issues, can be analyzed from the 
+perspective of users of social networks. 
+Reference 
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+ 
+
diff --git a/RNAyT4oBgHgl3EQft_lA/content/tmp_files/load_file.txt b/RNAyT4oBgHgl3EQft_lA/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf,len=1117
+page_content='In submission Russia-Ukraine war: Modeling and Clustering the Sentiments Trends of Various Countries Hamed Vahdat-Nejad*, Mohammad Ghasem Akbari**, Fatemeh Salmani*, Faezeh Azizi*, Hamid-Reza Nili-Sani** * PerLab, Faculty of Electrical and Computer Engineering, University of Birjand, Birjand, Iran **Department of Statistics, University of Birjand, Birjand, Iran Email: vahdatnejad@birjand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='ir, g_z_akbari@birjand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='ir, salmani_fatemeh98@birjand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='ir, faezeh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='azizi1995@birjand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='ir, hnilisani@birjand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='ir Abstract With Twitter’s growth and popularity, a huge number of views are shared by users on various topics, making this platform a valuable information source on various political, social, and economic issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' This paper investigates English tweets on the Russia-Ukraine war to analyze trends reflecting users’ opinions and sentiments regarding the conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The tweets’ positive and negative sentiments are analyzed using a BERT-based model, and the time series associated with the frequency of positive and negative tweets for various countries is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Then, we propose a method based on the neighborhood average for modeling and clustering the time series of countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The clustering results provide valuable insight into public opinion regarding this conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='  Among other things, we can mention the similar thoughts of users from the United States, Canada, the United Kingdom, and most Western European countries versus the shared views of Eastern European, Scandinavian, Asian, and South American nations toward the conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Keywords: Social network mining, Russia-Ukraine war, Sentiment analysis, Clustering 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Introduction  Social networks are currently one of the most significant platforms for population information extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Users of social networks share innumerable posts in many domains, including social, economic, and political ones, that reflect their perspectives and sentiments regarding global events and incidents (Crannell, Clark, Jones, James, & Moore, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' As a social network, Twitter is a key source of users’ sentiments and perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Users’ views can be analyzed using natural language processing and data retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Natural language processing (NLP) is “focused on the design and analysis of computational algorithms and representations for processing natural human language” (Eisenstein, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' It aims to provide “new computational capabilities around human language” (Eisenstein, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Indeed, we can extract information from a mass of texts using NLP techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Sentiment analysis is a promising technique used in natural language processing to extract emotional states and subjective information from a text (Zad, Heidari, Jones, & Uzuner, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' It allows for the extraction of sentiments from users’ tweets regarding a variety of events and issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The outbreak of COVID-19, for instance, was a globally important event about which many users have expressed their opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='  Alongside this, there have been sentiment analyses of tweets related to COVID-19 (Vahdat-Nejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', 2022), the economy during COVID-19 (Salmani, Vahdat- Nejad, & Hajiabadi, 2021), and education during the epidemic of COVID-19 (Jamalian, Vahdat- Nejad, & Hajiabadi, 2022), among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Other events, including wars and conflicts, have also prompted users to post and analyze tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' For instance, the sentiment analysis of Turkish and English tweets about Syrian refugees has revealed a predominantly positive tone in Turkish tweets but a negative tone in English tweets (Öztürk & Ayvaz, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Similarly, an analysis of opinions regarding recent events in Afghanistan has revealed that tweets containing negative terms such as terrorist, attack, destroy, and violence have dominated  (Aggarwal, Khan, & Kakkar, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Furthermore, the ratio of negative to positive tweets was identical across all eight investigated countries, indicating that these nations shared similar sentiments regarding the recent events in Afghanistan (Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Likewise, the sentiment analysis of tweets about the Syrian chemical war reveals that users negatively view the Syrian chemical attack (Bashir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The Russia-Ukraine war has probably been the most prominent event of 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Recently, the main topics discussed on the Chinese Weibo social network regarding this conflict have been extracted (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' To our best knowledge, no study has analyzed tweets regarding this war to investigate and categorize the attitude of various countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' To this end, this research collects and analyzes tweets on this conflict to cluster various countries according to their population’s attitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' For this purpose, 140,000 tweets posted in the first month of the war (March 2022) were collected using the keyword Ukraine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' In order to analyze the sentiments of tweets, the language- based RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', 2019) model is employed, which has superior accuracy and performance compared to similar models (Briskilal & Subalalitha, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Following this, the time series of the frequency of positive and negative sentiments are calculated and modeled for several countries that have a sufficient number of English tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='  Next, the time series models of the countries are clustered based on the neighborhood average method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Results indicate that countries such as the United States, England, Canada, and those of Western Europe are in one cluster, while Eastern Europe, Scandinavian, South American, and Asian countries are in another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' In addition, Ukraine was placed alone in one cluster, indicating a divergent trend in public opinion between this country and the rest of the world in the first month of the conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The main contributions of the paper are as follows:  To the best of our knowledge, it is the first work investigating and clustering the attitudes of nations on Twitter regarding the Ukraine war.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='  It proposes a new method for modeling and clustering the time series of countries’ sentiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The article’s remaining sections are as follows: Section 2 reviews previous research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Section 3 describes the proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Implementation and results are described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Lastly, the fifth section summarizes the conclusions and limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 2  Related work As a result of their growth and popularity, social networks have become an important and valuable information source on users’ attitudes in various fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Techniques such as big data and natural language processing are required to analyze this unstructured textual information of social network users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Twitter is one of the most popular social networks on which users share their thoughts and feelings about current events and issues (Mathur, Kubde, & Vaidya, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Numerous analyses of user opinions have been conducted in a variety of fields, including politics (Ansari, Aziz, Siddiqui, Mehra, & Singh, 2020), stock return forecasting (Sul, Dennis, & Yuan, 2017), sustainable energy (Corbett & Savarimuthu, 2022), and tourism (Abbasi  Moud, Vahdat  Nejad, & Mansoor, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Asani, Vahdatnejad, Hosseinabadi, & Sadri, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' For instance, with the spread of COVID  19 over the past few years, researchers have analyzed the sentiments of COVID  19  related tweets (Salmani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Vahdat  Nejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', 2022) and extracted the primary topics discussed (Azizi, Vahdat  Nejad, Hajiabadi, & Khosravi, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The incidence of wars and conflicts is among the occasions that prompt users to post their opinions on social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' In this regard, users’ views in various countries regarding recent events in  Afghanistan were analyzed from August to November 2021 (Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' For this purpose, the eight nations with the most tweets were investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' One of the findings of this study was the high number of tweets containing negative terms such as terrorist, attack, destruction, and violence, as well as the fact that all eight investigated countries shared a similar perspective regarding the recent events in Afghanistan (Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Likewise, a sentiment analysis (positive, neutral, negative, very positive, and very negative) was conducted on English and Turkish tweets related to the Syrian civil war in April 2017 (Öztürk & Ayvaz, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The R- Sentiment package was used to analyze the sentiments of tweets written in English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Due to the substantial influx of Syrian refugees into Turkey, Turkish tweets were also analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' In the absence of a model for sentiment analysis of tweets written in Turkish, a sentiment lexicon of Turkish vocabulary was presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Accordingly, there was a large number of positive tweets in Turkish and a large number of neutral, negative, and extremely negative tweets in English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Additionally, English tweets were more political and focused more on the legality of immigrants, whereas the Turkish-language tweets focused more on the war’s details, as the conflict in proximity to the Turkish border was of interest to the Turkish community (Öztürk & Ayvaz, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='  Similar sentiment analyses have been conducted for the ten countries with the highest volume of tweets regarding Syria’s chemical warfare (Bashir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Overall, a greater proportion of tweets were negative, indicating that users had a negative view regarding the Syrian chemical attack (Bashir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The Russia-Ukraine war is one of the most important events of 2022, about which countless tweets have been published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Recently, Chinese Weibo texts have been investigated to extract the main topics discussed by Chinese users about this conflict(Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' We aim to continue that research by modeling the sentiment trends of various countries and clustering them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' To the best of  our knowledge, no study has been conducted to investigate and classify the sentiments trends of users from different countries regarding this war.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 3- Proposed Method Numerous tweets have been posted regarding the Russia-Ukraine war, which is one of the most significant events in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' To the best of our knowledge, no research has been conducted to analyze the sentiments of tweets related to this war and to examine and classify the sentiments of users from various countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' This research aims to collect and analyze tweets pertaining to the Ukrainian dispute in order to classify the sentiment trends of users from varying countries over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 3-1- Tweet collection and processing In March 2022, the war between Ukraine and Russia was the prominent topic of tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' First, tweets related to the war in Ukraine are collected and separated by location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Afterward, positive and negative tweets are identified using sentiment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The final step concerns clustering the time series of tweet sentiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' We collected 140,000 English tweets associated with the Russia-Ukraine war using the keyword Ukraine during March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The geotag is then applied to separate tweets from different countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Thus, tweets with an unknown location are deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The process of determining whether a tweet is positive, negative, or neutral is known as tweet sentiment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='  A large share of sentiment analysis research is conducted on social network posts, such as those on Twitter, since these posts contain users’ views and feelings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' We utilize sentiment analysis to determine the trends of public sentiment regarding this war over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' To calculate the sentiment scores of tweets, we employed the promising language-based RoBERTa  (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', 2019) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Recommended by Google, RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', 2019) is the optimized version of the language-based BERT (Devlin, Chang, Lee, & Toutanova, 2019) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Identifying positive and negative tweets gives us insight into the direction of public opinion regarding the Ukraine conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The number of positive and negative tweets for each country (for which there are sufficient tweets) was then calculated weekly to yield a time series for each country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The analysis and clustering of time series pertaining to various nations unveiled intriguing information regarding public opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=" 3-2- Clustering countries' sentiments trends Various methods have been presented for modeling and clustering users' opinions in social networks (Asadolahi, Akbari, Hesamian, & Arefi, 2021;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Hesamian & M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Akbari, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Hesamian & Akbari, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Hesamian & M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Akbari, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Hesamian & Akbari, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' This research draws on the neighborhood average method to cluster countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Because the number of tweets in different countries varies, with the majority of these differences being substantial, the weekly data for each country is first normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Subsequently, the weekly frequency of the number of positive and negative tweets are computed for each country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Therefore, four positive and four negative frequencies corresponding to the first four weeks of the war’s onset are calculated for each country and modeled as a vector with eight features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='  The neighborhood average method is then used to cluster the coefficients obtained from the proposed model for the nations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The distance between countries is calculated using the Euclidean meter, and the countries with a smaller distance are put in one cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The proposed model, which uses the support vector method (SVM), is elaborated on below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' First, we consider the following model for time data {𝑍𝑡:𝑡 =  1,2,…, 𝑇   (T = 8)}:  𝑍𝑡 = ∑ 𝑊𝑗𝑍𝑡−𝑗 + 𝑊0 = 𝑊 ⃗⃗⃗ 𝑇𝑍𝑡 ⃗⃗⃗ + 𝑊0               𝑍 𝑡 = ( 𝑍𝑡−1, … , 𝑍𝑡−𝑝) 𝑝 𝑗=1  To estimate the vector 𝑊 ⃗⃗⃗  and 𝑊0, we use the SVM as per figure1 and following equations:  𝑍𝑡 =  ∑ 𝑊𝑗𝑍𝑡−𝑗 𝑝 𝑗=1 + 𝑊0  ± 𝑒  𝑍𝑡 =  ∑ 𝑊𝑗𝑍𝑡−𝑗 𝑝 𝑗=1 + 𝑊0  ± 𝑒  𝜀𝑡 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Using SVM for estimation min𝑊 ⃗⃗⃗ ,𝑊0,𝜀⃗ ,𝜀⃗ ∗ 𝐽(𝑊, ⃗⃗⃗⃗⃗ 𝑊0, 𝜀 , 𝜀 ∗) = 1 2 𝑊 ⃗⃗⃗  𝑇𝑊 ⃗⃗⃗ + 𝑐 ∑ (𝜀𝑡 + 𝜀𝑡 ∗)             (1) 𝑇 𝑡=𝑝+1  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='T 𝑍𝑡 − 𝑊 ⃗⃗⃗ 𝑇𝑍𝑡 ⃗⃗⃗ − 𝑊0 ≤ 𝑒 + 𝜀𝑡                     𝑡 = 𝑝 + 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' … ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝑇 𝑊 ⃗⃗⃗ 𝑇𝑍𝑡 ⃗⃗⃗ + 𝑊0 − 𝑍𝑡 ≤ 𝑒 + 𝜀𝑡 ∗ 𝑡 = 𝑝 + 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' … ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝑇 𝜀𝑡 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝜀𝑡 ∗  ≥ 0 Support vectors  𝜀𝑡  Lagrange’s equation for the above optimization problem with coefficients 𝛼𝑡 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝛼𝑡 ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝜇𝑡 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝜇𝑡 ∗ is as follows: 𝐿(𝑊,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' ⃗⃗⃗⃗⃗ 𝑊0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝜀  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝜀 ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝛼  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝛼 ∗𝜇 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝜇 ∗) =  𝐽(𝑊,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' ⃗⃗⃗⃗⃗ 𝑊0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝜀 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝜀 ∗) − ∑ 𝛼𝑡 (𝑒 + 𝜀  − 𝑍𝑡 + 𝑊 ⃗⃗⃗ 𝑇𝑍𝑡 ⃗⃗⃗ + 𝑊0) 𝑇 𝑡=𝑝+1 − ∑ 𝛼𝑡 ∗ (𝑒 + 𝜀 ∗ + 𝑍𝑡 − 𝑊 ⃗⃗⃗ 𝑇𝑍𝑡 ⃗⃗⃗ − 𝑊0) − ∑  (𝜇𝑡𝜀𝑡 + 𝜇𝑡 ∗𝜀𝑡 ∗) 𝑇 𝑡=𝑝+1 𝑇 𝑡=𝑝+1 (2) Where 𝐿 is used in Eq (2) to reach the optimal value 𝐽 in Eq (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' max(𝛼 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝛼 ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝜇 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝜇 ∗)      min(𝑊,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' ⃗⃗⃗⃗ 𝑊0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝜀 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝜀 ∗)      𝐿(𝑊,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' ⃗⃗⃗⃗ 𝑊0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝜀  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝜀 ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝛼⃗  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝛼⃗ ∗𝜇 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝜇⃗ ∗) We have: 𝜕𝐿 𝜕𝑊 ⃗⃗⃗ = 0  → 𝑊 ⃗⃗⃗ = ∑ (𝛼𝑡 − 𝛼𝑡 ∗) 𝑇 𝑡=𝑝+1 𝑍𝑡 𝜕𝐿 𝜕𝑊0 = 0  → ∑ (𝛼𝑡 − 𝛼𝑡 ∗) 𝑇 𝑡=𝑝+1 = 0                                                         (3) 𝜕𝐿 𝜕𝜀𝑡 = 0  →   𝑐 − 𝛼𝑡−𝜇𝑡 = 0           𝜕𝐿 𝜕𝜀𝑡 ∗ = 0   → 𝑐 − 𝛼𝑡 ∗ − 𝜇𝑡 ∗ = 0 By embedding Eq (3) in (2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' the dual (1) is yielded as follows:  m𝑎𝑥𝛼⃗⃗ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='𝛼⃗⃗ ∗ 𝐽𝐷(𝛼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝛼∗) = − 1 2 𝑊 ⃗⃗⃗  𝑇𝑊 ⃗⃗⃗ − 𝑒 ∑ (𝛼𝑡 + 𝛼𝑡 ∗) 𝑇 𝑡=𝑝+1 + ∑ 𝑍𝑡(𝛼𝑡 − 𝛼𝑡 ∗) 𝑇 𝑡=𝑝+1 = ∑ ∑ (𝛼𝑡 − 𝛼𝑡 ∗)(𝛼𝑘 − 𝛼𝑘 ∗) 𝑇 𝑘=𝑝+1  𝑇 𝑡=𝑝+1 𝑍𝑡 ⃗⃗⃗ 𝑇𝑍𝑘 ⃗⃗⃗⃗ − 𝑒   ∑ (𝛼𝑡 + 𝛼𝑡 ∗) + 𝑇 𝑡=𝑝+1 ∑ 𝑍𝑡(𝛼𝑡 − 𝛼𝑡 ∗) 𝑇 𝑡=𝑝+1  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='T ∑ (𝛼𝑡 − 𝛼𝑡 ∗) 𝑇 𝑡=𝑝+1 = 0          0 ≤ 𝛼𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝛼𝑡 ∗  ≤   𝑐 The KKT method is used to obtain 𝑊0 as follows: 𝛼𝑡 (𝑒 + 𝜀𝑡 − 𝑍𝑡 + 𝑊 ⃗⃗⃗⃗  𝑇𝑍𝑡 ⃗⃗⃗⃗ + 𝑊0) = 0 𝛼𝑡 ∗ (𝑒 + 𝜀𝑡 ∗ + 𝑍𝑡 − 𝑊 ⃗⃗⃗⃗  𝑇𝑍𝑡 ⃗⃗⃗⃗ − 𝑊0) = 0 As a result,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' with a few simple calculations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' we have: 𝑊0 = 𝑍𝑡 − 𝑊 ⃗⃗⃗⃗  𝑇𝑍𝑡 ⃗⃗⃗⃗ − 𝑒                   𝛼𝑡𝜖[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝑐]           𝜀𝑡 = 0 𝑊0 = 𝑍𝑡 − 𝑊 ⃗⃗⃗⃗  𝑇𝑍𝑡 ⃗⃗⃗⃗ + 𝑒                   𝛼𝑡 ∗𝜖[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' 𝑐]           𝜀𝑡 ∗ = 0  Therefore: 𝑊0 ̂ = 1 |𝑆| ∑ (𝑍𝑡 − 𝑊 ⃗⃗⃗⃗  𝑇𝑍𝑡 ⃗⃗⃗⃗ − 𝑆𝑖𝑔𝑛(𝛼𝑡 − 𝛼𝑡 ∗)𝑒) 𝑡𝜖𝑆  𝑆 = {𝑡:  0 < 𝛼𝑡 − 𝛼𝑡 ∗ < 𝑐} If so,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' we will have: 𝑍𝑡̂ = ∑ (𝛼̂𝑘 − 𝛼̂𝑘 ∗) 𝑇 𝑘=𝑝+1 𝑍𝑘 ⃗⃗⃗⃗⃗  𝑇𝑍𝑡 ⃗⃗⃗⃗ + Ŵ0 Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' if we put 𝑍𝑡 = 𝑊 ⃗⃗⃗  𝑇𝜑(𝑍 𝑡) + 𝑊0 where φ: 𝑅𝑝 → 𝑅𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' we will similarly have: 𝑍𝑡̂ = ∑ (𝛼𝑘 − 𝛼𝑘 ∗) 𝑇 𝑘=𝑝+1 𝜑 𝑇 (𝑍⃗⃗ 𝑘)𝜑 (𝑍⃗⃗ 𝑡) = ∑ (𝛼𝑘 − 𝛼𝑘 ∗) 𝑇 𝑘=𝑝+1 𝐾 (𝑍⃗⃗ 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='𝑍⃗⃗ 𝑡)                (4) in which k is a kernel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Instead of model (4), we consider the following model: 𝑍𝑡 = ∑ 𝑊𝑗 𝐾ℎ(𝑍 𝑗, 𝑍 𝑡) + 𝑊0 𝑇 𝑗=𝑝+1  This model has many characteristics, including: • The target equation is derived from the SVM method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' • Instead of using errors 𝜀𝑡 , 𝜀𝑡 ∗, the loss function 𝜌 is used to estimate coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' • The kernel function K h is used in dual equations in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' • There are fewer time twists when programming to obtain the values of the coefficients of the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='  The smoothing constant can be calculated using the trial  and  error method or the generalized Wasserman cross  validation criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='  The optimal value of the smoothing constant h in the loss function makes it possible to consider many outlier data and alterations in linear or non  linear 𝑍𝑡 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' All 𝑍𝑝, … , 𝑍𝑝+1 observations via function ∑ 𝑊𝑗 𝐾ℎ(𝑍 𝑗, 𝑍 𝑡) 𝑇 𝑗=𝑝+1 , which is the weighted sum of the neighborhood to the center 𝑍𝑡 and radius h, are used to estimate 𝑍𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Besides, h is the smoothing constant obtained by the GCV criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Moreover, we use the following objective equation to obtain 𝑊0 and 𝑊 ⃗⃗⃗ : (𝑊̂ , 𝑊0) = min𝑊 ⃗⃗⃗ ,𝑊0 𝑊 ⃗⃗⃗  𝑇𝑊 2 + 𝑐 ∑ 𝜌(𝑍𝑡 − ∑ 𝑊𝑗 𝑘 (𝑍 𝑗, 𝑍 𝑡) 𝑇 𝑗=𝑝+1 ) 𝑇 𝑡=𝑝+1  , where 𝜌 is the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The mean least squared error (𝜌𝐿𝑆), which measures the distance between predicted and actual values, is one of the most well-known and widely used loss functions in the analysis and modeling of time-dependent data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Occasionally, the data is arranged such that the predicted values tend towards the outlier data and are so-called “crooked”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' In this case, the loss function mentioned above leads to problems when estimating parameters and predicting the response variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Huber’s loss function (𝜌𝐻) is utilized to solve such a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' In most modeling problems involving real- world data, we must determine whether predictions are certain or uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Knowledge of the range of variations for predicted values is crucial for solving real-world issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Using the quantile loss function (𝜌𝑄) has the property of providing an interval for the response variable rather than a specific value as a prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The forms of the aforementioned functions are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='  4-Evaluation The dataset contains 140,000 tweets related to the Russia-Ukraine war, collected during the first month of the war (March 2022) using the keyword Ukraine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The Python programming language was utilized for all implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Geotags were used to determine the tweets posted from each country, and sentiment analysis was performed on each country’s tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' At this stage, retweets were deleted to avoid duplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Figure 2 depicts the frequency of tweets with positive and negative sentiments during the initial four weeks of the war per country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' In all countries, the number of negative tweets exceeds the number of positive tweets, indicating that users have a negative view of the conflict in Ukraine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' In addition, the proportion of countries that shared sufficient tweets about the conflict is larger in Europe, with European states accounting for 50 percent of the countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Asian nations came in second place and made up nearly 30% of the nations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Switzerland, Singapore, Portugal, Ukraine, Spain, Italy, Austria, and Turkey have a higher negative to positive ratio of tweets, indicating a more negative attitude towards the events of the Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Loss functions loss function  methods  𝝆𝑳𝑺(𝒆) = 𝒆𝟐  Least Squares  𝝆𝑯(𝒆) = { 𝟏 𝟐 𝒆𝟐                  𝒇𝒐𝒓 |𝒆| ≤ 𝒌 𝒌|𝒆| − 𝟏 𝟐 𝒌𝟐    𝒇𝒐𝒓 |𝒆| > 𝒌  Huber  𝝆𝑸(𝒆) = 𝒆(𝑸 − 𝑰(𝒆<𝟎)), 𝟎 ≤ 𝑸 ≤ 𝟏 Quantile  Russia-Ukraine war.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' In contrast, the positive to negative tweet ratio is higher in Belgium, Denmark, China, Argentina, the Philippines, and Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' This ratio indicates that the citizens’ opinions of these nations were less negative, in the first month of the conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content='1-4- Clustering time series of countries The proposed clustering model is applied to the 34 countries with the highest number of tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' These 34 nations are ultimately classified into five clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Figure 3 illustrates the clustering of the countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The greatest number of countries are located in clusters 1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Cluster 1 consists of the United States, Canada, England, India, and the majority of Western European nations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Except for India, all other states have fully backed Ukraine during the war.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Cluster 2 only contains Ukraine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Given that this nation has been attacked, it is understandable that their attitude regarding this war differs from other nations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Although the Japanese government sanctioned Russia, the model obtained for Japan is surprisingly unlike any other country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Cluster 4 is composed of Australia, Italy, and Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Ultimately, Cluster 5 comprises the United Arab Emirates, Estonia, Russia, Singapore, Finland, Portugal, Brazil, Argentina, Mexico, China, Denmark, Belgium, the Czech Republic, Poland, South Africa, Switzerland, the Philippines, and Sweden, all of which held relatively similar opinions  with Russia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Most of these nations are in Asia, southern and central Africa, eastern Europe, and Scandinavia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' When interpreting the clustering, it is important to consider the following two points, which also reflect the limitations of the work: • The clustering of countries was performed according to the users’ tweets in the first month of the war.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='  Naturally, the type of clustering may change as the war continued and other events occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' • The dynamic trend of the time series of positive and negative tweets was clustered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' As a result, the time parameter and the importance of the topic over time have been somehow involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Clustering of 34 countries based on neighborhood average values 14 34 30 23 25 22 28 21 18 19 27 17 31 20 29 26 33 32 24 16 12 10 8 9 13 11 5 3 4 7 15 2 6 1 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='15 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='43 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='72 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='00 Observations Similarity Dendrogram Average Linkage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Euclidean Distance  8  7  6  5  4  3  2  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content='5  Index  UKRAINE  8  7  6  5  4  3  2  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content='5  Index  Data  NETHERLAND  USA  INDIA  GERMANY  AUSTRIA  FRANCE  CANADA  TURKEY  IRELAND  UK  Variable  8  7  6  5  4  3  2  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content='5  Index  Data  AUSTRALIA  SPAIN  ITALY  Variable  8  7  6  5  4  3  2  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content='Index  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='JAPAN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content='Country  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content='DENMARK  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content='BELGIUM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='TURCKY  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='BRAZIL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='CANADA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content='CZECHIA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content='IRELAND  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content='13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='NETHERLAND  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='JAPAN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='31  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='AUSTRIA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='RUSSIA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='UAE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='33  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='SINGAPORE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='17  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='ARGENTINA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='SWEDEN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='0 1 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='Index  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='CHINA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='DENMARK  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='BELGIUM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='CZECHIA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='POLAND  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='SOUTH AFRICA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='Switzerland  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='PHILIPPINES  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='SWEDEN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='UAE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='ESTONIA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='RUSSIA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='SINGAPORE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='FINLAND  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='PORTUGAL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='BRAZIL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='Argentina  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='MEXICO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='Variable  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='Cluster 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='Cluster 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='Cluster 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='Cluster 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='Cluster 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Conclusion This article examines user sentiments regarding the Russia-Ukraine conflict during the first month of the conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' For this purpose, 140,000 related English tweets were collected through the keyword Ukraine in March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Then, the location of each tweet was determined based on its geotag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' RoBERTa, a language-based model with superior performance to similar models, was used to analyze the sentiments of tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Afterward, the weekly time series of frequencies of positive and negative tweets from countries with a sufficient number of tweets were analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The analysis of these time series yielded significant insight into users’ perspectives regarding the Russia- Ukraine conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Because the number of negative tweets was greater than the number of positive tweets in all countries, it is safe to conclude that most users have a negative view of this war.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Furthermore, the trend of positive and negative tweets of the countries was clustered, which accounted for the topic’s importance over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' The findings of this study allow us to draw attention to the similarity of views held by users in the United States, Canada, and Western Europe during the first month of the war, as well as the similarity of opinions held by users based in Eastern Europe, South America, Asia, and Scandinavia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' In addition, users’ views in Ukraine and Japan were distinct and unlike those of any other nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='  One of the limitations of this article is that it does not consider tweets in languages other than English, as users in many countries do not speak English fluently and publish tweets in other languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Additionally, we have only used Twitter for polling purposes, and other social networks were omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Lastly, only tweets containing the keyword Ukraine were collected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' therefore, all relevant tweets might not have been captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' In future work, the coefficients derived from the proposed model for each country can be clustered based on various methods, and the results can be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Besides, the effects of this war on  international relations, as well as economic, political, and other issues, can be analyzed from the perspective of users of social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Reference Abbasi-Moud, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', Vahdat-Nejad, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', & Mansoor, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=" Detecting Tourist's Preferences by Sentiment Analysis in Smart Cities." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Paper presented at the Global Conference on Internet of Things, Dubai, United Arab Emirates, United Arab Emirates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Aggarwal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', Khan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', & Kakkar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Analyzing opinion of different countries for recent events in Afghanistan using text mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Paper presented at the 12th International Conference on Cloud Computing, Data Science & Engineering India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Ansari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', Aziz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', Siddiqui, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', Mehra, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', & Singh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Analysis of political sentiment orientations on twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Paper presented at the Computational Intelligence and Data Science, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Asadolahi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', Akbari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content=', & Arefi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content=' A robust support vector regression with exact predictors and fuzzy responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' International Journal of Approximate Reasoning, 132(1), 206-225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Asani, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content=" Extracting User's Food Preferences by Sentiment Analysis." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Paper presented at the 8th Iranian Joint Congress on Fuzzy and intelligent Systems, Mashhad, Iran, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Azizi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', Vahdat-Nejad, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', Hajiabadi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content=' Paper presented at the International Conference on Computer Engineering and Knowledge, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='  Bashir, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content=' Twitter chirps for Syrian people: Sentiment analysis of tweets related to Syria Chemical Attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content=' Stoyanov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Roberta: A robustly optimized bert pretraining approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' arXiv preprint arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='11692.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Mathur, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', Kubde, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', & Vaidya, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Emotional analysis using twitter data during pandemic situation: COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Paper presented at the Communication and Electronics Systems India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Öztürk, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', & Ayvaz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Sentiment analysis on Twitter: A text mining approach to the Syrian refugee crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Telematics and Informatics, 35, 136-147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Salmani, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', Vahdat-Nejad, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', & Hajiabadi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Analyzing the Impact of COVID-19 on Economy from the Perspective of User’s Reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='  Paper presented at the International Conference on Computer Engineering and Knowledge, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Sul, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', & Yuan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Trading on twitter: Using social media sentiment to predict stock returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Decision Sciences, 48, 454-488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content='  Vahdat-Nejad, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', Salmani, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', Hajiabadi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', Azizi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', Abbasi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', Jamalian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Hajiabadi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Extracting Feelings of People Regarding COVID-19 by Social Network Mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Journal of Information & Knowledge Management, 1, 2240008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Zad, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', Heidari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', Jones, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=', & Uzuner, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' A survey on concept-level sentiment analysis techniques of textual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
+page_content=' Paper presented at the IEEE World AI IoT Congress Virtual conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAyT4oBgHgl3EQft_lA/content/2301.00604v1.pdf'}
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+Triple Graph Grammars for Multi-version Models
+Matthias Barkowsky
+matthias.barkowsky@hpi.de
+Holger Giese
+holger.giese@hpi.de
+January 3, 2023
+Abstract
+Like conventional software projects, projects in model-driven software
+engineering require adequate management of multiple versions of devel-
+opment artifacts, importantly allowing living with temporary inconsisten-
+cies. In the case of model-driven software engineering, employed version-
+ing approaches also have to handle situations where different artifacts,
+that is, different models, are linked via automatic model transformations.
+In this report, we propose a technique for jointly handling the transfor-
+mation of multiple versions of a source model into corresponding versions
+of a target model, which enables the use of a more compact representa-
+tion that may afford improved execution time of both the transformation
+and further analysis operations. Our approach is based on the well-known
+formalism of triple graph grammars and a previously introduced encod-
+ing of model version histories called multi-version models.
+In addition
+to showing the correctness of our approach with respect to the standard
+semantics of triple graph grammars, we conduct an empirical evaluation
+that demonstrates the potential benefit regarding execution time perfor-
+mance.
+1
+Introduction
+In model-driven software development, models are treated as primary develop-
+ment artifacts. Complex projects can involve multiple models, which describe
+the system under development at different levels of abstraction or with respect
+to different system aspects and can be edited independently by a team of de-
+velopers. In this case, consistency of the holistic system description is ensured
+by model transformations that automatically derive concrete models from more
+abstract ones or propagate changes to a model describing one aspect of the
+system to other models concerned with different but overlapping aspects [20].
+Similarly to program code in conventional software development, the evolu-
+tion of models via changes made by different developers requires management
+of the resulting versions of the software description. In particular, version man-
+agement has to support parallel development activities of multiple developers
+working on the same development artifact, where living with inconsistencies of a
+1
+arXiv:2301.00623v1  [cs.SE]  2 Jan 2023
+
+single artifact may temporarily be necessary to avoid loss of information [8]. In
+[2], we have introduced multi-version models as a means of managing multiple
+versions of the same model that also enables monitoring the consistency of the
+individual model versions and potential merge results of versions developed in
+parallel.
+However, with model transformations effectively linking multiple models
+via consistency relationships, considering only the evolution of a single model
+without its context is insufficient for larger model-driven software development
+projects. Thus, a mechanism for establishing consistency of different versions of
+such linked models that simultaneously allows parallel development of multiple
+versions is required.
+Such a mechanism would allow working with more compact representations
+that also enable further analysis operations as described in [2]. In addition, an
+integrated handling of multiple model versions may afford an improved execution
+time performance of the transformation.
+In this report, we propose a first step in the direction of model transforma-
+tions working on multi-version models by adapting the well-known formalism
+of triple graph grammars, which enables the implementation of single-version
+model transformations, to the multi-version case.
+The remainder of the report is structured as follows: In Section 2, we briefly
+reiterate the basic concepts of graphs, graph transformations, triple graph gram-
+mars, and multi-version models, as used in this report. Subsequently, we present
+our approach for deriving transformation rules that work on multi-version mod-
+els from single-version model transformation specifications in the form of triple
+graph grammars in Section 3. In Section 4, we describe how the derived rules
+can be used to realize the joint transformation of all individual model versions
+encoded in a multi-version model and prove the correctness of our technique
+with respect to the semantics of triple graph grammars. Section 5 reports on
+the results of an initial evaluation of the presented solution’s performance re-
+garding execution time, which is based on an application scenario in the software
+development domain. Related work is discussed in Section 6, before Section 7
+concludes the report.
+2
+Preliminaries
+In this section, we give a brief overview of required preliminaries regarding
+graphs and graph transformations, triple graph grammars and multi-version
+models.
+2.1
+Graphs and Graph Transformations
+We briefly reiterate the concepts of graphs, graph morphisms and graph transfor-
+mations and their typed analogs as defined in [6] and required in the remainder
+of the report.
+2
+
+TypeAccess
+ClassDecl
+FieldDecl
+type
+access
+declaration
+c1:ClassDecl
+f1:FieldDecl
+c2:ClassDecl
+t1:TypeAccess
+G
+TG
+declaration
+type
+access
+Figure 1: example graph and type graph
+A graph G = (V G, EG, sG, tG) consists of a set of nodes V G, a set of edges
+EG and two functions sG : EG → V G and tG : EG → V G assigning each edge
+its source and target, respectively. A graph morphism m : G → H is given by a
+pair of functions mV : V G → V H and mE : EG → EH that map elements from
+G to elements from H such that sH ◦ mE = mV ◦ sG and tH ◦ mE = mV ◦ tG.
+We also call mV the vertex morphism and mE the edge morphism.
+A graph G can be typed over a type graph TG via a typing morphism
+type : G → TG, forming the typed graph GT = (G, typeG). In this report, we
+consider a model to be a typed graph, with the type graph defining a modeling
+language by acting as a metamodel.
+A typed graph morphism between two typed graphs GT = (G, typeG) and
+HT = (H, typeH) with the same type graph then denotes a graph morphism
+mT : G → H such that typeG = typeH ◦ mT . A (typed) graph morphism m is
+a monomorphism iff its functions mV and mE are injective.
+Figure 1 shows an example typed graph G from the software development
+domain along with the corresponding type graph TG. The typing morphism
+is encoded by the node’s labels.
+G represents an abstract syntax graph of
+a program written in an object-oriented programming language, where nodes
+may represent class declarations (ClassDecl), field declarations (FieldDecl) or
+type accesses (TypeAccess). Class declarations may contain field declarations
+via edges of type declaration, whereas field declarations can reference a class
+declaration as the field type via a TypeAccess node and edges of type access
+and type.
+The example graph contains two class declarations, one of which
+contains a field declaration, the field type of which is given by the other class
+declaration.
+A (typed) graph transformation rule r is characterized by a span of (typed)
+graph monomorphisms L
+l←− K
+r−→ R and can be applied to a graph G via a
+monomorphism m : L → G called match that satisfies the so-called dangling
+condition [6]. The result graph H of the rule application is then formally defined
+by a double pushout over an intermediate graph [6]. Intuitively, the application
+of r deletes the elements in m(L) that do not have a corresponding element in
+R and creates new elements for elements in R that do not have a corresponding
+element in L. The graph L is also called the rule’s left-hand side, K is called
+the rule’s glueing graph, and R is called the right-hand side.
+r is called a graph production if it does not delete any elements, that is, l
+3
+
+c1:ClassDecl
+f1:FieldDecl
+c2:ClassDecl
+t1:TypeAccess
+declaration
+type
+access
+++
+++
+++
+++
+++
+Figure 2: example graph transformation rule in shorthand notation
+is surjective. In this case, since L and K are isomorphic with l an isomorphism
+and we only distinguish graphs up to isomorphism, we also use the simplified
+representation L
+r−→ R.
+Figure 2 shows an example graph production in shorthand notation, where
+preserved elements are colored black, whereas created elements are colored green
+and marked by an additional “++” label. For two existing classes, the produc-
+tion creates a field declaration in one of them that references the other class
+declaration as the field type.
+We denote a sequence of applications of rules from a set of rules R to a graph
+G with resulting graph G′ by G →R G′. We say that such a rule application
+sequence is maximal if it cannot be extended by any application of a rule from
+R.
+Definition 1. Maximal Rule Application Sequence A sequence of rule appli-
+cations G →R G′ with a set of (multi-version or original) forward rules R is
+maximal if no rule from R is applicable to G′.
+2.2
+Triple Graph Grammars
+Triple graph grammars were initially presented by Schuerr [19]. This report is
+based on the slightly adapted version introduced in [9].
+In [9], a triple graph grammar (TGG) relates a source and a target modeling
+language via a correspondence modeling language and is characterized by a set
+of TGG rules. A TGG rule is defined by a graph production that simultane-
+ously transforms connected graphs from the source, correspondence and target
+modeling language into a consistently modified graph triplet. The set of TGG
+rules has to include a dedicated axiom rule, which has a triplet of empty graphs
+as its left-hand side and practically defines a triplet of starting graphs via its
+right-hand side.
+The left-hand side of a TGG rule r = L
+r−→ R can be divided into the source,
+correspondence, and target domains LS, LC, and LT respectively, with LS ⊆ L,
+LC ⊆ L, and LR ⊆ L and LS ⊎ LC ⊎ LR = L. The right-hand side can similarly
+be divided into three domains RS, RC, and RT .
+The type graph for graph
+triplets and TGG rules is hence given by the union of the type graphs defining
+the source, correspondence, and target language along with additional edges
+connecting nodes in the correspondence language to nodes in the source and
+target language.
+4
+
+c1:ClassDecl
+f1:FieldDecl
+c2:ClassDecl
+t1:TypeAccess
+declaration
+type
+access
+++
+++
+++
+++
+++
+uc1:Class
+uc2:Class
+a1:Association
+source
+target
+++
+++
+++
+cc1:CorrClass
+cc2:CorrClass
+cf1:CorrField
+S
+++
+++
+++
+++
+C
+T
+Figure 3: example TGG rule in shorthand notation
+CorrClass
+CorrField
+Class
+Association
+source
+target
+TC
+TT
+Figure 4: example type graphs for the TGG rule in Figure 3
+Figure 3 shows a TGG rule for linking the language for abstract syntax
+graphs given by the type graph in Figure 1 to a modeling language for class
+diagrams given by the type graphs TT in Figure 4, using the correspondence
+language TC from Figure 4. The rule simultaneously creates a FieldDecl and
+TypeAccess along with associated edges in the source domain (labeled S) and a
+corresponding Association with associated edges in the target domain (labeled
+T), which are linked via a newly created correspondence node of type CorrField
+in the correspondence domain (labeled C).
+TGGs can be employed to transform a model of the source language into a
+model of the target language. This requires the derivation of so-called forward
+rules from the set of TGG rules. A forward rule for a TGG rule r = L
+r−→ R
+can be constructed as rF = LF
+id
+←− LF
+rF
+−−→ R, where LF = L ∪ (RS \ r(L))
+and rF = r ∪ id, with id the identity morphism. Intuitively, rF already requires
+the existence of the elements in the source domain that would be created by
+an application of r and only creates elements in the correspondence and target
+domain. In the following, we also denote the subgraph of a forward rule that
+corresponds to the subgraph that is newly transformed by the rule by LT =
+LF \ L.
+Additionally, the derivation of a forward rule requires a technical extension
+to avoid redundant translation of the same element.
+Therefore, a dedicated
+bookkeeping node, which is connected to every currently untranslated source
+element via a bookkeeping edge, is introduced. Then, a bookkeeping node and
+bookkeeping edges to all elements in LT are added to the forward rule’s left-
+hand side. The bookkeeping node is also added to the rule’s glueing graph and
+right-hand side. Additionally, negative application conditions are added to LF ,
+5
+
+c1:ClassDecl
+f1:FieldDecl
+c2:ClassDecl
+t1:TypeAccess
+declaration
+type
+access
+uc1:Class
+uc2:Class
+a1:Association
+source
+target
+++
+++
+++
+cc1:CorrClass
+cc2:CorrClass
+cf1:CorrField
+S
+++
+++
+++
+++
+C
+T
+Figure 5: example forward rule derived from the TGG rule in Figure 3, with
+the bookkeeping mechanism omitted for readability reasons
+which ensure that for a match m from LF into SCT, ∀x ∈ LF \ LT : ∄b ∈
+BSCT : tSCT
+B
+= m(x).
+The application of the forward rule via m thus requires that elements in
+m(LT ) are untranslated, as indicated by the existence of bookkeeping edges,
+and marks these elements as translated by deleting the adjacent bookkeeping
+edges. Elements in m(LF \LT ) are in contrast required to already be translated.
+Note that, in order to allow bookkeeping edges between the bookkeeping node
+and regular edges, a slightly extended graph model is used, which is detailed in
+[10].
+Figure 5 shows the forward rule derived from the TGG rule in Figure 3.
+The elements f1 and t1 and adjacent edges are no longer created but preserved
+instead. Also, the rule requires bookkeeping edges to f1, t1, and adjacent edges,
+and contains NACs that forbid the existence of bookkeeping edges to c1 and c2.
+However, this bookkeeping mechanism is omitted in the figure for readability
+reasons. The rule’s application then deletes the bookkeeping edges to f1, t1,
+and their adjacent edges, and creates the corresponding elements in the target
+domain along with the linking node cf1 in the correspondence domain.
+TGGs can also be used to perform a transformation from the target to the
+source language by means of similarly derived backward rules. In the following,
+we will focus on the forward case. However, the backward case simply works
+analogously.
+A TGG without any critical pairs [6] among its rules is called deterministic
+[9]. A forward transformation with a deterministic TGG can be executed via an
+operation transF , which simply applies the TGG’s forward rules for as long as
+there is a match for any of them, with the order of rule applications not affecting
+the final result due to the absence of critical pairs. Specifically, for a determin-
+istic TGG with a set of forward rules R and a starting model triple SCT, any
+maximal rule transformation sequence SCT →R SCT ′ constitutes a correct
+model transformation if it deletes all bookkeeping edges in SCT. Note that, if
+SCT →R SCT ′ satisfies this bookkeeping criterion, every other possible maxi-
+mal rule transformation sequence for SCT and R also satisfies the bookkeeping
+criterion. In this report, we will focus on such deterministic TGGs, which allow
+for efficient practical implementations that avoid potentially expensive undoing
+6
+
+TypeAccess
+ClassDecl
+FieldDecl
+sdeclaration
+suc
+TG (adapted)
+access
+declaration
+type
+version
+tdeclaration
+stype
+ttype
+taccess
+saccess
+Figure 6: example adapted type graph derived from the type graph in Figure
+1, with cv and dv edges omitted for readability reasons
+of forward rule applications and backtracking [9].
+2.3
+Multi-version Models
+In this report, we consider models in the form of typed graphs. A model modi-
+fications can in this context be represented by a span of morphisms M ← K →
+M ′, where M ′ is the original model, which is modified into a changed model M ′
+via an intermediate model K, similar to a graph transformation step [21]. A ver-
+sion history of a model is then given by a set of model modifications ∆M{1,...,n}
+between models M1, M2, ..., Mn with type graph TM. We call a version his-
+tory with a unique initial version and acyclic model modification relationships
+between the individual versions a correct version history.
+In [2], we have introduced multi-version models as a means of encoding such
+a version history in a single consolidated graph. Therefore, an adapted version
+of TM, TMmv, is created. To represent model structure, TMmv contains a node
+for each node and each edge in TM. Source and target relationships of edges
+in TM are represented by edges in TMmv. In addition, a version node with
+a reflexive suc edge is added to TMmv, which allows the materialization of the
+version history’s version graph. The version graph and the model structure are
+linked via cvv and dvv edges from each node v in TMmv to the version node.
+The result of the adaptation of the type graph from Figure 1 is displayed in
+Figure 6. Note that cv and dv edges are omitted for readability reasons.
+TMmv allows the translation of ∆M{1,...,n} into a single typed graph MV M
+conforming to TMmv, which is called a multi-version model, via a procedure
+comb. This yields a bijective function origin : V MV M → �
+i∈{1,2,...,n} V Mi∪EMi
+mapping the vertices in MV M to their respective original element. An individ-
+ual model versions can be extracted from MV M via the projection operation
+proj(MV M, i) = Mi. Finally for a vertex vmv ∈ V MV M, the set of model ver-
+sions that include the element origin(vmv) can be computed via the function p,
+with p(vmv) = {Mi ∈ {M1, M2, ..., Mn}|origin(vmv) ∈ Mi}.
+7
+
+3
+Derivation of Multi-version Transformation
+Rules from Triple Graph Grammars
+The transformation of the individual model versions encoded in a multi-version
+model with a triple graph grammar can trivially be realized via the projection
+operation proj.
+However, the multi-version model may in practice afford a
+more compact representation compared to an explicit enumeration of all model
+versions, as derived via proj.
+In such practical application scenarios, operations concerning all model ver-
+sions that directly work on the multi-version model may therefore also perform
+better regarding execution time than the corresponding operations on individ-
+ual model versions, as we have already demonstrated for the case of pattern
+matching for checking the well-formedness of all model versions in a version
+history [2]. Since pattern matching also constitutes an important task in model
+transformation via triple graph grammars, a direct, joint translation of all model
+versions based on the multi-version model representation seems desirable.
+Given a triple graph grammar TGG, graph transformation rules for the joint
+translation of all source or target model versions encoded in a multi-version
+model can be derived from the regular translation rules in a straightforward
+manner. In the following, we will discuss the deriviation for forward translation.
+Rules for the backward case can be derived analogously.
+First, the adapted multi-version type graph for the TGG’s merged source,
+correspondence and target type graph is created via the translation procedure
+described in [2].
+The resulting adapted type graph TGmv for multi-version
+models is extended by two additional edges, ucvv and udvv, for each node v in the
+source domain of the merged type graph. Source and target of these edges are
+given by sT Gmv(ucvv) = sT Gmv(udvv) = v and tT Gmv(ucvv) = tT Gmv(udvv) =
+version, where version is the dedicated version node in the adapted type graph.
+Analogously to the bookkeeping edges in the original typegraph, these edges
+will be used in the translation process to encode in which versions an element
+represented by a node vmv with type v has already been translated. We there-
+fore define the set of versions where vmv has not been translated yet u(vmv)
+analogously to the set of versions p(vmv) where vmv is present, except that ucvv
+and udvv replace cvv and dvv in the definition.
+Then, for each forward rule r = L
+l←− K
+r−→ R a corresponding multi-version
+forward rule is created via a procedure adapt, with adapt(r) = trans′(L)
+lmv
+←−−
+trans′(K)
+rmv
+−−→ trans′(R).
+The vertex morphism of lmv is given by lV
+mv =
+origin−1 ◦ l ◦ origin and the edge morphisms by lE
+mv = s ◦ origin−1 ◦ lE ◦
+origin ◦ s−1 and lE
+mv = t ◦ origin−1 ◦ lE ◦ origin ◦ t−1 for all edges representing
+source respectively target relationships. rmv is constructed analogously.
+The trans′ procedure is a minor adaptation of the trans procedure in [2],
+which ignores the bookkeeping node, bookkeeping edges, and negative applica-
+tion conditions, but otherwise works analogously. The bookkeeping mechanism
+is instead translated into the additional constraint P ̸= ∅ over trans′(L), where
+P = (�
+vmv∈V trans′(L) p(vmv)∩�
+vmv∈origin−1(LT ) u(vmv))\�
+vmv∈V trans′(L) u(vmv).
+8
+
+The application of the adapted rule additionally creates outgoing cv and dv
+edges for all vertices vC
+mv ∈ V trans(R)\(origin−1 ◦ r ◦ origin)(trans(K)) to real-
+ize the assignment p(vC
+mv) := P. Furthermore, for vmv ∈ origin−1(r(l−1(LT ))),
+the application also adds and deletes outgoing ucv and udv edges to realize the
+modification u(vmv) := u(vmv) \ P.
+Note that, since the computation of the p and u sets requires considedring
+paths of arbitrary length, these computations cannot technically be defined as
+part of the graph transformation but have to be realized externally.
+For the set of forward rules R, the corresponding set of multi-version forward
+rules is then simply given by Rmv = {adapt(r)|r ∈ R}.
+4
+Execution of Multi-version Transformations
+The forward transformation of all model versions encoded in a multi-version
+model MV M according to a specified TGG can jointly be performed via the
+TGG’s set of multi-version forward rules.
+In a first step, all ucv and udv edges present in MV M are removed. Then,
+for each edge ecv ∈ EMV M with type(ecv) = cvx and sMV M(ecv), an edge
+eucv with type(eucv) = ucvx and sMV M(ecv) = sMV M(eucv) and tMV M(ecv) =
+tMV M(eucv) is created. For all dv edges, corresponding udv edges are created
+analogously. Thus, after the creation of the ucv and udv edges, it holds that
+∀vvm ∈ V MV M : u(vvm) = p(vvm).
+Subsequently, the simultaneous transformation of all model versions encoded
+in MV M is performed similarly to the regular transformation of a single model
+version via the TGG. More specifically, the adapted forward rules of the TGG
+are simply applied to MV M until no such rule is applicable anymore.
+In the following, we will argue that this transformation approach is correct
+in the sense that it yields the same result as the transformation of an individual
+model version via the regular forward rules.
+Therefore, we extend the projection operation proj from [2] to a bookkeeping-
+sensitive variant.
+Definition 2. (Bookkeeping-sensitive Projection) For a multi-version model
+MV M with version graph V and version t ∈ V V , the bookkeeping-sensitive
+projection operation works similarly to the regular projection operation proj,
+except that it also adds a bookkeeping node and bookkeeping edges to an ele-
+ment origin(v) iff t /∈ u(v) for all v ∈ V MV M. We also denote the result of the
+bookkeeping-sensitive projection operation by MV M[t] = projM(MV M, t).
+We also define two sets that represent the bookkeeping during the transfor-
+mation process.
+Definition 3. (Bookkeeping Set) For a model M, we denote the set of translated
+elements (vertices and edges) by B(M) = {x ∈ M|∄b ∈ E′M : t′M = x},
+with E′M the set of bookkeeping edges in M and t′M the target function for
+bookkeeping edges. We also call B(M) the bookkeeping set of M.
+9
+
+Definition 4. (Projection Bookkeeping Set) For a multi-version model MV M
+and version t ∈ V V , with V the version graph, we denote the set of already
+handled elements (vertices and edges) in MV M[t] by Bmv(MV M[t]) = {x ∈
+MV M[t]|t /∈ u(proj−1(x))}. We also call Bmv(MV M[t]) the projection book-
+keeping set of MV M[t].
+The following theorem states that, at the start of the transformation process
+via adapted forward rules, the prepared multi-version model via the bookkeeping-
+sensitive projection correctly encodes the starting situation for the translation
+of the individual model versions.
+Theorem 1. Given a multi-version model MV M encoding a version history
+with model versions M1, M2, ..., Mn such that ∀vvm ∈ V MV M : u(vvm) =
+p(vvm), it holds that ∀t ∈ {1, 2, ..., n} : MV M[t] = initF (Mt) up to isomor-
+phism, where initF (SCTt) denotes the graph with bookkeeping resulting from
+the preparation of Mt for the regular forward transformation process, that is,
+the graph Mt with an added bookkeeping node and bookkeeping edges to all
+elements in Mt.
+Proof. Follows directly from the fact that ∀t ∈ {1, 2, ..., n} : proj(MV M, t) =
+Mt, which has been shown in [2], and the definition of the bookkeeping-sensitive
+projection operation.
+By Theorem 1, we also get the following corollary:
+Corollary 1. Given a multi-version model MV M encoding a version history
+with model versions M1, M2, ..., Mn such that ∀vvm ∈ V MV M : u(vvm) =
+p(vvm), it holds that ∀t ∈ {1, 2, ..., n} : Bmv(MV M[t]) = B(initF (Mt)) up to
+isomorphism, where initF (SCTt) denotes the graph with bookkeeping resulting
+from the preparation of Mt for the regular forward transformation process, that
+is, the graph Mt with an added bookkeeping node and bookkeeping edges to all
+elements in Mt.
+Proof. Follows directly from Theorem 1 and the definition of bookkeeping set
+and projection bookkeeping set.
+We now show that a multi-version rule is applicable to a multi-version model
+iff the corresponding regular rule is applicable to all individual model versions
+affected by the rule application.
+Theorem 2. A multi-version forward rule rmv = Lmv ← Kmv → Rmv is
+applicable to a multi-version model triple SCTmv with bookkeeping via match
+m, if and only if for all t ∈ P, the associated original forward rule r = L ← K →
+R is applicable to SCTmv[t] via match trans(m), with P = �
+v∈V Lmv p(m(v)) ∩
+�
+v∈V LT
+mv u(m(v)).
+Proof. For a version t, as we have already shown in [2], the match m : Lmv →
+SCTmv has a corresponding match trans(m) : L → SCTmv[t] if and only if
+10
+
+t ∈ �
+v∈V Lmv p(m(v)). Furthermore, due to the definition of P and the con-
+struction of rmv, all elements in m(trans(m)(LT )) have an adjacent book-
+keeping edge in SCTmv[t] iff t ∈ �
+v∈V LT
+mv u(m(v)).
+Similarly, all elements
+in m(trans(m)(L \ LT )) have no adjacent bookkeeping edge in SCTmv[t] iff
+t /∈ �
+v∈V Lmv\LT
+mv u(m(v)). Since r and rmv delete no vertices, the dangling
+condition is trivially satisfied for r and the match trans(m).
+rmv is hence
+applicable to SCTmv via m, with t ∈ P, iff r is applicable to SCTmv[t] via
+trans(m).
+We can now show the equivalence of a single multi-version rule application
+to a multi-version model to the application of the corresponding regular rule to
+all affected model versions.
+Theorem 3. For an application SCTmv →rmv
+m
+SCT ′
+mv of a multi-version for-
+ward rule rmv = Lmv ← Kmv → Rmv with original forward rule r = L ←
+K → R to a multi-version model triple SCTmv with bookkeeping and ver-
+sion graph V via match m, it holds that ∀t ∈ P : SCT ′
+mv[t] = SCT ′ ∧
+Bmv(SCT ′
+mv[t]) = B(SCT ′) up to isomorphism, with the corresponding ap-
+plication SCTmv[t] →r
+trans(m) SCT ′. Furthermore, ∀t ∈ V V \ P : SCT ′
+mv[t] =
+SCTmv[t] ∧ Bmv(SCT ′
+mv[t]) = B(SCTmv[t]) up to isomorphism, where P =
+�
+v∈V Lmv p(m(v)) ∩ �
+v∈V LT
+mv u(m(v)).
+Proof. Disregarding bookkeeping edges, all forward rules and thus also the
+adapted forward rules are productions. Due to the construction of the adapted
+forward rules, all elements created by the rule’s application are only mv-present
+in SCT ′
+mv for the versions in P. Therefore, for all remaining versions, SCTmv[t]
+contains the same elements as SCT ′
+mv[t]. An isomorphism iso : SCTmv[t] →
+SCT ′
+mv[t] is hence trivially given by the identity in this case. Since the appli-
+cation of rmv only changes the projection bookkeeping sets for versions in P,
+Bmv(SCT ′
+mv[t]) = B(SCTmv[t]) with isomorphism iso.
+It thus holds up to isomorphism that ∀t ∈ V V \ P : SCT ′
+mv[t] = SCTmv[t] ∧
+Bmv(SCT ′
+mv[t]) = B(SCTmv[t]).
+The application of rmv to SCTmv yields a comatch n : Rmv → SCT ′
+mv and
+the associated application of r to SCTmv[t] similarly yields a comatch n′ : R →
+SCT ′ for any t ∈ P.
+An isomorphism iso : SCT ′
+mv[t] → SCT ′ can then be constructed as follows:
+Since rmv is a production, SCTmv is a subgraph of SCT ′
+mv and hence SCTmv[t]
+is also a subgraph of SCT ′
+mv[t]. Since r is a production, SCTmv[t] is also a
+subgraph of SCT ′. Isomorphic mappings for SCTmv[t] between SCT ′
+mv[t] and
+SCT ′ are thus simply given by the identity. This leaves only the elements in
+n(Rmv \Lmv) and the elements in n′(R\L) unmapped. Due to the construction
+of rmv being unique up to isomorphism, n and n′ being monomorphisms, and
+trans and origin being bijections, the remaining isomorphic mappings are given
+by n′ ◦ trans ◦ n−1 ◦ origin. Note that for elements in n(Lmv), the definition of
+iso via identity and n′ ◦ trans ◦ n−1 ◦ origin is redundant but compatible.
+Due to the definition of bookkeeping-sensitive projection, bookkeeping set,
+and projection bookkeeping set, it holds that B(SCTmv[t]) = Bmv(SCTmv[t])
+11
+
+and thus Bmv(SCTmv[t]) = B(SCTmv[t])). Compared to Bmv(SCTmv[t]), the
+application of rmv only changes the projection bookkeeping set Bmv(SCT ′
+mv[t])
+by adding the elements in trans(m(LT
+mv)). The modification to Bmv(SCT ′
+mv[t])
+hence corresponds to the modification of the bookkeeping set B(SCT ′) by the
+application of r via trans(m) for the isomorphism iso due to the construction
+of rmv.
+It thus holds that ∀t ∈ P : SCT ′
+mv[t] = SCT ′ ∧Bmv(SCT ′
+mv[t]) = B(SCT ′).
+Based on Theorem 3 for individual rule applications, we get the following
+corollary for sequences of rule applications:
+Corollary 2. For a TGG with associated set of forward rules R and multi-
+version forward rules Rmv and a multi-version model triple SCTmv with book-
+keeping and version graph V , there is a sequence of rule applications SCTmv →Rmv
+SCT ′
+mv if and only if for all t ∈ V V , there is a sequence of rule applications
+SCTmv[t] →R SCT ′ with SCT ′
+mv[t] = SCT ′∧iso(Bmv(SCT ′
+mv[t])) = B(SCT ′),
+where iso is an isomorphism from SCT ′
+mv[t] into SCT ′.
+Proof. We prove the corollary by induction over the length of the multi-version
+rule application sequence.
+For the base case of application sequences of length 0, the identity morphism
+and empty application sequences trivially satisfy the corollary.
+If there is a sequence of rule applications SCTmv →Rmv SCT ′
+mv if and
+only if for all t ∈ V V , there is a sequence of rule applications SCTmv[t] →R
+SCT ′ with SCT ′
+mv[t] = SCT ′ ∧ iso(Bmv(SCT ′
+mv[t])) = B(SCT ′), by Theorem
+3 we have an extended multi-version sequence SCTmv →Rmv SCT ′
+mv →rmv
+m
+SCT ′′
+mv and all t ∈ V V if and only if for all t ∈ V V , there is a sequence
+of regular rule applications SCTmv[t] →R SCT ′′ with SCT ′′
+mv[t] = SCT ′′ ∧
+iso(Bmv(SCT ′′
+mv[t])) = B(SCT ′′).
+For all t ∈ V V \ P, where P = �
+v∈V Lmv p(m(v)) ∩ �
+v∈V LT
+mv u(m(v)), the
+corresponding regular rule application sequence SCTmv[t] →R SCT ′ and iso-
+morphism iso : SCT ′
+mv[t] → SCT ′ are also valid for SCT ′′
+mv[t] and satisfy the
+condition on bookkeeping sets, since SCT ′ = SCT ′
+mv[t] = SCT ′′
+mv[t] (up to
+isomorphism).
+In accordance with Theorem 3, there is an extended sequence SCTmv →Rmv
+SCT ′
+mv →rmv
+m
+SCT ′′
+mv if and only if for all t ∈ P, the regular rule applica-
+tion sequence SCTmv[t] →R SCT ′
+mv[t] can be extended by a rule application
+SCT ′
+mv[t] →r
+trans(m) SCT ′′
+mv[t] that satisfies the condition on bookkeeping sets.
+Thus, there is a sequence of rule applications SCTmv →Rmv SCT ′
+mv →rmv
+m
+SCT ′′
+mv if and only if for all t ∈ V V , there is a sequence of rule applica-
+tions SCTmv[t] →R SCT ′′ with SCT ′′
+mv[t] = SCT ′′ ∧ iso(Bmv(SCT ′′
+mv[t])) =
+B(SCT ′′).
+With the proof for the base case and the induction step, we have proven the
+validity of the corollary.
+12
+
+Intuitively, the multi-version forward rules perform an interleaved, parallel
+transformation of all model versions encoded in SCTmv. The application of a
+multi-version rule Lmv ← Kmv → Rmv corresponds to the application of the
+original rule to all model versions in P = �
+v∈V Lmv p(m(v)) ∩ �
+v∈V LT
+mv u(m(v))
+and leaves all other model versions unchanged. Thus, a multi-version rule appli-
+cation effectively extends the corresponding original rule application sequences
+for all versions in P by the associated original rule application, whereas it rep-
+resents the “skipping” of a step in the sequences of all versions not in P.
+For a deterministic TGG, a correct translation of source graph S is given by
+any maximal rule application sequence of forward rules that deletes all book-
+keeping edges in the source model. Note that because of the determinism crite-
+rion, either every maximal rule application sequences or none of them satisfies
+the bookkeeping criterion. Correctness of the joint translation of all individual
+versions via multi-version forward rules is hence given by the following corollary:
+Corollary 3. For a TGG with associated set of forward rules R and multi-
+version forward rules Rmv and a multi-version model triple SCTmv with book-
+keeping and version graph V , there is a maximal sequence of rule applications
+SCTmv →Rmv SCT ′
+mv if and only if for all t ∈ V V , there is a maximal se-
+quence of regular rule applications SCTmv[t] →R SCT ′ such that SCT ′
+mv[t] =
+SCT ′ ∧ Bmv(SCT ′
+mv[t]) = B(SCT ′).
+Proof. The existence of a sequence of original rule applications for a sequence of
+multi-version rule applications and all versions t ∈ V V and vice-versa is given by
+Corollary 2. From Theorem 2, it follows directly that the multi-version sequence
+is maximal if and only if the regular sequences are maximal for all t ∈ V V .
+Thus, for a deterministic TGG and by corollaries 1 and 3, the result of
+repeated application of adapted transformation rules to a multi-version model
+prepared for multi-version translation until a fixpoint is reached is equivalent
+to the results of repeated application of the original rules to the individual
+model versions prepared for translation, that is, the results of transforming the
+individual model versions using the TGG.
+We thereby have the correctness of the forward transformation using multi-
+version forward rules transF
+mv, which applies multi-version forward rules to a
+multi-version model with bookkeeping until a fixpoint is reached.
+Theorem 4. For a correct version history ∆M{1,...,n} and a triple graph gram-
+mar with set of forward rules R, it holds up to isomorphism that
+∀t ∈ {1, ..., n} : transF
+mv(initF (comb(∆M{1,...,n})), adapt(R))[t] = transF (Mt, R)
+(1)
+Proof. Follows directly from Theorem 1 and Corollary 3.
+13
+
+5
+Evaluation
+In order to evaluate our approach empirically with respect to execution time
+performance, we have realized the presented concepts in our MoTE2 tool [12] for
+TGG-based model transformation, which is implemented in the context of the
+Java-based Eclipse Modeling Framework [7] and has been shown to be efficient
+compared to other existing model transformation tools [12].
+As an application scenario, we consider the transformation of Java abstract
+syntax trees to class diagrams. We have therefore modeled this transformation
+as a TGG with MoTE2 and use the original and our adapted implementation
+to automatically derive the required forward rules respectively multi-version
+forward rules.
+To obtain realistic source models, we have extracted the version history of
+one small personal Java project (rete, around 50 versions) and one larger open
+source Java project (henshin [1], around 2000 versions) from their respective
+GitHub repositories and have constructed the corresponding history of abstract
+syntax trees using the MoDisco tool [3]. As input for the solution presented in
+Sections 3 and 4, we have consolidated both version histories into multi-version
+models using a mapping based on hierarchy and naming.
+Our implementation and the employed datasets are available under [22].
+Based on this, we run the following model transformations for both reposi-
+tories and measure the overall execution time1 for each of them:
+• SVM: individual forward transformation of all model versions (abstract
+syntax trees) in the repository using the original MoTE2 implementation
+• MVM: joint forward transformation of all model versions in the reposi-
+tory using a multi-version model encoding and our implementation of the
+technique presented in Sections 3 and 4
+Note that the SVM strategy would require initial projection operations and
+a final combination of transformation results to work within the framework of
+multi-version models. However, for fairness of comparison of the transformation,
+we do not consider these additional operations in our evaluation.
+Figure 7 shows the execution times of the transformations using the two
+strategies.
+For both repositories, the transformation based on multi-version
+models requires substantially less time than the transformation of the individ-
+ual model versions using the original MoTE2 tool, with a more pronounced
+improvement for the larger repository (factor 4 for the smaller and factor 74 for
+the larger repository).
+The improvement in efficiency and scalability can likely be explained by two
+factors: First, SVM has to perform a somewhat expensive initialization step for
+every indidvidual model version that is to be transformed, whereas MVM only
+1All experiments were performed on a Linux SMP Debian 4.19.67-2 machine with Intel
+Xeon E5-2630 CPU (2.3 GHz clock rate) and 386 GB system memory running OpenJDK
+version 11.0.6.
+Reported execution time measurements correspond to the mean execution
+time of 10 runs of the respective experiment.
+14
+
+1
+10
+100
+1000
+10000
+100000
+1000000
+10000000
+rete
+henshin
+execution time (ms)
+Transformation Time
+SVM
+MVM
+Figure 7: execution time measurements for the transformation of all model
+versions in two different software repositories (logarithmic axis)
+requires one such initialization. Second, many elements in the abstract syntax
+trees of the repositories are shared between many versions. SVM has to perform
+a separate transformation, including separate pattern matching, for each model
+version. In contrast, MVM only performs a transformation including pattern
+matching over a single multi-version model, the size of which is much smaller
+than the combined sizes of the encoded model versions, along with efficient
+search operations over the version graph. Since pattern matching is efficient in
+this example, that is, pattern matching has a runtime complexity that is linear
+in the size of the model for the derived forward rules, this results in an improved
+overall efficiency.
+Threats to the internal validity of our experimental results include unex-
+pected behavior of the Java virtual machine such as garbage collection.
+To
+address this threat, we have performed multiple runs of all experiments and re-
+port the mean execution time of these runs, with the standard deviation always
+below 5% of the execution time. To minimize the impact of the concrete imple-
+mentation on our measurements, we have realized our solution in the framework
+of the transformation tool we use for comparison and thereby largely use the
+same execution mechanism.
+To mitigate threats to external validity, we use real-world models as the
+source models of the transformation. However, we remark that our results are
+not necessarily generalizable to different examples or application domains and
+make no quantitative claims regarding the performance of our approach.
+6
+Related Work
+The general problem of model versioning has already been studied extensively,
+both formally [5, 17] and in the form of concrete tool implementations [14, 13].
+Several solutions employ a unified representation of a model’s version history
+15
+
+similar to multi-version models [17, 14]. However, due to the problem definition
+focusing on the management of different versions of a single model, realising
+model transformation based on a unified encoding is out of scope for these
+approaches.
+There is also a significant body of previous work on synchronization of con-
+currently modified pairs of models using triple graph grammars [24, 15]. The
+focus of these works is the derivation of compatible versions of source and tar-
+get model that respect the modifications to either of. This report aims to make
+a step in an orthogonal direction, namely towards allowing living with incon-
+sistencies by enabling developers to temporarily work with multiple modified,
+possibly conflicting versions of source and target model.
+In the context of software product lines, so-called 150% models are em-
+ployed to encode different configurations of a software system [4, 16]. In this
+context, Greiner and Westfechtel present an approach for propagating so-called
+variability annotations along trace links created by model transformations [23],
+explicitly considering the case of transformations implemented via triple graph
+grammars. A similar approach could also be employed to propagate versioning
+information and would have the advantage of not requiring any adaptation of
+the employed rules, type graph, or transformation process. However, not inte-
+grating this propagation with the transformation process and only propagating
+versioning information after the transformation has been executed would mean
+that certain cases that are covered by our approach could not be handled. The
+occurence of such cases may hence prevent a possible correct transformation. For
+instance, under standard TGG semantics, such cases include a model element
+being translated differently in different model versions based on its context.
+In previous work in our group, the joint execution of queries over multiple
+versions of an evolving model has been considered for both the case with [2] and
+without [11, 18] parallel, branching development. This report builds on these
+results, but instead of focusing on pure queries without side-effects considers
+the case of writing operations in the form of model transformations.
+7
+Conclusion
+In this report, we have presented a first step in the direction of model transfor-
+mation on multi-version models in the form of an adaptation of the well-known
+triple graph grammar formalism that enables the joint transformation of all ver-
+sions encoded in a multi-version model. The presented approach is correct with
+respect to the translation semantics of deterministic triple graph grammars for
+individual model versions, that is, it produces equivalent results. Initial exper-
+iments for evaluating the efficiency of our approach demonstrate that our tech-
+nique can improve performance compared to a na¨ıve realization, which simply
+translates all model versions individually according to a triple graph grammar
+specification, in a realistic application scenario.
+In future work, we plan to build on the presented approach to realize model
+synchronization for multi-version models, that is, incremental propagation of
+16
+
+changes to one or more versions of a source model to the corresponding target
+model versions. Furthermore, we want to explore the possibility of improving
+the efficiency of multi-version model transformations via incremental pattern
+matching for multi-version models.
+Another interesting direction for future
+work is the integration of advanced application conditions for the specification
+of triple graph grammar rules such as nested graph conditions into our approach.
+Finally, a more extensive evaluation can be conducted to study the scalability
+of the presented technique in more detail.
+Acknowledgements
+This work was developed mainly in the course of the project modular and incre-
+mental Global Model Management (project number 336677879), which is funded
+by the Deutsche Forschungsgemeinschaft.
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+
diff --git a/V9AyT4oBgHgl3EQfu_l-/content/tmp_files/load_file.txt b/V9AyT4oBgHgl3EQfu_l-/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf,len=529
+page_content='Triple Graph Grammars for Multi-version Models  Matthias Barkowsky  matthias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='barkowsky@hpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='de  Holger Giese  holger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='giese@hpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='de  January 3, 2023  Abstract  Like conventional software projects, projects in model-driven software  engineering require adequate management of multiple versions of devel-  opment artifacts, importantly allowing living with temporary inconsisten-  cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' In the case of model-driven software engineering, employed version-  ing approaches also have to handle situations where different artifacts,  that is, different models, are linked via automatic model transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  In this report, we propose a technique for jointly handling the transfor-  mation of multiple versions of a source model into corresponding versions  of a target model, which enables the use of a more compact representa-  tion that may afford improved execution time of both the transformation  and further analysis operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Our approach is based on the well-known  formalism of triple graph grammars and a previously introduced encod-  ing of model version histories called multi-version models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  In addition  to showing the correctness of our approach with respect to the standard  semantics of triple graph grammars, we conduct an empirical evaluation  that demonstrates the potential benefit regarding execution time perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  1  Introduction  In model-driven software development, models are treated as primary develop-  ment artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Complex projects can involve multiple models, which describe  the system under development at different levels of abstraction or with respect  to different system aspects and can be edited independently by a team of de-  velopers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' In this case, consistency of the holistic system description is ensured  by model transformations that automatically derive concrete models from more  abstract ones or propagate changes to a model describing one aspect of the  system to other models concerned with different but overlapping aspects [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Similarly to program code in conventional software development, the evolu-  tion of models via changes made by different developers requires management  of the resulting versions of the software description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' In particular, version man-  agement has to support parallel development activities of multiple developers  working on the same development artifact, where living with inconsistencies of a  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='00623v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='SE]  2 Jan 2023  single artifact may temporarily be necessary to avoid loss of information [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' In  [2], we have introduced multi-version models as a means of managing multiple  versions of the same model that also enables monitoring the consistency of the  individual model versions and potential merge results of versions developed in  parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  However, with model transformations effectively linking multiple models  via consistency relationships, considering only the evolution of a single model  without its context is insufficient for larger model-driven software development  projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Thus, a mechanism for establishing consistency of different versions of  such linked models that simultaneously allows parallel development of multiple  versions is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Such a mechanism would allow working with more compact representations  that also enable further analysis operations as described in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' In addition, an  integrated handling of multiple model versions may afford an improved execution  time performance of the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  In this report, we propose a first step in the direction of model transforma-  tions working on multi-version models by adapting the well-known formalism  of triple graph grammars, which enables the implementation of single-version  model transformations, to the multi-version case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  The remainder of the report is structured as follows: In Section 2, we briefly  reiterate the basic concepts of graphs, graph transformations, triple graph gram-  mars, and multi-version models, as used in this report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Subsequently, we present  our approach for deriving transformation rules that work on multi-version mod-  els from single-version model transformation specifications in the form of triple  graph grammars in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' In Section 4, we describe how the derived rules  can be used to realize the joint transformation of all individual model versions  encoded in a multi-version model and prove the correctness of our technique  with respect to the semantics of triple graph grammars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Section 5 reports on  the results of an initial evaluation of the presented solution’s performance re-  garding execution time, which is based on an application scenario in the software  development domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Related work is discussed in Section 6, before Section 7  concludes the report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  2  Preliminaries  In this section, we give a brief overview of required preliminaries regarding  graphs and graph transformations, triple graph grammars and multi-version  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='1  Graphs and Graph Transformations  We briefly reiterate the concepts of graphs, graph morphisms and graph transfor-  mations and their typed analogs as defined in [6] and required in the remainder  of the report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  2  TypeAccess  ClassDecl  FieldDecl  type  access  declaration  c1:ClassDecl  f1:FieldDecl  c2:ClassDecl  t1:TypeAccess  G  TG  declaration  type  access  Figure 1: example graph and type graph  A graph G = (V G, EG, sG, tG) consists of a set of nodes V G, a set of edges  EG and two functions sG : EG → V G and tG : EG → V G assigning each edge  its source and target, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' A graph morphism m : G → H is given by a  pair of functions mV : V G → V H and mE : EG → EH that map elements from  G to elements from H such that sH ◦ mE = mV ◦ sG and tH ◦ mE = mV ◦ tG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  We also call mV the vertex morphism and mE the edge morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  A graph G can be typed over a type graph TG via a typing morphism  type : G → TG, forming the typed graph GT = (G, typeG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' In this report, we  consider a model to be a typed graph, with the type graph defining a modeling  language by acting as a metamodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  A typed graph morphism between two typed graphs GT = (G, typeG) and  HT = (H, typeH) with the same type graph then denotes a graph morphism  mT : G → H such that typeG = typeH ◦ mT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' A (typed) graph morphism m is  a monomorphism iff its functions mV and mE are injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Figure 1 shows an example typed graph G from the software development  domain along with the corresponding type graph TG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' The typing morphism  is encoded by the node’s labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  G represents an abstract syntax graph of  a program written in an object-oriented programming language, where nodes  may represent class declarations (ClassDecl), field declarations (FieldDecl) or  type accesses (TypeAccess).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Class declarations may contain field declarations  via edges of type declaration, whereas field declarations can reference a class  declaration as the field type via a TypeAccess node and edges of type access  and type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  The example graph contains two class declarations, one of which  contains a field declaration, the field type of which is given by the other class  declaration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  A (typed) graph transformation rule r is characterized by a span of (typed)  graph monomorphisms L  l←− K  r−→ R and can be applied to a graph G via a  monomorphism m : L → G called match that satisfies the so-called dangling  condition [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' The result graph H of the rule application is then formally defined  by a double pushout over an intermediate graph [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Intuitively, the application  of r deletes the elements in m(L) that do not have a corresponding element in  R and creates new elements for elements in R that do not have a corresponding  element in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' The graph L is also called the rule’s left-hand side, K is called  the rule’s glueing graph, and R is called the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  r is called a graph production if it does not delete any elements, that is, l  3  c1:ClassDecl  f1:FieldDecl  c2:ClassDecl  t1:TypeAccess  declaration  type  access  ++  ++  ++  ++  ++  Figure 2: example graph transformation rule in shorthand notation  is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' In this case, since L and K are isomorphic with l an isomorphism  and we only distinguish graphs up to isomorphism, we also use the simplified  representation L  r−→ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Figure 2 shows an example graph production in shorthand notation, where  preserved elements are colored black, whereas created elements are colored green  and marked by an additional “++” label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' For two existing classes, the produc-  tion creates a field declaration in one of them that references the other class  declaration as the field type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  We denote a sequence of applications of rules from a set of rules R to a graph  G with resulting graph G′ by G →R G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' We say that such a rule application  sequence is maximal if it cannot be extended by any application of a rule from  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Maximal Rule Application Sequence A sequence of rule appli-  cations G →R G′ with a set of (multi-version or original) forward rules R is  maximal if no rule from R is applicable to G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='2  Triple Graph Grammars  Triple graph grammars were initially presented by Schuerr [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' This report is  based on the slightly adapted version introduced in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  In [9], a triple graph grammar (TGG) relates a source and a target modeling  language via a correspondence modeling language and is characterized by a set  of TGG rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' A TGG rule is defined by a graph production that simultane-  ously transforms connected graphs from the source, correspondence and target  modeling language into a consistently modified graph triplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' The set of TGG  rules has to include a dedicated axiom rule, which has a triplet of empty graphs  as its left-hand side and practically defines a triplet of starting graphs via its  right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  The left-hand side of a TGG rule r = L  r−→ R can be divided into the source,  correspondence, and target domains LS, LC, and LT respectively, with LS ⊆ L,  LC ⊆ L, and LR ⊆ L and LS ⊎ LC ⊎ LR = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' The right-hand side can similarly  be divided into three domains RS, RC, and RT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  The type graph for graph  triplets and TGG rules is hence given by the union of the type graphs defining  the source, correspondence, and target language along with additional edges  connecting nodes in the correspondence language to nodes in the source and  target language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='c1:ClassDecl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='f1:FieldDecl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='c2:ClassDecl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='t1:TypeAccess  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='declaration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='type  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='access  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='++  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='++  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='++  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='++  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='++  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='uc1:Class  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='uc2:Class  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='a1:Association  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='source  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='++  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='++  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='++  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='cc1:CorrClass  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='cc2:CorrClass  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='cf1:CorrField  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='S  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='++  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='++  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='++  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='++  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='Figure 3: example TGG rule in shorthand notation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='CorrClass  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='CorrField  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='Class  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='Association  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='source  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='TC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='TT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='Figure 4: example type graphs for the TGG rule in Figure 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='Figure 3 shows a TGG rule for linking the language for abstract syntax  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='graphs given by the type graph in Figure 1 to a modeling language for class  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='diagrams given by the type graphs TT in Figure 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' using the correspondence  language TC from Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' The rule simultaneously creates a FieldDecl and  TypeAccess along with associated edges in the source domain (labeled S) and a  corresponding Association with associated edges in the target domain (labeled  T), which are linked via a newly created correspondence node of type CorrField  in the correspondence domain (labeled C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  TGGs can be employed to transform a model of the source language into a  model of the target language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' This requires the derivation of so-called forward  rules from the set of TGG rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' A forward rule for a TGG rule r = L  r−→ R  can be constructed as rF = LF  id  ←− LF  rF  −−→ R, where LF = L ∪ (RS \\ r(L))  and rF = r ∪ id, with id the identity morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Intuitively, rF already requires  the existence of the elements in the source domain that would be created by  an application of r and only creates elements in the correspondence and target  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' In the following, we also denote the subgraph of a forward rule that  corresponds to the subgraph that is newly transformed by the rule by LT =  LF \\ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Additionally, the derivation of a forward rule requires a technical extension  to avoid redundant translation of the same element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Therefore, a dedicated  bookkeeping node, which is connected to every currently untranslated source  element via a bookkeeping edge, is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Then, a bookkeeping node and  bookkeeping edges to all elements in LT are added to the forward rule’s left-  hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' The bookkeeping node is also added to the rule’s glueing graph and  right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Additionally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' negative application conditions are added to LF ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  5  c1:ClassDecl  f1:FieldDecl  c2:ClassDecl  t1:TypeAccess  declaration  type  access  uc1:Class  uc2:Class  a1:Association  source  target  ++  ++  ++  cc1:CorrClass  cc2:CorrClass  cf1:CorrField  S  ++  ++  ++  ++  C  T  Figure 5: example forward rule derived from the TGG rule in Figure 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' with  the bookkeeping mechanism omitted for readability reasons  which ensure that for a match m from LF into SCT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' ∀x ∈ LF \\ LT : ∄b ∈  BSCT : tSCT  B  = m(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  The application of the forward rule via m thus requires that elements in  m(LT ) are untranslated, as indicated by the existence of bookkeeping edges,  and marks these elements as translated by deleting the adjacent bookkeeping  edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Elements in m(LF \\LT ) are in contrast required to already be translated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Note that, in order to allow bookkeeping edges between the bookkeeping node  and regular edges, a slightly extended graph model is used, which is detailed in  [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Figure 5 shows the forward rule derived from the TGG rule in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  The elements f1 and t1 and adjacent edges are no longer created but preserved  instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Also, the rule requires bookkeeping edges to f1, t1, and adjacent edges,  and contains NACs that forbid the existence of bookkeeping edges to c1 and c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  However, this bookkeeping mechanism is omitted in the figure for readability  reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' The rule’s application then deletes the bookkeeping edges to f1, t1,  and their adjacent edges, and creates the corresponding elements in the target  domain along with the linking node cf1 in the correspondence domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  TGGs can also be used to perform a transformation from the target to the  source language by means of similarly derived backward rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' In the following,  we will focus on the forward case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' However, the backward case simply works  analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  A TGG without any critical pairs [6] among its rules is called deterministic  [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' A forward transformation with a deterministic TGG can be executed via an  operation transF , which simply applies the TGG’s forward rules for as long as  there is a match for any of them, with the order of rule applications not affecting  the final result due to the absence of critical pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Specifically, for a determin-  istic TGG with a set of forward rules R and a starting model triple SCT, any  maximal rule transformation sequence SCT →R SCT ′ constitutes a correct  model transformation if it deletes all bookkeeping edges in SCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Note that, if  SCT →R SCT ′ satisfies this bookkeeping criterion, every other possible maxi-  mal rule transformation sequence for SCT and R also satisfies the bookkeeping  criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' In this report, we will focus on such deterministic TGGs, which allow  for efficient practical implementations that avoid potentially expensive undoing  6  TypeAccess  ClassDecl  FieldDecl  sdeclaration  suc  TG (adapted)  access  declaration  type  version  tdeclaration  stype  ttype  taccess  saccess  Figure 6: example adapted type graph derived from the type graph in Figure  1, with cv and dv edges omitted for readability reasons  of forward rule applications and backtracking [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='3  Multi-version Models  In this report, we consider models in the form of typed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' A model modi-  fications can in this context be represented by a span of morphisms M ← K →  M ′, where M ′ is the original model, which is modified into a changed model M ′  via an intermediate model K, similar to a graph transformation step [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' A ver-  sion history of a model is then given by a set of model modifications ∆M{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=',n}  between models M1, M2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=', Mn with type graph TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' We call a version his-  tory with a unique initial version and acyclic model modification relationships  between the individual versions a correct version history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  In [2], we have introduced multi-version models as a means of encoding such  a version history in a single consolidated graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Therefore, an adapted version  of TM, TMmv, is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' To represent model structure, TMmv contains a node  for each node and each edge in TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Source and target relationships of edges  in TM are represented by edges in TMmv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' In addition, a version node with  a reflexive suc edge is added to TMmv, which allows the materialization of the  version history’s version graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' The version graph and the model structure are  linked via cvv and dvv edges from each node v in TMmv to the version node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  The result of the adaptation of the type graph from Figure 1 is displayed in  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Note that cv and dv edges are omitted for readability reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  TMmv allows the translation of ∆M{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=',n} into a single typed graph MV M  conforming to TMmv, which is called a multi-version model, via a procedure  comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' This yields a bijective function origin : V MV M → �  i∈{1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=',n} V Mi∪EMi  mapping the vertices in MV M to their respective original element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' An individ-  ual model versions can be extracted from MV M via the projection operation  proj(MV M, i) = Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Finally for a vertex vmv ∈ V MV M, the set of model ver-  sions that include the element origin(vmv) can be computed via the function p,  with p(vmv) = {Mi ∈ {M1, M2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=', Mn}|origin(vmv) ∈ Mi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  7  3  Derivation of Multi-version Transformation  Rules from Triple Graph Grammars  The transformation of the individual model versions encoded in a multi-version  model with a triple graph grammar can trivially be realized via the projection  operation proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  However, the multi-version model may in practice afford a  more compact representation compared to an explicit enumeration of all model  versions, as derived via proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  In such practical application scenarios, operations concerning all model ver-  sions that directly work on the multi-version model may therefore also perform  better regarding execution time than the corresponding operations on individ-  ual model versions, as we have already demonstrated for the case of pattern  matching for checking the well-formedness of all model versions in a version  history [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Since pattern matching also constitutes an important task in model  transformation via triple graph grammars, a direct, joint translation of all model  versions based on the multi-version model representation seems desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Given a triple graph grammar TGG, graph transformation rules for the joint  translation of all source or target model versions encoded in a multi-version  model can be derived from the regular translation rules in a straightforward  manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' In the following, we will discuss the deriviation for forward translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Rules for the backward case can be derived analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  First, the adapted multi-version type graph for the TGG’s merged source,  correspondence and target type graph is created via the translation procedure  described in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  The resulting adapted type graph TGmv for multi-version  models is extended by two additional edges, ucvv and udvv, for each node v in the  source domain of the merged type graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Source and target of these edges are  given by sT Gmv(ucvv) = sT Gmv(udvv) = v and tT Gmv(ucvv) = tT Gmv(udvv) =  version, where version is the dedicated version node in the adapted type graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Analogously to the bookkeeping edges in the original typegraph, these edges  will be used in the translation process to encode in which versions an element  represented by a node vmv with type v has already been translated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' We there-  fore define the set of versions where vmv has not been translated yet u(vmv)  analogously to the set of versions p(vmv) where vmv is present, except that ucvv  and udvv replace cvv and dvv in the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Then, for each forward rule r = L  l←− K  r−→ R a corresponding multi-version  forward rule is created via a procedure adapt, with adapt(r) = trans′(L)  lmv  ←−−  trans′(K)  rmv  −−→ trans′(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  The vertex morphism of lmv is given by lV  mv =  origin−1 ◦ l ◦ origin and the edge morphisms by lE  mv = s ◦ origin−1 ◦ lE ◦  origin ◦ s−1 and lE  mv = t ◦ origin−1 ◦ lE ◦ origin ◦ t−1 for all edges representing  source respectively target relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' rmv is constructed analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  The trans′ procedure is a minor adaptation of the trans procedure in [2],  which ignores the bookkeeping node, bookkeeping edges, and negative applica-  tion conditions, but otherwise works analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' The bookkeeping mechanism  is instead translated into the additional constraint P ̸= ∅ over trans′(L), where  P = (�  vmv∈V trans′(L) p(vmv)∩�  vmv∈origin−1(LT ) u(vmv))\\�  vmv∈V trans′(L) u(vmv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  8  The application of the adapted rule additionally creates outgoing cv and dv  edges for all vertices vC  mv ∈ V trans(R)\\(origin−1 ◦ r ◦ origin)(trans(K)) to real-  ize the assignment p(vC  mv) := P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Furthermore, for vmv ∈ origin−1(r(l−1(LT ))),  the application also adds and deletes outgoing ucv and udv edges to realize the  modification u(vmv) := u(vmv) \\ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Note that, since the computation of the p and u sets requires considedring  paths of arbitrary length, these computations cannot technically be defined as  part of the graph transformation but have to be realized externally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  For the set of forward rules R, the corresponding set of multi-version forward  rules is then simply given by Rmv = {adapt(r)|r ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  4  Execution of Multi-version Transformations  The forward transformation of all model versions encoded in a multi-version  model MV M according to a specified TGG can jointly be performed via the  TGG’s set of multi-version forward rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  In a first step, all ucv and udv edges present in MV M are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Then,  for each edge ecv ∈ EMV M with type(ecv) = cvx and sMV M(ecv), an edge  eucv with type(eucv) = ucvx and sMV M(ecv) = sMV M(eucv) and tMV M(ecv) =  tMV M(eucv) is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' For all dv edges, corresponding udv edges are created  analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Thus, after the creation of the ucv and udv edges, it holds that  ∀vvm ∈ V MV M : u(vvm) = p(vvm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Subsequently, the simultaneous transformation of all model versions encoded  in MV M is performed similarly to the regular transformation of a single model  version via the TGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' More specifically, the adapted forward rules of the TGG  are simply applied to MV M until no such rule is applicable anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  In the following, we will argue that this transformation approach is correct  in the sense that it yields the same result as the transformation of an individual  model version via the regular forward rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Therefore, we extend the projection operation proj from [2] to a bookkeeping-  sensitive variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' (Bookkeeping-sensitive Projection) For a multi-version model  MV M with version graph V and version t ∈ V V , the bookkeeping-sensitive  projection operation works similarly to the regular projection operation proj,  except that it also adds a bookkeeping node and bookkeeping edges to an ele-  ment origin(v) iff t /∈ u(v) for all v ∈ V MV M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' We also denote the result of the  bookkeeping-sensitive projection operation by MV M[t] = projM(MV M, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  We also define two sets that represent the bookkeeping during the transfor-  mation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' (Bookkeeping Set) For a model M, we denote the set of translated  elements (vertices and edges) by B(M) = {x ∈ M|∄b ∈ E′M : t′M = x},  with E′M the set of bookkeeping edges in M and t′M the target function for  bookkeeping edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' We also call B(M) the bookkeeping set of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  9  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' (Projection Bookkeeping Set) For a multi-version model MV M  and version t ∈ V V , with V the version graph, we denote the set of already  handled elements (vertices and edges) in MV M[t] by Bmv(MV M[t]) = {x ∈  MV M[t]|t /∈ u(proj−1(x))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' We also call Bmv(MV M[t]) the projection book-  keeping set of MV M[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  The following theorem states that, at the start of the transformation process  via adapted forward rules, the prepared multi-version model via the bookkeeping-  sensitive projection correctly encodes the starting situation for the translation  of the individual model versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Given a multi-version model MV M encoding a version history  with model versions M1, M2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=', Mn such that ∀vvm ∈ V MV M : u(vvm) =  p(vvm), it holds that ∀t ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=', n} : MV M[t] = initF (Mt) up to isomor-  phism, where initF (SCTt) denotes the graph with bookkeeping resulting from  the preparation of Mt for the regular forward transformation process, that is,  the graph Mt with an added bookkeeping node and bookkeeping edges to all  elements in Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Follows directly from the fact that ∀t ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=', n} : proj(MV M, t) =  Mt, which has been shown in [2], and the definition of the bookkeeping-sensitive  projection operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  By Theorem 1, we also get the following corollary:  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Given a multi-version model MV M encoding a version history  with model versions M1, M2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=', Mn such that ∀vvm ∈ V MV M : u(vvm) =  p(vvm), it holds that ∀t ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=', n} : Bmv(MV M[t]) = B(initF (Mt)) up to  isomorphism, where initF (SCTt) denotes the graph with bookkeeping resulting  from the preparation of Mt for the regular forward transformation process, that  is, the graph Mt with an added bookkeeping node and bookkeeping edges to all  elements in Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Follows directly from Theorem 1 and the definition of bookkeeping set  and projection bookkeeping set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  We now show that a multi-version rule is applicable to a multi-version model  iff the corresponding regular rule is applicable to all individual model versions  affected by the rule application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' A multi-version forward rule rmv = Lmv ← Kmv → Rmv is  applicable to a multi-version model triple SCTmv with bookkeeping via match  m, if and only if for all t ∈ P, the associated original forward rule r = L ← K →  R is applicable to SCTmv[t] via match trans(m), with P = �  v∈V Lmv p(m(v)) ∩  �  v∈V LT  mv u(m(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' For a version t, as we have already shown in [2], the match m : Lmv →  SCTmv has a corresponding match trans(m) : L → SCTmv[t] if and only if  10  t ∈ �  v∈V Lmv p(m(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Furthermore, due to the definition of P and the con-  struction of rmv, all elements in m(trans(m)(LT )) have an adjacent book-  keeping edge in SCTmv[t] iff t ∈ �  v∈V LT  mv u(m(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Similarly, all elements  in m(trans(m)(L \\ LT )) have no adjacent bookkeeping edge in SCTmv[t] iff  t /∈ �  v∈V Lmv\\LT  mv u(m(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Since r and rmv delete no vertices, the dangling  condition is trivially satisfied for r and the match trans(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  rmv is hence  applicable to SCTmv via m, with t ∈ P, iff r is applicable to SCTmv[t] via  trans(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  We can now show the equivalence of a single multi-version rule application  to a multi-version model to the application of the corresponding regular rule to  all affected model versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' For an application SCTmv →rmv  m  SCT ′  mv of a multi-version for-  ward rule rmv = Lmv ← Kmv → Rmv with original forward rule r = L ←  K → R to a multi-version model triple SCTmv with bookkeeping and ver-  sion graph V via match m, it holds that ∀t ∈ P : SCT ′  mv[t] = SCT ′ ∧  Bmv(SCT ′  mv[t]) = B(SCT ′) up to isomorphism, with the corresponding ap-  plication SCTmv[t] →r  trans(m) SCT ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Furthermore, ∀t ∈ V V \\ P : SCT ′  mv[t] =  SCTmv[t] ∧ Bmv(SCT ′  mv[t]) = B(SCTmv[t]) up to isomorphism, where P =  �  v∈V Lmv p(m(v)) ∩ �  v∈V LT  mv u(m(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Disregarding bookkeeping edges, all forward rules and thus also the  adapted forward rules are productions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Due to the construction of the adapted  forward rules, all elements created by the rule’s application are only mv-present  in SCT ′  mv for the versions in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Therefore, for all remaining versions, SCTmv[t]  contains the same elements as SCT ′  mv[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' An isomorphism iso : SCTmv[t] →  SCT ′  mv[t] is hence trivially given by the identity in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Since the appli-  cation of rmv only changes the projection bookkeeping sets for versions in P,  Bmv(SCT ′  mv[t]) = B(SCTmv[t]) with isomorphism iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  It thus holds up to isomorphism that ∀t ∈ V V \\ P : SCT ′  mv[t] = SCTmv[t] ∧  Bmv(SCT ′  mv[t]) = B(SCTmv[t]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  The application of rmv to SCTmv yields a comatch n : Rmv → SCT ′  mv and  the associated application of r to SCTmv[t] similarly yields a comatch n′ : R →  SCT ′ for any t ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  An isomorphism iso : SCT ′  mv[t] → SCT ′ can then be constructed as follows:  Since rmv is a production, SCTmv is a subgraph of SCT ′  mv and hence SCTmv[t]  is also a subgraph of SCT ′  mv[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Since r is a production, SCTmv[t] is also a  subgraph of SCT ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Isomorphic mappings for SCTmv[t] between SCT ′  mv[t] and  SCT ′ are thus simply given by the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' This leaves only the elements in  n(Rmv \\Lmv) and the elements in n′(R\\L) unmapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Due to the construction  of rmv being unique up to isomorphism, n and n′ being monomorphisms, and  trans and origin being bijections, the remaining isomorphic mappings are given  by n′ ◦ trans ◦ n−1 ◦ origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Note that for elements in n(Lmv), the definition of  iso via identity and n′ ◦ trans ◦ n−1 ◦ origin is redundant but compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Due to the definition of bookkeeping-sensitive projection, bookkeeping set,  and projection bookkeeping set, it holds that B(SCTmv[t]) = Bmv(SCTmv[t])  11  and thus Bmv(SCTmv[t]) = B(SCTmv[t])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Compared to Bmv(SCTmv[t]), the  application of rmv only changes the projection bookkeeping set Bmv(SCT ′  mv[t])  by adding the elements in trans(m(LT  mv)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' The modification to Bmv(SCT ′  mv[t])  hence corresponds to the modification of the bookkeeping set B(SCT ′) by the  application of r via trans(m) for the isomorphism iso due to the construction  of rmv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  It thus holds that ∀t ∈ P : SCT ′  mv[t] = SCT ′ ∧Bmv(SCT ′  mv[t]) = B(SCT ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Based on Theorem 3 for individual rule applications, we get the following  corollary for sequences of rule applications:  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' For a TGG with associated set of forward rules R and multi-  version forward rules Rmv and a multi-version model triple SCTmv with book-  keeping and version graph V , there is a sequence of rule applications SCTmv →Rmv  SCT ′  mv if and only if for all t ∈ V V , there is a sequence of rule applications  SCTmv[t] →R SCT ′ with SCT ′  mv[t] = SCT ′∧iso(Bmv(SCT ′  mv[t])) = B(SCT ′),  where iso is an isomorphism from SCT ′  mv[t] into SCT ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' We prove the corollary by induction over the length of the multi-version  rule application sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  For the base case of application sequences of length 0, the identity morphism  and empty application sequences trivially satisfy the corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  If there is a sequence of rule applications SCTmv →Rmv SCT ′  mv if and  only if for all t ∈ V V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' there is a sequence of rule applications SCTmv[t] →R  SCT ′ with SCT ′  mv[t] = SCT ′ ∧ iso(Bmv(SCT ′  mv[t])) = B(SCT ′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' by Theorem  3 we have an extended multi-version sequence SCTmv →Rmv SCT ′  mv →rmv  m  SCT ′′  mv and all t ∈ V V if and only if for all t ∈ V V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' there is a sequence  of regular rule applications SCTmv[t] →R SCT ′′ with SCT ′′  mv[t] = SCT ′′ ∧  iso(Bmv(SCT ′′  mv[t])) = B(SCT ′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  For all t ∈ V V \\ P, where P = �  v∈V Lmv p(m(v)) ∩ �  v∈V LT  mv u(m(v)), the  corresponding regular rule application sequence SCTmv[t] →R SCT ′ and iso-  morphism iso : SCT ′  mv[t] → SCT ′ are also valid for SCT ′′  mv[t] and satisfy the  condition on bookkeeping sets, since SCT ′ = SCT ′  mv[t] = SCT ′′  mv[t] (up to  isomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  In accordance with Theorem 3, there is an extended sequence SCTmv →Rmv  SCT ′  mv →rmv  m  SCT ′′  mv if and only if for all t ∈ P, the regular rule applica-  tion sequence SCTmv[t] →R SCT ′  mv[t] can be extended by a rule application  SCT ′  mv[t] →r  trans(m) SCT ′′  mv[t] that satisfies the condition on bookkeeping sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Thus, there is a sequence of rule applications SCTmv →Rmv SCT ′  mv →rmv  m  SCT ′′  mv if and only if for all t ∈ V V , there is a sequence of rule applica-  tions SCTmv[t] →R SCT ′′ with SCT ′′  mv[t] = SCT ′′ ∧ iso(Bmv(SCT ′′  mv[t])) =  B(SCT ′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  With the proof for the base case and the induction step, we have proven the  validity of the corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  12  Intuitively, the multi-version forward rules perform an interleaved, parallel  transformation of all model versions encoded in SCTmv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' The application of a  multi-version rule Lmv ← Kmv → Rmv corresponds to the application of the  original rule to all model versions in P = �  v∈V Lmv p(m(v)) ∩ �  v∈V LT  mv u(m(v))  and leaves all other model versions unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Thus, a multi-version rule appli-  cation effectively extends the corresponding original rule application sequences  for all versions in P by the associated original rule application, whereas it rep-  resents the “skipping” of a step in the sequences of all versions not in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  For a deterministic TGG, a correct translation of source graph S is given by  any maximal rule application sequence of forward rules that deletes all book-  keeping edges in the source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Note that because of the determinism crite-  rion, either every maximal rule application sequences or none of them satisfies  the bookkeeping criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Correctness of the joint translation of all individual  versions via multi-version forward rules is hence given by the following corollary:  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' For a TGG with associated set of forward rules R and multi-  version forward rules Rmv and a multi-version model triple SCTmv with book-  keeping and version graph V , there is a maximal sequence of rule applications  SCTmv →Rmv SCT ′  mv if and only if for all t ∈ V V , there is a maximal se-  quence of regular rule applications SCTmv[t] →R SCT ′ such that SCT ′  mv[t] =  SCT ′ ∧ Bmv(SCT ′  mv[t]) = B(SCT ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' The existence of a sequence of original rule applications for a sequence of  multi-version rule applications and all versions t ∈ V V and vice-versa is given by  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' From Theorem 2, it follows directly that the multi-version sequence  is maximal if and only if the regular sequences are maximal for all t ∈ V V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Thus, for a deterministic TGG and by corollaries 1 and 3, the result of  repeated application of adapted transformation rules to a multi-version model  prepared for multi-version translation until a fixpoint is reached is equivalent  to the results of repeated application of the original rules to the individual  model versions prepared for translation, that is, the results of transforming the  individual model versions using the TGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  We thereby have the correctness of the forward transformation using multi-  version forward rules transF  mv, which applies multi-version forward rules to a  multi-version model with bookkeeping until a fixpoint is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' For a correct version history ∆M{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=',n} and a triple graph gram-  mar with set of forward rules R, it holds up to isomorphism that  ∀t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=', n} : transF  mv(initF (comb(∆M{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=',n})), adapt(R))[t] = transF (Mt, R)  (1)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Follows directly from Theorem 1 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  13  5  Evaluation  In order to evaluate our approach empirically with respect to execution time  performance, we have realized the presented concepts in our MoTE2 tool [12] for  TGG-based model transformation, which is implemented in the context of the  Java-based Eclipse Modeling Framework [7] and has been shown to be efficient  compared to other existing model transformation tools [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  As an application scenario, we consider the transformation of Java abstract  syntax trees to class diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' We have therefore modeled this transformation  as a TGG with MoTE2 and use the original and our adapted implementation  to automatically derive the required forward rules respectively multi-version  forward rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  To obtain realistic source models, we have extracted the version history of  one small personal Java project (rete, around 50 versions) and one larger open  source Java project (henshin [1], around 2000 versions) from their respective  GitHub repositories and have constructed the corresponding history of abstract  syntax trees using the MoDisco tool [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' As input for the solution presented in  Sections 3 and 4, we have consolidated both version histories into multi-version  models using a mapping based on hierarchy and naming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Our implementation and the employed datasets are available under [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Based on this,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' we run the following model transformations for both reposi-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='tories and measure the overall execution time1 for each of them:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='SVM: individual forward transformation of all model versions (abstract  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='syntax trees) in the repository using the original MoTE2 implementation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='MVM: joint forward transformation of all model versions in the reposi-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='tory using a multi-version model encoding and our implementation of the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='technique presented in Sections 3 and 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='Note that the SVM strategy would require initial projection operations and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='a final combination of transformation results to work within the framework of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='multi-version models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' However, for fairness of comparison of the transformation,  we do not consider these additional operations in our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Figure 7 shows the execution times of the transformations using the two  strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  For both repositories, the transformation based on multi-version  models requires substantially less time than the transformation of the individ-  ual model versions using the original MoTE2 tool, with a more pronounced  improvement for the larger repository (factor 4 for the smaller and factor 74 for  the larger repository).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  The improvement in efficiency and scalability can likely be explained by two  factors: First, SVM has to perform a somewhat expensive initialization step for  every indidvidual model version that is to be transformed, whereas MVM only  1All experiments were performed on a Linux SMP Debian 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='67-2 machine with Intel  Xeon E5-2630 CPU (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='3 GHz clock rate) and 386 GB system memory running OpenJDK  version 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Reported execution time measurements correspond to the mean execution  time of 10 runs of the respective experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  14  1  10  100  1000  10000  100000  1000000  10000000  rete  henshin  execution time (ms)  Transformation Time  SVM  MVM  Figure 7: execution time measurements for the transformation of all model  versions in two different software repositories (logarithmic axis)  requires one such initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Second, many elements in the abstract syntax  trees of the repositories are shared between many versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' SVM has to perform  a separate transformation, including separate pattern matching, for each model  version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' In contrast, MVM only performs a transformation including pattern  matching over a single multi-version model, the size of which is much smaller  than the combined sizes of the encoded model versions, along with efficient  search operations over the version graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Since pattern matching is efficient in  this example, that is, pattern matching has a runtime complexity that is linear  in the size of the model for the derived forward rules, this results in an improved  overall efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Threats to the internal validity of our experimental results include unex-  pected behavior of the Java virtual machine such as garbage collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  To  address this threat, we have performed multiple runs of all experiments and re-  port the mean execution time of these runs, with the standard deviation always  below 5% of the execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' To minimize the impact of the concrete imple-  mentation on our measurements, we have realized our solution in the framework  of the transformation tool we use for comparison and thereby largely use the  same execution mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  To mitigate threats to external validity, we use real-world models as the  source models of the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' However, we remark that our results are  not necessarily generalizable to different examples or application domains and  make no quantitative claims regarding the performance of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  6  Related Work  The general problem of model versioning has already been studied extensively,  both formally [5, 17] and in the form of concrete tool implementations [14, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Several solutions employ a unified representation of a model’s version history  15  similar to multi-version models [17, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' However, due to the problem definition  focusing on the management of different versions of a single model, realising  model transformation based on a unified encoding is out of scope for these  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  There is also a significant body of previous work on synchronization of con-  currently modified pairs of models using triple graph grammars [24, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' The  focus of these works is the derivation of compatible versions of source and tar-  get model that respect the modifications to either of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' This report aims to make  a step in an orthogonal direction, namely towards allowing living with incon-  sistencies by enabling developers to temporarily work with multiple modified,  possibly conflicting versions of source and target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  In the context of software product lines, so-called 150% models are em-  ployed to encode different configurations of a software system [4, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' In this  context, Greiner and Westfechtel present an approach for propagating so-called  variability annotations along trace links created by model transformations [23],  explicitly considering the case of transformations implemented via triple graph  grammars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' A similar approach could also be employed to propagate versioning  information and would have the advantage of not requiring any adaptation of  the employed rules, type graph, or transformation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' However, not inte-  grating this propagation with the transformation process and only propagating  versioning information after the transformation has been executed would mean  that certain cases that are covered by our approach could not be handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' The  occurence of such cases may hence prevent a possible correct transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' For  instance, under standard TGG semantics, such cases include a model element  being translated differently in different model versions based on its context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  In previous work in our group, the joint execution of queries over multiple  versions of an evolving model has been considered for both the case with [2] and  without [11, 18] parallel, branching development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' This report builds on these  results, but instead of focusing on pure queries without side-effects considers  the case of writing operations in the form of model transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  7  Conclusion  In this report, we have presented a first step in the direction of model transfor-  mation on multi-version models in the form of an adaptation of the well-known  triple graph grammar formalism that enables the joint transformation of all ver-  sions encoded in a multi-version model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' The presented approach is correct with  respect to the translation semantics of deterministic triple graph grammars for  individual model versions, that is, it produces equivalent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Initial exper-  iments for evaluating the efficiency of our approach demonstrate that our tech-  nique can improve performance compared to a na¨ıve realization, which simply  translates all model versions individually according to a triple graph grammar  specification, in a realistic application scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  In future work, we plan to build on the presented approach to realize model  synchronization for multi-version models, that is, incremental propagation of  16  changes to one or more versions of a source model to the corresponding target  model versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' Furthermore, we want to explore the possibility of improving  the efficiency of multi-version model transformations via incremental pattern  matching for multi-version models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Another interesting direction for future  work is the integration of advanced application conditions for the specification  of triple graph grammar rules such as nested graph conditions into our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Finally, a more extensive evaluation can be conducted to study the scalability  of the presented technique in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  Acknowledgements  This work was developed mainly in the course of the project modular and incre-  mental Global Model Management (project number 336677879), which is funded  by the Deutsche Forschungsgemeinschaft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
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+page_content=' Last accessed 14  October 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
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+page_content='  [24]  Yingfei Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' “Synchronizing concurrent model updates based on  bidirectional transformation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' In: Software & Systems Modeling 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='1 (2013),  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content=' 89–104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
+page_content='  19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfu_l-/content/2301.00623v1.pdf'}
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+arXiv:2301.00165v1  [math.AP]  31 Dec 2022
+EFFECTIVE VISCOSITY OF SEMI-DILUTE SUSPENSIONS
+MITIA DUERINCKX AND ANTOINE GLORIA
+Abstract. This review is devoted to the large-scale rheology of suspensions of rigid
+particles in Stokes fluid. After describing recent results on the definition of the effective
+viscosity of such systems in the framework of homogenization theory, we turn to our new
+results on the asymptotic expansion of the effective viscosity in the dilute regime. This
+includes a new optimal proof of Einstein’s viscosity formula for the first-order expansion,
+as well as the continuation of this expansion to higher orders. The essential difficulty orig-
+inates in the long-range nature of hydrodynamic interactions: suitable renormalizations
+are needed and are captured by means of diagrammatic expansions.
+1. Introduction
+Suspensions of rigid particles in fluids are omnipresent in natural phenomena and in
+practical applications. They are known to display complex rheological behaviors on large
+scales, including possible non-Newtonian effects, e.g. [22], which we aim to understand
+and describe from a rigorous micro-macro perspective. More precisely, we consider the
+macroscopic limit for a large number N ≫ 1 of small particles of size ε ≪ 1 in a given
+tank U ⊂ Rd. Neglecting both particle and fluid inertia, we assume that particles follow
+the fluid velocity and that the latter is instantaneously determined by the steady Stokes
+equations with no-slip conditions at particle boundaries. Denoting by {In
+ε,N}n the set of
+particles in U, with respective barycenters {xn
+ε,N}n and orientations {rn
+ε,N}n, the steady
+Stokes equations for the fluid velocity uε,N ∈ H1
+0(U)d take the form
+− △uε,N + ∇pε,N = h,
+div(uε,N) = 0,
+in U \ ∪nIn
+ε,N,
+(1.1)
+where h ∈ L2(U)d stands for some internal force in U. In view of no-slip conditions, we
+implicitly extend the fluid velocity uε,N inside the particles, where it corresponds to the
+velocity of the particles. The rigidity of the particles then translates into
+D(uε,N) = 0,
+in ∪nIn
+ε,N,
+(1.2)
+where D(u) = 1
+2(∇u + (∇u)′) stands for the symmetric gradient. Equivalently, this means
+that for all n,
+un
+ε,N = V n
+ε,N + W n
+ε,N(x − xn
+ε,N),
+in In
+ε,N,
+for some translational velocity V n
+ε,N ∈ Rd and angular velocity tensor W n
+ε,N ∈ Rd×d
+skew. As
+we neglect the inertia of the particles, Newton’s equations of motion reduce to the balance
+of forces and torques, which take the form of complementary boundary conditions,
+εe +
+ffl
+∂In
+ε,N σ(uε,N, pε,N)ν
+=
+0,
+ffl
+∂In
+ε,N (x − xn
+ε,N) × σ(uε,N, pε,N)ν
+=
+0,
+for all n,
+(1.3)
+where εe ∈ Rd stands for some (weak) sedimentation force and σ(u, p) := 2 D(u) − p Id
+is the Cauchy stress tensor.
+Given particle positions {In
+ε,N}n, the instantaneous fluid
+1
+
+2
+M. DUERINCKX AND A. GLORIA
+velocity uε,N is obtained as the unique solution of the above Stokes problem (1.1)–(1.3).
+Particle positions and orientations are then updated according to
+∂txn
+ε,N = V n
+ε,N,
+∂trn
+ε,N = W n
+ε,Nrn
+ε,N,
+for all n.
+In this way, particles interact via the fluid flow that they generate, and the resulting
+dynamics is reputedly complex in view of the multi-body, long-range, and singular nature
+of these hydrodynamic interactions.
+Heuristically, while particles constitute small rigid inclusions in the fluid and hinder its
+flow, we expect homogenization to hold on large scales, leading to a notion of effective
+viscosity for the suspension.
+This effective viscosity naturally depends on the spatial
+arrangement of the particles, i.e. on the microstructure, which evolves with the fluid
+flow and can thus adapt in time to external forces.
+This creates a possibly nonlinear
+response, hence non-Newtonian effects, which are indeed well-known in applications (e.g.
+shear thinning of suspensions like ketchup).
+The mathematical understanding of such
+behaviors requires to couple homogenization with microstructure dynamics, which is still
+a fascinating open problem. Taking inspiration from the physics literature, we focus here
+on semi-dilute regimes and split the analysis into three steps:
+• ‘Instantaneous’ effective viscosity:
+Given particle positions {In
+ε,N}n, we expect the Stokes problem (1.1)–(1.3) defining
+the fluid velocity to be approximated on large scales by an effective Stokes problem
+with some effective viscosity ¯B. This is by now well-understood in the framework of
+homogenization theory, and we review in Section 2 our main results [10, 7, 9, 12] on the
+topic. Keeping in mind the question of coupling homogenization with microstructure
+dynamics, we emphasize the importance of proving homogenization under the weakest
+possible assumptions on the microstructure.
+• Semi-dilute expansion of the effective viscosity:
+In the dilute regime, particles are sparse and interact little, hence the details of the mi-
+crostructure should no longer be so relevant: we expect to expand the effective viscosity
+as ¯B = Id +ϕ ¯B(1) + . . . at low volume fraction ϕ ≪ 1, where the different terms would
+involve only reduced information on the microstructure that would be easier to track
+along the dynamics. To first order, this expansion is the celebrated Einstein formula,
+which has attracted considerable interest recently in the mathematical community. In
+Section 3, we describe our new work [11] on the topic.
+• Coupling to semi-dilute microstructure dynamics:
+Coupling homogenization to microstructure dynamics remains an open problem even in
+the semi-dilute regime. Only partial results are available for now, limited to mean-field
+regimes so dilute that they miss the description of non-Newtonian effects, cf. [25, 28, 26].
+This is the subject of ongoing work that will not be discussed further in this review.
+2. Effective viscosity problem
+We review recent results on the homogenization of the steady Stokes problem (1.1)–(1.3)
+for given particle positions {In
+ε,N}n. More precisely, we shall consider a random ensemble
+for the set of particles, which is obtained by ε-rescaling of a given ensemble: given a
+random family {In}n of disjoint bounded subsets of Rd, with respective barycenters {xn}n,
+we consider the following random set of rescaled inclusions in the reference domain U,
+Iε(U) := �
+n∈Nε(U) εIn,
+Nε(U) := {n : εIn + εB ⊂ U},
+(2.1)
+
+EFFECTIVE VISCOSITY OF SEMI-DILUTE SUSPENSIONS
+3
+where B stands for the unit ball, and we then consider the solution uε ∈ H1
+0(U)d of the
+corresponding Stokes problem
+
+
+
+
+
+
+
+−△uε + ∇pε = h, div(uε) = 0,
+in U \ Iε(U),
+D(uε) = 0,
+in Iε(U),
+|εIn|e +
+´
+ε∂In σ(uε, pε)ν = 0,
+∀n ∈ Nε(U),
+´
+ε∂In(x − εxn) × σ(uε, pε)ν = 0,
+∀n ∈ Nε(U).
+(2.2)
+2.1. Structure of the equations. Before moving on to the effective viscosity problem,
+we briefly comment on the Stokes problem (2.2) and emphasize that the different boundary
+conditions are natural for rigid particles: indeed, the weak formulation of (2.2) takes on
+the simple guise
+2
+ˆ
+U
+D(w) : D(uε) =
+ˆ
+U
+w ·
+�
+h1U\Iε(U) + e1Iε(U)
+�
+,
+for all w ∈ H1
+0(U)d with div(w) = 0 and D(w)|Iε(U) = 0.
+Here, h1U\Iε(U)+e1Iε(U) corresponds to the total force, combining the internal force in the
+fluid domain and the (weak) sedimentation force on the particles. This weak formulation
+allows to represent
+uε = πεvε,
+(2.3)
+where vε ∈ H1
+0(U)d is the solution of the following Stokes problem without rigid particles,
+−△vε + ∇qε = h1U\Iε(U) + e1Iε(U),
+div(vε) = 0,
+in U,
+and where πε stands for the orthogonal projection
+πε : H1
+0(U)d ։
+�
+w ∈ H1
+0(U)d : div(w) = 0, D(w)|Iε(U) = 0
+�
+,
+with respect to the scalar product (v, w) =
+´
+U D(v) : D(w). Another useful way to think of
+the above equations is to view rigid particles as droplets of diverging viscosity, cf. e.g. [7]:
+we have uθ
+ε ⇀ uε in H1
+0(U)d as θ ↑ ∞, where uθ
+ε ∈ H1
+0(U)d is the solution of the following
+Stokes problem with droplets of viscosity θ,
+− div
+�
+2(1 + θ1Iε(U)) D(uθ
+ε)
+�
++ ∇pθ
+ε = h1U\Iε(U) + e1Iε(U),
+div(uθ
+ε) = 0,
+in U. (2.4)
+2.2. Homogenization. In the macroscopic limit ε ↓ 0, provided that the ensemble of
+particles {In}n is stationary and ergodic, we expect the fluid velocity to be approximated
+by some effective Stokes problem, uε ⇀ ¯u in H1
+0(U)d,
+− div(2 ¯B D(¯u)) + ∇¯p = (1 − ϕ)h + ϕe,
+div(¯u) = 0,
+in U,
+(2.5)
+where ¯B is the effective viscosity tensor and where (1 − ϕ)h + ϕe is the weak limit of the
+total force h1U\Iε(U) + e1Iε(U), in terms of the particle volume fraction ϕ = E [1∪nIn].
+The effective viscosity is naturally computed in terms of a cell problem: given a strain
+rate E ∈ Rd×d
+sym with tr(E) = 0, we consider the correction ΨE := Ex + ψE of the linear
+straining flow Ex that satisfies the following whole-space Stokes problem associated with
+the infinite family {In}n of rigid particles,
+
+
+
+
+
+
+
+−△ΨE + ∇ΣE = 0, div(ΨE) = 0,
+in Rd \ ∪nIn,
+D(ΨE) = 0,
+in ∪nIn,
+´
+∂In σ(ΨE, ΣE)ν = 0,
+∀n,
+´
+∂In(x − xn) × σ(ΨE, ΣE)ν = 0,
+∀n,
+(2.6)
+
+4
+M. DUERINCKX AND A. GLORIA
+and the effective viscosity in direction E is then given by the averaged dissipation rate
+E : ¯BE = E
+�
+|D(ψE) + E|2�
+.
+(2.7)
+As usual for cell problems in stochastic homogenization, the corrector ψE solving the
+infinite-volume problem (2.6) is to be constructed in the class of random fields ψ that are
+almost surely in H1
+loc(Rd)d such that the gradient ∇ψ is stationary, centered E [∇ψ] = 0,
+and has finite second moments E[|∇ψ|2] < ∞. In particular, this entails that ψE is a.s.
+sublinear at infinity and is thus indeed a “correction” to the linear straining flow Ex.
+Before our first work [10] on this problem, the qualitative justification of this homog-
+enization limit was still missing, surprisingly even in the periodic setting, as it required
+to deal at the same time with rigidity constraints and with incompressibility. Clearly, ho-
+mogenization cannot hold without further geometric assumptions on the set of particles: if
+particles were to form an infinite chain, it would create macroscopic rigidity and ¯B would
+be infinite in the corresponding direction. The simplest way to prohibit such behaviors is
+the following, which has been largely used in previous work on the topic:
+(H1) Uniform condition: assume ρn := infm:m̸=n dist(In, Im) ≥ ρ > 0 a.s. for all n.
+This condition is quite restrictive and would not be preserved under microstructure dy-
+namics, which motivates further generalizations. In [7], we showed that it can be replaced
+by a finer moment condition on interparticle distances:
+(H2) Moment condition: assume E[�
+n ρ−r0
+n
+1In] < ∞ for some r0 = r0(d) large enough.
+(We can choose r0(d) = 3
+2 in dimension d = 3, and r0(d) = 0 for d > 5.)
+This condition is still not preserved under microstructure dynamics in 3D, cf. [5].
+To
+weaken it further, the main difficulty is that we have poor control of the pressure in the
+vicinity of close particles, which hinders the standard proof of homogenization by Tartar’s
+oscillating test function method, cf. [7]. Intuitively, however, the presence of close particles
+should not be problematic for the homogenization result: only long chains of close particles
+would be. In [9], we finally succeeded in avoiding the need for any detailed control at close
+particles, using instead a variational Γ-convergence approach. It holds under the following
+subcritical percolation type condition, which is essentially necessary for homogenization.
+(H3) Cluster condition: Given some ρ > 0, let {Kq}q be the family of connected compo-
+nents of the fattened set �
+n In + ρB, and assume that E[�
+q(diam Kq)r01Kq] < ∞
+for some r0 = r0(d) large enough.
+We can now state homogenization under any of the above conditions (H1)–(H3). Note
+that homogenization of stiff inclusions under (H3) is new even in the scalar setting.
+Theorem 1 (Qualitative homogenization; see [10, 7, 9]). Assume that the ensemble of
+particles {In}n is stationary and ergodic, that particles are disjoint, uniformly bounded
+by In ⊂ B(xn), and uniformly C2 almost surely. Further assume that either (H1), (H2),
+or (H3) holds.
+Then the effective viscosity (2.7) is well-defined and we have uε ⇀ ¯u
+in H1
+0(U)d, where ¯u satisfies the effective Stokes problem (2.5).
+♦
+2.3. Further results and extensions. In [12], under suitable mixing assumptions for
+the ensemble of particles {In}n, we further prove quantitative error estimates for the
+homogenization limit, in form of
+��uε − ¯u − ε �
+E ψE( ·
+ε)∂E ¯u
+��
+L2(Ω;H1(U)) ≲h ε
+1
+2 ×
+� 1
+:
+d > 2,
+|log ε|
+1
+4
+:
+d = 2,
+
+EFFECTIVE VISCOSITY OF SEMI-DILUTE SUSPENSIONS
+5
+where �
+E runs over an orthonormal basis of {E ∈ Rd×d
+sym : tr(E) = 0}. We also obtain
+large-scale regularity results for the heterogeneous Stokes problem (2.2), which play a key
+role in our work on sedimentation [13]. This builds on the quantitative homogenization
+theory recently developed for divergence-form uniformly elliptic problems, e.g. [2, 21]. A
+particular difficulty in our setting is that the field/flux structure of the Stokes prob-
+lem is apparently destroyed due to rigid inclusions, in the sense that the na¨ıve flux
+σ(ΨE, ΣE)1Rd\∪nIn is not divergence-free globally: instead, we take advantage of the rep-
+resentation (2.4) to recover the relevant notion of flux; see [7, Remark 4.2].
+We also mention [6] (see also [20]), where we generalize the above homogenization results
+to the case of suspensions of active swimmers. Swimming particles generate dipole forces
+in the fluid at small scales, which can lead to surprising rheological effects, such as a
+possible drastic reduction of the effective viscosity [30].
+3. Semi-dilute expansion
+While the effective viscosity ¯B depends on the full microstructure {In}n, cf. (2.7), we
+expect this dependence to simplify perturbatively in the dilute regime. This idea takes
+its roots in the so-called effective medium expansions that emerged in the second half of
+the 19th century in the physics community; see e.g. the historical account in [27]. For the
+effective viscosity problem, this was pioneered by Einstein [15] in his PhD thesis, where he
+predicted that the first correction to the plain fluid viscosity is proportional to the particle
+volume fraction ϕ and is given by the universal formula
+¯B = Id +ϕd+2
+2 Id +o(ϕ),
+as ϕ ↓ 0,
+(3.1)
+in case of spherical particles. Formally, this is obtained by summing single-particle con-
+tributions to the effective viscosity. The rigorous justification has attracted considerable
+interest in the mathematical community over the last decade, e.g. [1, 23, 29, 24, 18], but
+two main questions have remained open:
+— While previous contributions start from the (unphysical) uniform condition (H1) on
+interparticle distances, can one also establish Einstein’s formula (3.1) under weaker
+assumptions of the form (H2) or (H3)?
+— What is the optimal error bound in (3.1)? In particular, under what minimal assump-
+tion is the error indeed o(ϕ)?
+These questions are fully answered in our recent work [11], as described in Section 3.3
+below. In link with the error bound in (3.1), there has also been interest in describing
+the next-order correction to Einstein’s formula. Formally, the correction corresponds to
+summing pair contributions and the main difficulty is that this sum is not absolutely
+convergent: a renormalization is needed and a formal understanding was first achieved
+by Batchelor and Green [4]. A few rigorous contributions have recently been devoted to
+this topic [17, 19, 16], although limited to special regimes. The general description of
+higher-order corrections to Einstein’s formula was first achieved in our recent work [11]
+in form of a cluster expansion, as described in Sections 3.4–3.5 below. In the sequel, we
+focus on spherical particles In = B(xn) for simplicity, although it is not essential.
+3.1. Cluster expansion. In the dilute regime, particles are typically well-separated and
+their interactions can thus formally be neglected perturbatively in the corrector prob-
+lem (2.6). The so-called cluster expansion corresponds to summing contributions of inter-
+actions between subsets of particles (or ‘clusters’) of increasing cardinality. For a finite
+
+6
+M. DUERINCKX AND A. GLORIA
+index set S ⊂ Rd, let ψS
+E be the solution of the corrector problem (2.6) associated with
+the finite set of particles {B(x)}x∈S; in particular note that ψ∅
+E = 0. To first order, we
+then write
+E : ¯BE ∼ |E|2 + ≪ E
+� �
+n
+�
+|D(ψ{xn}
+E
+) + E|2 − |E|2��
+≫ + . . .
+(We use quotation marks to indicate that this quantity is a priori not well defined, as
+indeed explained below.) Using the short-hand notation
+δ{xn}|D(ψ#
+E ) + E|2 := |D(ψ{xn}
+E
+) + E|2 − |E|2
+for the first-order difference, and similarly defining higher-order differences, the formal
+cluster expansion takes the form
+¯B ∼ Id +
+∞
+�
+k=1
+¯B(k),
+E : ¯B(k)E =
+≪ E
+�
+�
+S⊂{xn}n
+♯S=k
+δS|D(ψ#
+E ) + E|2
+�
+≫.
+(3.2)
+The main difficulty is that for all k ≥ 1 the kth cluster term ¯B(k) is given by a series that
+is not absolutely convergent due to the long-range nature of hydrodynamic interactions.
+More precisely, to first order, we have
+δ{xn}|D(ψ#
+E ) + E|2 = 2E : D(ψ{xn}
+E
+) + |D(ψ{xn}
+E
+)|2,
+(3.3)
+where the single-particle solution satisfies |D(ψ{xn}
+E
+)| ≃ ⟨· − xn⟩−d, which entails that the
+definition of the first cluster term is indeed not absolutely convergent,
+E
+� �
+n
+��δ{xn}|D(ψ#
+E ) + E|2��
+�
+= ∞.
+The same holds for all higher-order cluster formulas, cf. (3.2), and suitable renormalization
+procedures are thus required. Note, however, that divergence issues are only borderline
+and that cluster formulas can in fact be viewed as combinations of Calder´on–Zygmund
+kernels — this is explicit at first order in terms of the single-particle solution, cf. (3.3),
+but the structure is much more complicated and non-explicit in general.
+3.2. Correct scaling of cluster terms. As the kth cluster term ¯B(k) involves contribu-
+tions of k-tuples of particles, we might intuitively expect it to be of order O(ϕk), but we
+argue that this cannot be true in general. By definition of the k-point density fk of the
+point process, the sum over k-tuples in the cluster formula (3.2) can formally be written
+as a multiple integral with respect to fk,
+E : ¯B(k)E =
+≪
+ˆ
+(Rd)k
+�
+δ{x1,...,xk}|D(ψ#
+E ) + E|2(0)
+�
+fk(x1, . . . , xk) dx1 . . . dxk ≫,
+or equivalently, by stationarity,
+E : ¯B(k)E =
+≪
+ˆ
+(Rd)k−1
+� ˆ
+Rd δ{0,x1,...,xk}|D(ψ#
+E ) + E|2�
+× fk(0, x1, . . . , xk−1) dx1 . . . dxk−1 ≫.
+(3.4)
+Forgetting for now divergence issues at infinity, and defining the kth-order intensity of the
+point process as
+λk := ∥fk∥L∞((Rd)k)
+(3.5)
+
+EFFECTIVE VISCOSITY OF SEMI-DILUTE SUSPENSIONS
+7
+(we refer to [11, Section 1.3.2] for a more careful definition with local averages), we are
+led to expect, if the cluster formula makes any sense,
+¯B(k) = O(λk).
+In fact, due to long-range issues, the correct scaling is rather ¯B(k) = O(λk|log λk|k−1) in
+general. At first order, λ1 = λ is the intensity of the point process and is of order O(ϕ) as
+in Einstein’s formula (3.1). For a Poisson point process, due to tensorization, higher-order
+intensities are given by λk = λk, but this fails in general: for a strongly mixing point
+process, we only have
+λj ≤ λj ≤ λj−1.
+In particular, the correction to Einstein’s formula is of order O(λ2|log λ|) in general, so
+the approximation (3.1) is only valid provided λ2|log λ| = o(λ), which amounts to some
+weak form of local independence for the point process.
+3.3. Einstein’s formula. With the above notation, we can now state our new main re-
+sult on Einstein’s formula. The novelty is the optimality of the error estimate, as well as
+the generality of the result, which holds under the weakest hypotheses for which homoge-
+nization is established, cf. Theorem 1.
+Theorem 2 (Einstein’s formula; see Theorem 1 in [11]). Under the same assumptions as
+in Theorem 1, we have
+| ¯B − Id − ¯B(1)| ≲ λ2|log λ| +
+�
+0,
+under (H1),
+λ
+1− 1
+κ
+2
+λ
+1
+κ ,
+under (H2) or (H3) with κ = κ(r0, d),
+where the first cluster term ¯B
+(1) satisfies | ¯B
+(1)| ≃ λ and is given by the well-defined renor-
+malized cluster formula
+E : ¯B(1)E := E
+� �
+n
+|D(ψ(xn)
+E
+)|2
+�
+.
+(3.6)
+In case of spherical particles, we recover Einstein’s formula ¯B(1) = ϕd+2
+2 Id.
+♦
+The renormalized formula (3.6) is obtained from the cluster formula (3.2) by removing
+the linear term in (3.3). This is indeed natural: starting from (3.4) and expanding the
+difference, the cluster formula takes the form
+E : ¯B(1)E =
+≪ λ
+ˆ
+Rd 2E : D(ψ{0}
+E ) ≫ + λ
+ˆ
+Rd |D(ψ{0}
+E )|2,
+where the first linear term can only be meant to vanish as the integral of a gradient. As
+we shall see, this is to be understood for instance in a finite-volume approximation.
+We now briefly describe our proof of the above result. The idea is to neglect particle
+interactions, approximating the corrector locally by single-particle solutions, and we pro-
+ceed by energy comparison. For that purpose, we denote by {Vn}n the Voronoi tessellation
+associated with the point process {xn}n, that is,
+Vn :=
+�
+z ∈ Rd : |z − xn| <
+inf
+m:m̸=n |z − xm|
+�
+,
+and for all n we consider the solutions ψ{xn}
+E;D and ψ{xn}
+E;N of the Dirichlet and Neumann
+problems in Vn, respectively, associated with the single particle In. Comparing energies in
+
+8
+M. DUERINCKX AND A. GLORIA
+Voronoi cells, we can bound the effective viscosity from above and below by infinite-volume
+averages of Dirichlet and Neumann single-particle energies, respectively,
+�
+E : ¯BE ≤ |E|2 + limR↑∞ |BR|−1 �
+n:In⊂BR
+´
+Vn |D(ψ{n}
+E;D)|2,
+E : ¯BE ≥ |E|2 + limR↑∞ |BR|−1 �
+n:In⊂BR
+´
+Vn |D(ψ{n}
+E;N)|2.
+As whole-space single-particle energies can also be bounded from above and below by
+Dirichlet and Neumann energies, we infer
+���E : ¯BE − |E|2 − lim
+R↑∞ |RB|−1
+�
+n:In⊂RB
+ˆ
+Rd |D(ψ{n}
+E )|2���
+≤ lim
+R↑∞ |RB|−1
+�
+n:In⊂RB
+� ˆ
+Vn
+|D(ψ{n}
+E;D)|2 −
+ˆ
+Vn
+|D(ψ{n}
+E;N)|2�
+.
+By elliptic regularity, the gap between Dirichlet and Neumann energies in Vn can be con-
+trolled by O(ρ−d
+n ), so that the above is bounded by E[�
+n ρ−d
+n 1In]. This can be evaluated
+in terms of the second-order intensity, and the conclusion follows. Note that the generality
+of this proof makes it applicable to dilute systems in many other contexts; see also [14].
+3.4. Higher-order cluster expansion. The main difficulty to higher-order cluster ex-
+pansions is that there is no simple way to make sense of cluster formulas (3.2) due to
+divergence issues. A natural idea is to start by considering finite-volume approximations
+of the effective viscosity,
+E : ¯BLE := E
+�  
+QL
+|D(ψE;L) + E|2
+�
+,
+where ψE;L satisfies the corresponding corrector problem with periodic boundary condi-
+tions in the cube QL := (− 1
+2L, 1
+2L)d. The homogenization result, cf. Theorem 1, yields in
+particular ¯BL → ¯B in the infinite-volume limit L ↑ ∞. For fixed L, as there is at most a
+finite number of particles in QL, cluster formulas make sense for all k ≥ 1,
+E : ¯B(k)
+L E := E
+�
+�
+S⊂{xn}n∩QL
+♯S=k
+ 
+QL
+δS|D(ψ#
+E;L) + E|2
+�
+,
+(3.7)
+and we can investigate the associated cluster expansion. It remains to prove good enough
+estimates on the different terms to pass to the infinite-volume limit. In [11, Section 3], we
+prove two types of estimates,
+• Direct estimates: By a direct bound on cluster formulas (3.7), we can capture the scaling
+in {λk}k as in Section 3.2, but the long-range decay of hydrodynamic interactions a priori
+leads to a logarithmic divergence in the infinite-volume limit,
+| ¯B(k)
+L | ≲ λk(log L)k−1.
+(3.8)
+• Uniform-in-L estimates: Thinking of cluster formulas as complicated non-explicit com-
+binations of Calder´on–Zygmund kernels, we may expect to estimate them better by
+means of suitable energy estimates, carefully avoiding to take absolute values of the
+kernels.
+Taking inspiration from our previous work [8] on the effective conductivity
+
+EFFECTIVE VISCOSITY OF SEMI-DILUTE SUSPENSIONS
+9
+problem, this can indeed be achieved by means of a hierarchy of interpolating energy
+estimates. Yet, in this way, we necessarily miss the scaling in {λk}k,
+| ¯B(k)
+L | ≲ Ck.
+(3.9)
+As a direct consequence of uniform-in-L estimates, we can deduce that infinite-volume
+limits ¯B
+(k) := limL↑∞ ¯B
+(k)
+L
+exist, with convergence controlled by
+| ¯B(k)
+L − ¯B(k)| ≲ | ¯BL − ¯B|2−k → 0,
+as L ↑ ∞.
+(3.10)
+This is shown in the same way as the fact that a sequence of functions that converge
+uniformly and have uniformly bounded derivatives also have converging derivatives.
+It remains to interpolate between (3.8) and (3.9) to prove corresponding estimates in the
+infinite-volume limit. For this, we can think of the following shortcut: as the divergence
+in (3.8) is only logarithmic, it could be compensated by any algebraic rate in (3.10). Now,
+appealing to quantitative homogenization theory, e.g. [2], under an algebraic α-mixing
+condition, finite-volume approximations of the effective viscosity satisfy | ¯BL − ¯B| ≲ L−γ
+for some γ > 0. Combined with (3.10), this yields
+| ¯B(k)
+L − ¯B(k)| ≲ L−2−kγ,
+(3.11)
+so that (3.8) implies
+| ¯B(k)| ≲ | ¯B(k)
+L | + | ¯B(k)
+L − ¯B(k)| ≲ λk(log L)k−1 + L−2−kγ,
+and we deduce by optimization | ¯B(k)| ≲ λk|log λ|k−1. In this way, we get the following.
+Theorem 3 (Higher-order cluster expansion; see Theorem 5 in [11]). Let the same as-
+sumptions hold as in Theorem 1, assume (H1) for simplicity, and assume that the ensemble
+of particles {In}n satisfies an α-mixing condition with algebraic rate. Then, for all K ≥ 1,
+the cluster expansion of the effective viscosity holds in form of
+��� ¯B − Id −
+K
+�
+k=1
+¯B(k)��� ≲
+2K
+�
+k=K
+λk+1|log λ|k,
+(3.12)
+where cluster terms are defined by infinite-volume approximation (3.10) and satisfy
+| ¯B(k)| ≲ λk|log λ|k−1.
+♦
+3.5. Explicit renormalization. Although Theorem 3 essentially solves the problem of
+higher-order cluster expansions, it has several drawbacks:
+— Cluster terms are defined by infinite-volume approximation, cf. (3.10), which only
+provides an implicit renormalization of cluster formulas (3.2). In particular, from this
+point of view, it is unclear whether logarithmic corrections in the estimates are actually
+optimal in the above statement.
+— The above relies on α-mixing and quantitative homogenization theory as a black box,
+which seems however quite disconnected from the question of dilute expansions.
+— The convergence rate (3.11) deteriorates dramatically for large k, which is not expected
+to be optimal.
+All those criticisms led us to look for an explicit understanding of renormalization of
+cluster formulas, which was still open in the physics community beyond second order.
+
+10
+M. DUERINCKX AND A. GLORIA
+As explained after Theorem 2, the renormalization of the first-order cluster formula
+follows from the simple cancellation of the integral of a gradient in a finite-volume ap-
+proximation, in form of
+´
+QL D(ψ{0}
+E;L) = 0. At second order, the situation is already more
+complicated and we briefly recall the formal argument by Batchelor and Green [4]: the
+cluster formula (3.4) takes the form
+E : ¯B(2)E =
+≪
+ˆ
+Rd
+� ˆ
+Rd δ{0,y}|D(ψ#
+E ) + E|2�
+f2(0, y) dy ≫,
+or equivalently, after using corrector equations, cf. [11, Theorem 3],
+E : ¯B(2)E =
+≪
+ˆ
+Rd
+� ˆ
+∂B
+ψ{y}
+E
+· σ{0,y}
+E
+)ν
+�
+f2(0, y) dy ≫,
+with the short-hand notation σS
+E := σ(ψS
+E + Ex, ΣS
+E). This can now be decomposed as
+E : ¯B(2)E =
+ˆ
+Rd
+� ˆ
+∂B
+ψ{y}
+E
+·
+�
+σ{0,y}
+E
+− σ{0}
+E
+�
+ν
+�
+f2(0, y) dy
++ ≪
+ˆ
+Rd
+� ˆ
+∂B
+ψ{y}
+E
+· σ{0}
+E ν
+�
+f2(0, y) dy ≫,
+where the first integrand has pointwise decay |
+´
+∂B ψ{y}
+E
+· (σ{0,y}
+E
+− σ{0}
+E )ν| = O(⟨y⟩−2d),
+so the first integral is absolutely convergent. Now noting that the second integrand has
+vanishing integral in a finite-volume approximation,
+ˆ
+QL
+� ˆ
+∂B
+ψ{y}
+E;L · σ{0}
+E;Lν
+�
+dy =
+� ˆ
+QL
+ψ{y}
+E;L dy
+�
+·
+ˆ
+∂B
+σ{0}
+E;Lν = 0,
+(3.13)
+we can formally replace f2(0, y) by the two-point correlation h2(0, y) = f2(0, y) − λ2, to
+the effect of
+E : ¯B(2)E =
+ˆ
+Rd
+� ˆ
+∂B
+ψ{y}
+E
+·
+�
+σ{0,y}
+E
+− σ{0}
+E
+�
+ν
+�
+f2(0, y) dy
++
+ˆ
+Rd
+� ˆ
+∂B
+ψ{y}
+E
+· σ{0}
+E ν
+�
+h2(0, y) dy.
+(3.14)
+The second integral is now absolutely convergent provided that h2 has Dini decay (so that
+y �→ ⟨y⟩−d|h2(0, y)| be integrable). This renormalized formula can be justified rigorously
+and leads to a fine understanding of the second cluster term.
+Theorem 4 (Batchelor–Green renormalization; see Proposition 4.6 in [11]). Let the same
+assumptions hold as in Theorem 1, assume (H1) for simplicity, and assume that the two-
+point correlation function satisfies |h2(0, y)| ≤ C⟨y⟩−γ for some C, γ > 0. Then the second
+cluster term defined in Theorem 3 is equivalently given by (3.14). Moreover, examination
+of this renormalized formula yields the following.
+• The bound | ¯B
+(2)| ≲ λ2|log λ| is optimal in general, in the sense that it is attained by
+some point process. Yet, the logarithmic correction can be removed in some cases:
+— if the point process is isotropic, we have | ¯B
+(2)| ≲ λ2;
+— if the interparticle distance is ℓ := infn̸=m |xn − xm| ≥ 1, we have | ¯B
+(2)| ≲ ℓ−2d.
+• The convergence rate (3.11) for finite-volume approximations can be improved to
+| ¯B(2)
+L − ¯B(2)| ≲ L−γ + log L
+L .
+♦
+
+EFFECTIVE VISCOSITY OF SEMI-DILUTE SUSPENSIONS
+11
+The optimality of the logarithmic correction in the bound | ¯B
+(2)| ≲ λ2|log λ| is under-
+stood as follows: the second term in (3.14) can be written as
+´
+K(y)h2(0, y) dy for some
+Calder´on–Zygmund kernel K. As λ2 is the sup norm of h2, cf. (3.5), the validity of the
+bound | ¯B
+(2)| ≲ λ2 would require to control K∗ in L∞(Rd). This cannot be possible in
+general and we easily construct a correlation function h2 for which it fails.
+At higher orders, the renormalization of cluster formulas is more problematic as simple
+cancellations like (3.13) are no longer sufficient. Using corrector equations, the kth cluster
+formula (3.4) can be written as
+E : ¯B(k)E =
+≪ k+1
+2
+ˆ
+(Rd)k−1
+� ˆ
+∂B
+δ{x1,...,xk−1}ψ#
+E · σ{0}
+E ν
+�
+× fk(0, x1, . . . , xk−1) dx1 . . . dxk−1 ≫ + . . .
+(3.15)
+up to other similar terms that are slightly better behaved. In order to capture relevant
+cancellations, we introduce in [11] some new diagrammatic decomposition of corrector
+differences δ{x1,...,xk−1}ψ#
+E .
+Instead of using the classical method of reflections, which
+would yield an infinite series of terms only involving single-particle solutions, we introduce
+a suitable finite decomposition involving multiparticle solutions. For instance, we write
+∇δx1,x2,x3ψ(0) =
++
++
++
++
++
+,
+where the vertex
+stands for evaluation at 0, where other vertices correspond to integration
+variables x1, x2, x3, and where an edge between vertices x and y corresponds to a kernel
+with pointwise decay ⟨x − y⟩−d. More precisely, edges inside a given loop of the graph
+correspond to mutliparticle solutions (associated with the set of particles corresponding
+to the different vertices of the loop), while all the other edges correspond to single-particle
+solutions. This provides a crucial separation of variables. These diagrammatic expansions
+are obtained by iterating Green’s representation formula for corrector differences; see [11,
+Section 4.4.3] for details. We insert this decomposition into (3.15) and we further expand
+the k-point density function fk in terms of correlation functions, e.g.
+f3(0, x1, x2) = λ2 + λ
+�
+h2(0, x1) + h2(0, x2) + h2(x1, x2)
+�
++ h3(0, x1, x2).
+Assuming algebraic decay of correlations, this leads to additional couplings between the
+different integration variables, with some pointwise algebraic decay ⟨·⟩−γ, which we shall
+represent by dotted edges in the diagrams. Now the point is that our decomposition of
+corrector differences precisely allows to capture relevant cancellations: whenever a graph
+can be split into two subgraphs that are only connected by a simple edge, its contribution
+can be shown to vanish as a boundary term in the infinite-volume limit, e.g.
+,
+,
+= 0.
+Starting from (3.15) and removing all such contributions, we are left with terms that all
+correspond to absolutely convergent integrals, e.g.
+E : ¯B(4)E =
++
++
++
++
++
++
++
++
++
++ . . .
+The number of terms in such decompositions grows very quickly, which probably makes
+them useless in practice, but it leads us to the proof of the following improved version of
+Theorem 3, which is our main result in [11].
+
+12
+M. DUERINCKX AND A. GLORIA
+Theorem 5 (Higher-order cluster expansion; see Proposition 4.8 in [11]). Let the same
+assumptions hold as in Theorem 1, and assume (H1) for simplicity.
+• Given K ≥ 1, further assuming that correlation functions hK+1, . . . , h2K+1 have some
+decay C⟨·⟩−γ in each direction for some C, γ > 0 (which is implied by algebraic α-
+mixing), the cluster expansion of the effective viscosity holds in form of (3.12), where
+cluster terms are defined by infinite-volume approximation (3.10).
+• For all k ≥ 1, further assuming that the correlation function hk has some decay C⟨·⟩−γ
+in each direction for some C, γ > 0, then
+— the kth cluster term is equivalently given by a renormalized formula only involving a
+finite number of absolutely convergent integrals;
+— it satisfies the bound | ¯B(k)| ≲ λk|log λ|k−1, which is optimal in general;
+— the convergence rate (3.11) for finite-volume approximations can be improved to
+| ¯B(k)
+L − ¯B(k)| ≲ (log L)k−1
+Lγ∧1
+.
+♦
+3.6. Remark: a new elliptic regularity result. In order to prove the above, in [11], we
+make repeated use of decay properties of solutions to Stokes problems with finite numbers
+of rigid particles. More precisely, we use for instance the following mean-value property,
+which seems new and might be of independent interest.
+Theorem 6 (Mean-value property; see Appendix A in [11]). Let {In}n∈S be a finite collec-
+tion of disjoint subsets of BR, assume that they are uniformly C2 and that for some ρ > 0
+we have dist(In, Im) ≥ ρ and dist(In, ∂BR) ≥ ρ for all n ̸= m. Let u ∈ H1(BR)d satisfy
+the Stokes problem
+
+
+
+
+
+
+
+−△u + ∇p = 0, div(u) = 0,
+in BR \ ∪n∈SIn,
+D(u) = 0,
+in ∪n∈SIn,
+´
+∂In σ(u, p)ν = 0,
+∀n ∈ S,
+´
+∂In(x − xn) × σ(u, p)ν = 0,
+∀n ∈ S.
+Then we have
+ 
+B
+|∇u|2 ≤ C(ρ, ♯S)
+ 
+BR
+|∇u|2,
+where the multiplicative constant C(ρ, ♯S) only depends on ρ, ♯S, and on dimension d.
+♦
+Note that this result is false in general if ♯S = ∞, in link with classical counterexamples
+to Lipschitz regularity, but we show in [12] that a corresponding annealed estimate then
+holds upon taking a suitable ensemble average with respect to the set of particles.
+3.7. Further results and extensions. In [6], we derive similar results for the effective
+viscosity of active suspensions. In a work in preparation with Pertinand, we further study
+Batchelor’s dilute expansion [3] for the mean settling speed of a sedimenting suspension.
+The main difficulty is that, in the sedimentation problem, the long-range nature of hydro-
+dynamic interactions is even more drastic: the flow disturbance at x due to a particle at y
+then decays like ⟨x−y⟩1−d instead of ⟨x−y⟩−d. For this reason, a suitable renormalization
+is needed even to actually define the mean settling speed, which requires a nontrivial decay
+of correlations; see [13]. The dilute expansion is then particularly tricky, but the same type
+of diagrammatic decompositions proves to be of crucial use in capturing cancellations.
+
+EFFECTIVE VISCOSITY OF SEMI-DILUTE SUSPENSIONS
+13
+Acknowledgements
+MD acknowledges financial support from F.R.S.-FNRS, and AG from the European Re-
+search Council (ERC) under the European Union’s Horizon 2020 research and innovation
+programme (Grant Agreement n◦ 864066).
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+Simul. Sci. Eng. Technol., pages 1–162. Birkh¨auser Boston, Boston, MA, 2000.
+[28] A. Mecherbet. Sedimentation of particles in Stokes flow. Kinet. Relat. Models, 12(5):995–1044, 2019.
+[29] B. Niethammer and R. Schubert. A local version of Einstein’s formula for the effective viscosity of
+suspensions. SIAM J. Math. Anal., 52(3):2561–2591, 2020.
+[30] A. Sokolov and I. S. Aranson. Reduction of viscosity in suspension of swimming bacteria. Phys. Rev.
+Lett., 103(3):148101, 2009.
+(Mitia Duerinckx) Universit´e Libre de Bruxelles, D´epartement de Math´ematique, 1050 Brus-
+sels, Belgium
+Email address: mitia.duerinckx@ulb.be
+(Antoine Gloria) Sorbonne Universit´e, CNRS, Universit´e de Paris, Laboratoire Jacques-
+Louis Lions, 75005 Paris, France & Institut Universitaire de France & Universit´e Libre de
+Bruxelles, D´epartement de Math´ematique, 1050 Brussels, Belgium
+Email address: antoine.gloria@sorbonne-universite.fr
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf,len=660
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='00165v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='AP]  31 Dec 2022  EFFECTIVE VISCOSITY OF SEMI-DILUTE SUSPENSIONS  MITIA DUERINCKX AND ANTOINE GLORIA  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' This review is devoted to the large-scale rheology of suspensions of rigid  particles in Stokes fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' After describing recent results on the definition of the effective  viscosity of such systems in the framework of homogenization theory, we turn to our new  results on the asymptotic expansion of the effective viscosity in the dilute regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' This  includes a new optimal proof of Einstein’s viscosity formula for the first-order expansion,  as well as the continuation of this expansion to higher orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' The essential difficulty orig-  inates in the long-range nature of hydrodynamic interactions: suitable renormalizations  are needed and are captured by means of diagrammatic expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Introduction  Suspensions of rigid particles in fluids are omnipresent in natural phenomena and in  practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' They are known to display complex rheological behaviors on large  scales, including possible non-Newtonian effects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' [22], which we aim to understand  and describe from a rigorous micro-macro perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' More precisely, we consider the  macroscopic limit for a large number N ≫ 1 of small particles of size ε ≪ 1 in a given  tank U ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Neglecting both particle and fluid inertia, we assume that particles follow  the fluid velocity and that the latter is instantaneously determined by the steady Stokes  equations with no-slip conditions at particle boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Denoting by {In  ε,N}n the set of  particles in U, with respective barycenters {xn  ε,N}n and orientations {rn  ε,N}n, the steady  Stokes equations for the fluid velocity uε,N ∈ H1  0(U)d take the form  − △uε,N + ∇pε,N = h,  div(uε,N) = 0,  in U \\ ∪nIn  ε,N,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='1)  where h ∈ L2(U)d stands for some internal force in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' In view of no-slip conditions, we  implicitly extend the fluid velocity uε,N inside the particles, where it corresponds to the  velocity of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' The rigidity of the particles then translates into  D(uε,N) = 0,  in ∪nIn  ε,N,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='2)  where D(u) = 1  2(∇u + (∇u)′) stands for the symmetric gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Equivalently, this means  that for all n,  un  ε,N = V n  ε,N + W n  ε,N(x − xn  ε,N),  in In  ε,N,  for some translational velocity V n  ε,N ∈ Rd and angular velocity tensor W n  ε,N ∈ Rd×d  skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' As  we neglect the inertia of the particles, Newton’s equations of motion reduce to the balance  of forces and torques, which take the form of complementary boundary conditions,  εe +  ffl  ∂In  ε,N σ(uε,N, pε,N)ν  =  0,  ffl  ∂In  ε,N (x − xn  ε,N) × σ(uε,N, pε,N)ν  =  0,  for all n,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='3)  where εe ∈ Rd stands for some (weak) sedimentation force and σ(u, p) := 2 D(u) − p Id  is the Cauchy stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Given particle positions {In  ε,N}n, the instantaneous fluid  1  2  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' DUERINCKX AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' GLORIA  velocity uε,N is obtained as the unique solution of the above Stokes problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Particle positions and orientations are then updated according to  ∂txn  ε,N = V n  ε,N,  ∂trn  ε,N = W n  ε,Nrn  ε,N,  for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  In this way, particles interact via the fluid flow that they generate, and the resulting  dynamics is reputedly complex in view of the multi-body, long-range, and singular nature  of these hydrodynamic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Heuristically, while particles constitute small rigid inclusions in the fluid and hinder its  flow, we expect homogenization to hold on large scales, leading to a notion of effective  viscosity for the suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  This effective viscosity naturally depends on the spatial  arrangement of the particles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' on the microstructure, which evolves with the fluid  flow and can thus adapt in time to external forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  This creates a possibly nonlinear  response, hence non-Newtonian effects, which are indeed well-known in applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  shear thinning of suspensions like ketchup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  The mathematical understanding of such  behaviors requires to couple homogenization with microstructure dynamics, which is still  a fascinating open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Taking inspiration from the physics literature, we focus here  on semi-dilute regimes and split the analysis into three steps:  ‘Instantaneous’ effective viscosity:  Given particle positions {In  ε,N}n, we expect the Stokes problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='3) defining  the fluid velocity to be approximated on large scales by an effective Stokes problem  with some effective viscosity ¯B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' This is by now well-understood in the framework of  homogenization theory, and we review in Section 2 our main results [10, 7, 9, 12] on the  topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Keeping in mind the question of coupling homogenization with microstructure  dynamics, we emphasize the importance of proving homogenization under the weakest  possible assumptions on the microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Semi-dilute expansion of the effective viscosity:  In the dilute regime, particles are sparse and interact little, hence the details of the mi-  crostructure should no longer be so relevant: we expect to expand the effective viscosity  as ¯B = Id +ϕ ¯B(1) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' at low volume fraction ϕ ≪ 1, where the different terms would  involve only reduced information on the microstructure that would be easier to track  along the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' To first order, this expansion is the celebrated Einstein formula,  which has attracted considerable interest recently in the mathematical community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' In  Section 3, we describe our new work [11] on the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Coupling to semi-dilute microstructure dynamics:  Coupling homogenization to microstructure dynamics remains an open problem even in  the semi-dilute regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Only partial results are available for now, limited to mean-field  regimes so dilute that they miss the description of non-Newtonian effects, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' [25, 28, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  This is the subject of ongoing work that will not be discussed further in this review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Effective viscosity problem  We review recent results on the homogenization of the steady Stokes problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='3)  for given particle positions {In  ε,N}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' More precisely, we shall consider a random ensemble  for the set of particles, which is obtained by ε-rescaling of a given ensemble: given a  random family {In}n of disjoint bounded subsets of Rd, with respective barycenters {xn}n,  we consider the following random set of rescaled inclusions in the reference domain U,  Iε(U) := �  n∈Nε(U) εIn,  Nε(U) := {n : εIn + εB ⊂ U},  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='1)  EFFECTIVE VISCOSITY OF SEMI-DILUTE SUSPENSIONS  3  where B stands for the unit ball, and we then consider the solution uε ∈ H1  0(U)d of the  corresponding Stokes problem  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  −△uε + ∇pε = h, div(uε) = 0,  in U \\ Iε(U),  D(uε) = 0,  in Iε(U),  |εIn|e +  ´  ε∂In σ(uε, pε)ν = 0,  ∀n ∈ Nε(U),  ´  ε∂In(x − εxn) × σ(uε, pε)ν = 0,  ∀n ∈ Nε(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='2)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Structure of the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Before moving on to the effective viscosity problem,  we briefly comment on the Stokes problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='2) and emphasize that the different boundary  conditions are natural for rigid particles: indeed, the weak formulation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='2) takes on  the simple guise  2  ˆ  U  D(w) : D(uε) =  ˆ  U  w ·  �  h1U\\Iε(U) + e1Iε(U)  �  ,  for all w ∈ H1  0(U)d with div(w) = 0 and D(w)|Iε(U) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Here, h1U\\Iε(U)+e1Iε(U) corresponds to the total force, combining the internal force in the  fluid domain and the (weak) sedimentation force on the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' This weak formulation  allows to represent  uε = πεvε,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='3)  where vε ∈ H1  0(U)d is the solution of the following Stokes problem without rigid particles,  −△vε + ∇qε = h1U\\Iε(U) + e1Iε(U),  div(vε) = 0,  in U,  and where πε stands for the orthogonal projection  πε : H1  0(U)d ։  �  w ∈ H1  0(U)d : div(w) = 0, D(w)|Iε(U) = 0  �  ,  with respect to the scalar product (v, w) =  ´  U D(v) : D(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Another useful way to think of  the above equations is to view rigid particles as droplets of diverging viscosity, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' [7]:  we have uθ  ε ⇀ uε in H1  0(U)d as θ ↑ ∞, where uθ  ε ∈ H1  0(U)d is the solution of the following  Stokes problem with droplets of viscosity θ,  − div  �  2(1 + θ1Iε(U)) D(uθ  ε)  �  + ∇pθ  ε = h1U\\Iε(U) + e1Iε(U),  div(uθ  ε) = 0,  in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='4)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Homogenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' In the macroscopic limit ε ↓ 0, provided that the ensemble of  particles {In}n is stationary and ergodic, we expect the fluid velocity to be approximated  by some effective Stokes problem, uε ⇀ ¯u in H1  0(U)d,  − div(2 ¯B D(¯u)) + ∇¯p = (1 − ϕ)h + ϕe,  div(¯u) = 0,  in U,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='5)  where ¯B is the effective viscosity tensor and where (1 − ϕ)h + ϕe is the weak limit of the  total force h1U\\Iε(U) + e1Iε(U), in terms of the particle volume fraction ϕ = E [1∪nIn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  The effective viscosity is naturally computed in terms of a cell problem: given a strain  rate E ∈ Rd×d  sym with tr(E) = 0, we consider the correction ΨE := Ex + ψE of the linear  straining flow Ex that satisfies the following whole-space Stokes problem associated with  the infinite family {In}n of rigid particles,  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  −△ΨE + ∇ΣE = 0, div(ΨE) = 0,  in Rd \\ ∪nIn,  D(ΨE) = 0,  in ∪nIn,  ´  ∂In σ(ΨE, ΣE)ν = 0,  ∀n,  ´  ∂In(x − xn) × σ(ΨE, ΣE)ν = 0,  ∀n,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='6)  4  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' DUERINCKX AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' GLORIA  and the effective viscosity in direction E is then given by the averaged dissipation rate  E : ¯BE = E  �  |D(ψE) + E|2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='7)  As usual for cell problems in stochastic homogenization, the corrector ψE solving the  infinite-volume problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='6) is to be constructed in the class of random fields ψ that are  almost surely in H1  loc(Rd)d such that the gradient ∇ψ is stationary, centered E [∇ψ] = 0,  and has finite second moments E[|∇ψ|2] < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' In particular, this entails that ψE is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  sublinear at infinity and is thus indeed a “correction” to the linear straining flow Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Before our first work [10] on this problem, the qualitative justification of this homog-  enization limit was still missing, surprisingly even in the periodic setting, as it required  to deal at the same time with rigidity constraints and with incompressibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Clearly, ho-  mogenization cannot hold without further geometric assumptions on the set of particles: if  particles were to form an infinite chain, it would create macroscopic rigidity and ¯B would  be infinite in the corresponding direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' The simplest way to prohibit such behaviors is  the following, which has been largely used in previous work on the topic:  (H1) Uniform condition: assume ρn := infm:m̸=n dist(In, Im) ≥ ρ > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  This condition is quite restrictive and would not be preserved under microstructure dy-  namics, which motivates further generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' In [7], we showed that it can be replaced  by a finer moment condition on interparticle distances:  (H2) Moment condition: assume E[�  n ρ−r0  n  1In] < ∞ for some r0 = r0(d) large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  (We can choose r0(d) = 3  2 in dimension d = 3, and r0(d) = 0 for d > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=')  This condition is still not preserved under microstructure dynamics in 3D, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  To  weaken it further, the main difficulty is that we have poor control of the pressure in the  vicinity of close particles, which hinders the standard proof of homogenization by Tartar’s  oscillating test function method, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Intuitively, however, the presence of close particles  should not be problematic for the homogenization result: only long chains of close particles  would be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' In [9], we finally succeeded in avoiding the need for any detailed control at close  particles, using instead a variational Γ-convergence approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' It holds under the following  subcritical percolation type condition, which is essentially necessary for homogenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  (H3) Cluster condition: Given some ρ > 0, let {Kq}q be the family of connected compo-  nents of the fattened set �  n In + ρB, and assume that E[�  q(diam Kq)r01Kq] < ∞  for some r0 = r0(d) large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  We can now state homogenization under any of the above conditions (H1)–(H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Note  that homogenization of stiff inclusions under (H3) is new even in the scalar setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Theorem 1 (Qualitative homogenization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' see [10, 7, 9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Assume that the ensemble of  particles {In}n is stationary and ergodic, that particles are disjoint, uniformly bounded  by In ⊂ B(xn), and uniformly C2 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Further assume that either (H1), (H2),  or (H3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Then the effective viscosity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='7) is well-defined and we have uε ⇀ ¯u  in H1  0(U)d, where ¯u satisfies the effective Stokes problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  ♦  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Further results and extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' In [12], under suitable mixing assumptions for  the ensemble of particles {In}n, we further prove quantitative error estimates for the  homogenization limit, in form of  ��uε − ¯u − ε �  E ψE( ·  ε)∂E ¯u  ��  L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='H1(U)) ≲h ε  1  2 ×  � 1  :  d > 2,  |log ε|  1  4  :  d = 2,  EFFECTIVE VISCOSITY OF SEMI-DILUTE SUSPENSIONS  5  where �  E runs over an orthonormal basis of {E ∈ Rd×d  sym : tr(E) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' We also obtain  large-scale regularity results for the heterogeneous Stokes problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='2), which play a key  role in our work on sedimentation [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' This builds on the quantitative homogenization  theory recently developed for divergence-form uniformly elliptic problems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' [2, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' A  particular difficulty in our setting is that the field/flux structure of the Stokes prob-  lem is apparently destroyed due to rigid inclusions, in the sense that the na¨ıve flux  σ(ΨE, ΣE)1Rd\\∪nIn is not divergence-free globally: instead, we take advantage of the rep-  resentation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='4) to recover the relevant notion of flux;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' see [7, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  We also mention [6] (see also [20]), where we generalize the above homogenization results  to the case of suspensions of active swimmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Swimming particles generate dipole forces  in the fluid at small scales, which can lead to surprising rheological effects, such as a  possible drastic reduction of the effective viscosity [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Semi-dilute expansion  While the effective viscosity ¯B depends on the full microstructure {In}n, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='7), we  expect this dependence to simplify perturbatively in the dilute regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' This idea takes  its roots in the so-called effective medium expansions that emerged in the second half of  the 19th century in the physics community;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' the historical account in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' For the  effective viscosity problem, this was pioneered by Einstein [15] in his PhD thesis, where he  predicted that the first correction to the plain fluid viscosity is proportional to the particle  volume fraction ϕ and is given by the universal formula  ¯B = Id +ϕd+2  2 Id +o(ϕ),  as ϕ ↓ 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='1)  in case of spherical particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Formally, this is obtained by summing single-particle con-  tributions to the effective viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' The rigorous justification has attracted considerable  interest in the mathematical community over the last decade, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' [1, 23, 29, 24, 18], but  two main questions have remained open:  — While previous contributions start from the (unphysical) uniform condition (H1) on  interparticle distances, can one also establish Einstein’s formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='1) under weaker  assumptions of the form (H2) or (H3)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  — What is the optimal error bound in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='1)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' In particular, under what minimal assump-  tion is the error indeed o(ϕ)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  These questions are fully answered in our recent work [11], as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='3  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' In link with the error bound in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='1), there has also been interest in describing  the next-order correction to Einstein’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Formally, the correction corresponds to  summing pair contributions and the main difficulty is that this sum is not absolutely  convergent: a renormalization is needed and a formal understanding was first achieved  by Batchelor and Green [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' A few rigorous contributions have recently been devoted to  this topic [17, 19, 16], although limited to special regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' The general description of  higher-order corrections to Einstein’s formula was first achieved in our recent work [11]  in form of a cluster expansion, as described in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='4–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' In the sequel, we  focus on spherical particles In = B(xn) for simplicity, although it is not essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Cluster expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' In the dilute regime, particles are typically well-separated and  their interactions can thus formally be neglected perturbatively in the corrector prob-  lem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' The so-called cluster expansion corresponds to summing contributions of inter-  actions between subsets of particles (or ‘clusters’) of increasing cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' For a finite  6  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' DUERINCKX AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' GLORIA  index set S ⊂ Rd, let ψS  E be the solution of the corrector problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='6) associated with  the finite set of particles {B(x)}x∈S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' in particular note that ψ∅  E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' To first order, we  then write  E : ¯BE ∼ |E|2 + ≪ E  � �  n  �  |D(ψ{xn}  E  ) + E|2 − |E|2��  ≫ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  (We use quotation marks to indicate that this quantity is a priori not well defined, as  indeed explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=') Using the short-hand notation  δ{xn}|D(ψ#  E ) + E|2 := |D(ψ{xn}  E  ) + E|2 − |E|2  for the first-order difference, and similarly defining higher-order differences, the formal  cluster expansion takes the form  ¯B ∼ Id +  ∞  �  k=1  ¯B(k),  E : ¯B(k)E =  ≪ E  �  �  S⊂{xn}n  ♯S=k  δS|D(ψ#  E ) + E|2  �  ≫.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='2)  The main difficulty is that for all k ≥ 1 the kth cluster term ¯B(k) is given by a series that  is not absolutely convergent due to the long-range nature of hydrodynamic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  More precisely, to first order, we have  δ{xn}|D(ψ#  E ) + E|2 = 2E : D(ψ{xn}  E  ) + |D(ψ{xn}  E  )|2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='3)  where the single-particle solution satisfies |D(ψ{xn}  E  )| ≃ ⟨· − xn⟩−d, which entails that the  definition of the first cluster term is indeed not absolutely convergent,  E  � �  n  ��δ{xn}|D(ψ#  E ) + E|2��  �  = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  The same holds for all higher-order cluster formulas, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='2), and suitable renormalization  procedures are thus required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Note, however, that divergence issues are only borderline  and that cluster formulas can in fact be viewed as combinations of Calder´on–Zygmund  kernels — this is explicit at first order in terms of the single-particle solution, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='3),  but the structure is much more complicated and non-explicit in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Correct scaling of cluster terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' As the kth cluster term ¯B(k) involves contribu-  tions of k-tuples of particles, we might intuitively expect it to be of order O(ϕk), but we  argue that this cannot be true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' By definition of the k-point density fk of the  point process, the sum over k-tuples in the cluster formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='2) can formally be written  as a multiple integral with respect to fk,  E : ¯B(k)E =  ≪  ˆ  (Rd)k  �  δ{x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=',xk}|D(ψ#  E ) + E|2(0)  �  fk(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' , xk) dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' dxk ≫,  or equivalently, by stationarity,  E : ¯B(k)E =  ≪  ˆ  (Rd)k−1  � ˆ  Rd δ{0,x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=',xk}|D(ψ#  E ) + E|2�  × fk(0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' , xk−1) dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' dxk−1 ≫.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='4)  Forgetting for now divergence issues at infinity, and defining the kth-order intensity of the  point process as  λk := ∥fk∥L∞((Rd)k)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='5)  EFFECTIVE VISCOSITY OF SEMI-DILUTE SUSPENSIONS  7  (we refer to [11, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='2] for a more careful definition with local averages), we are  led to expect, if the cluster formula makes any sense,  ¯B(k) = O(λk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  In fact, due to long-range issues, the correct scaling is rather ¯B(k) = O(λk|log λk|k−1) in  general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' At first order, λ1 = λ is the intensity of the point process and is of order O(ϕ) as  in Einstein’s formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' For a Poisson point process, due to tensorization, higher-order  intensities are given by λk = λk, but this fails in general: for a strongly mixing point  process, we only have  λj ≤ λj ≤ λj−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  In particular, the correction to Einstein’s formula is of order O(λ2|log λ|) in general, so  the approximation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='1) is only valid provided λ2|log λ| = o(λ), which amounts to some  weak form of local independence for the point process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Einstein’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' With the above notation, we can now state our new main re-  sult on Einstein’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' The novelty is the optimality of the error estimate, as well as  the generality of the result, which holds under the weakest hypotheses for which homoge-  nization is established, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Theorem 2 (Einstein’s formula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' see Theorem 1 in [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Under the same assumptions as  in Theorem 1, we have  | ¯B − Id − ¯B(1)| ≲ λ2|log λ| +  �  0,  under (H1),  λ  1− 1  κ  2  λ  1  κ ,  under (H2) or (H3) with κ = κ(r0, d),  where the first cluster term ¯B  (1) satisfies | ¯B  (1)| ≃ λ and is given by the well-defined renor-  malized cluster formula  E : ¯B(1)E := E  � �  n  |D(ψ(xn)  E  )|2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='6)  In case of spherical particles, we recover Einstein’s formula ¯B(1) = ϕd+2  2 Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  ♦  The renormalized formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='6) is obtained from the cluster formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='2) by removing  the linear term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' This is indeed natural: starting from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='4) and expanding the  difference, the cluster formula takes the form  E : ¯B(1)E =  ≪ λ  ˆ  Rd 2E : D(ψ{0}  E ) ≫ + λ  ˆ  Rd |D(ψ{0}  E )|2,  where the first linear term can only be meant to vanish as the integral of a gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' As  we shall see, this is to be understood for instance in a finite-volume approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  We now briefly describe our proof of the above result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' The idea is to neglect particle  interactions, approximating the corrector locally by single-particle solutions, and we pro-  ceed by energy comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' For that purpose, we denote by {Vn}n the Voronoi tessellation  associated with the point process {xn}n, that is,  Vn :=  �  z ∈ Rd : |z − xn| <  inf  m:m̸=n |z − xm|  �  ,  and for all n we consider the solutions ψ{xn}  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='D and ψ{xn}  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='N of the Dirichlet and Neumann  problems in Vn, respectively, associated with the single particle In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Comparing energies in  8  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' DUERINCKX AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' GLORIA  Voronoi cells, we can bound the effective viscosity from above and below by infinite-volume  averages of Dirichlet and Neumann single-particle energies, respectively,  �  E : ¯BE ≤ |E|2 + limR↑∞ |BR|−1 �  n:In⊂BR  ´  Vn |D(ψ{n}  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='D)|2,  E : ¯BE ≥ |E|2 + limR↑∞ |BR|−1 �  n:In⊂BR  ´  Vn |D(ψ{n}  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='N)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  As whole-space single-particle energies can also be bounded from above and below by  Dirichlet and Neumann energies, we infer  ���E : ¯BE − |E|2 − lim  R↑∞ |RB|−1  �  n:In⊂RB  ˆ  Rd |D(ψ{n}  E )|2���  ≤ lim  R↑∞ |RB|−1  �  n:In⊂RB  � ˆ  Vn  |D(ψ{n}  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='D)|2 −  ˆ  Vn  |D(ψ{n}  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='N)|2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  By elliptic regularity, the gap between Dirichlet and Neumann energies in Vn can be con-  trolled by O(ρ−d  n ), so that the above is bounded by E[�  n ρ−d  n 1In].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' This can be evaluated  in terms of the second-order intensity, and the conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Note that the generality  of this proof makes it applicable to dilute systems in many other contexts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' see also [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Higher-order cluster expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' The main difficulty to higher-order cluster ex-  pansions is that there is no simple way to make sense of cluster formulas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='2) due to  divergence issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' A natural idea is to start by considering finite-volume approximations  of the effective viscosity,  E : ¯BLE := E  �  QL  |D(ψE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='L) + E|2  �  ,  where ψE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='L satisfies the corresponding corrector problem with periodic boundary condi-  tions in the cube QL := (− 1  2L, 1  2L)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' The homogenization result, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Theorem 1, yields in  particular ¯BL → ¯B in the infinite-volume limit L ↑ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' For fixed L, as there is at most a  finite number of particles in QL, cluster formulas make sense for all k ≥ 1,  E : ¯B(k)  L E := E  �  �  S⊂{xn}n∩QL  ♯S=k  QL  δS|D(ψ#  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='L) + E|2  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='7)  and we can investigate the associated cluster expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' It remains to prove good enough  estimates on the different terms to pass to the infinite-volume limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' In [11, Section 3], we  prove two types of estimates,  Direct estimates: By a direct bound on cluster formulas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='7), we can capture the scaling  in {λk}k as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='2, but the long-range decay of hydrodynamic interactions a priori  leads to a logarithmic divergence in the infinite-volume limit,  | ¯B(k)  L | ≲ λk(log L)k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='8)  Uniform-in-L estimates: Thinking of cluster formulas as complicated non-explicit com-  binations of Calder´on–Zygmund kernels, we may expect to estimate them better by  means of suitable energy estimates, carefully avoiding to take absolute values of the  kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Taking inspiration from our previous work [8] on the effective conductivity  EFFECTIVE VISCOSITY OF SEMI-DILUTE SUSPENSIONS  9  problem, this can indeed be achieved by means of a hierarchy of interpolating energy  estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Yet, in this way, we necessarily miss the scaling in {λk}k,  | ¯B(k)  L | ≲ Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='9)  As a direct consequence of uniform-in-L estimates, we can deduce that infinite-volume  limits ¯B  (k) := limL↑∞ ¯B  (k)  L  exist, with convergence controlled by  | ¯B(k)  L − ¯B(k)| ≲ | ¯BL − ¯B|2−k → 0,  as L ↑ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='10)  This is shown in the same way as the fact that a sequence of functions that converge  uniformly and have uniformly bounded derivatives also have converging derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  It remains to interpolate between (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='9) to prove corresponding estimates in the  infinite-volume limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' For this, we can think of the following shortcut: as the divergence  in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='8) is only logarithmic, it could be compensated by any algebraic rate in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Now,  appealing to quantitative homogenization theory, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' [2], under an algebraic α-mixing  condition, finite-volume approximations of the effective viscosity satisfy | ¯BL − ¯B| ≲ L−γ  for some γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Combined with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='10), this yields  | ¯B(k)  L − ¯B(k)| ≲ L−2−kγ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='11)  so that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='8) implies  | ¯B(k)| ≲ | ¯B(k)  L | + | ¯B(k)  L − ¯B(k)| ≲ λk(log L)k−1 + L−2−kγ,  and we deduce by optimization | ¯B(k)| ≲ λk|log λ|k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' In this way, we get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Theorem 3 (Higher-order cluster expansion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' see Theorem 5 in [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Let the same as-  sumptions hold as in Theorem 1, assume (H1) for simplicity, and assume that the ensemble  of particles {In}n satisfies an α-mixing condition with algebraic rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Then, for all K ≥ 1,  the cluster expansion of the effective viscosity holds in form of  ��� ¯B − Id −  K  �  k=1  ¯B(k)��� ≲  2K  �  k=K  λk+1|log λ|k,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='12)  where cluster terms are defined by infinite-volume approximation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='10) and satisfy  | ¯B(k)| ≲ λk|log λ|k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  ♦  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Explicit renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Although Theorem 3 essentially solves the problem of  higher-order cluster expansions, it has several drawbacks:  — Cluster terms are defined by infinite-volume approximation, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='10), which only  provides an implicit renormalization of cluster formulas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' In particular, from this  point of view, it is unclear whether logarithmic corrections in the estimates are actually  optimal in the above statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  — The above relies on α-mixing and quantitative homogenization theory as a black box,  which seems however quite disconnected from the question of dilute expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  — The convergence rate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='11) deteriorates dramatically for large k, which is not expected  to be optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  All those criticisms led us to look for an explicit understanding of renormalization of  cluster formulas, which was still open in the physics community beyond second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  10  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' DUERINCKX AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' GLORIA  As explained after Theorem 2, the renormalization of the first-order cluster formula  follows from the simple cancellation of the integral of a gradient in a finite-volume ap-  proximation, in form of  ´  QL D(ψ{0}  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' At second order, the situation is already more  complicated and we briefly recall the formal argument by Batchelor and Green [4]: the  cluster formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='4) takes the form  E : ¯B(2)E =  ≪  ˆ  Rd  � ˆ  Rd δ{0,y}|D(ψ#  E ) + E|2�  f2(0, y) dy ≫,  or equivalently, after using corrector equations, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' [11, Theorem 3],  E : ¯B(2)E =  ≪  ˆ  Rd  � ˆ  ∂B  ψ{y}  E  σ{0,y}  E  )ν  �  f2(0, y) dy ≫,  with the short-hand notation σS  E := σ(ψS  E + Ex, ΣS  E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' This can now be decomposed as  E : ¯B(2)E =  ˆ  Rd  � ˆ  ∂B  ψ{y}  E �  σ{0,y}  E  − σ{0}  E  �  ν  �  f2(0, y) dy  + ≪  ˆ  Rd  � ˆ  ∂B  ψ{y}  E  σ{0}  E ν  �  f2(0, y) dy ≫,  where the first integrand has pointwise decay |  ´  ∂B ψ{y}  E  (σ{0,y}  E  − σ{0}  E )ν| = O(⟨y⟩−2d),  so the first integral is absolutely convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Now noting that the second integrand has  vanishing integral in a finite-volume approximation,  ˆ  QL  � ˆ  ∂B  ψ{y}  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='L · σ{0}  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='Lν  �  dy =  � ˆ  QL  ψ{y}  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='L dy  � ˆ  ∂B  σ{0}  E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='Lν = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='13)  we can formally replace f2(0, y) by the two-point correlation h2(0, y) = f2(0, y) − λ2, to  the effect of  E : ¯B(2)E =  ˆ  Rd  � ˆ  ∂B  ψ{y}  E �  σ{0,y}  E  − σ{0}  E  �  ν  �  f2(0, y) dy  +  ˆ  Rd  � ˆ  ∂B  ψ{y}  E  σ{0}  E ν  �  h2(0, y) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='14)  The second integral is now absolutely convergent provided that h2 has Dini decay (so that  y �→ ⟨y⟩−d|h2(0, y)| be integrable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' This renormalized formula can be justified rigorously  and leads to a fine understanding of the second cluster term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Theorem 4 (Batchelor–Green renormalization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='6 in [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Let the same  assumptions hold as in Theorem 1, assume (H1) for simplicity, and assume that the two-  point correlation function satisfies |h2(0, y)| ≤ C⟨y⟩−γ for some C, γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Then the second  cluster term defined in Theorem 3 is equivalently given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Moreover, examination  of this renormalized formula yields the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  The bound | ¯B  (2)| ≲ λ2|log λ| is optimal in general, in the sense that it is attained by  some point process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Yet, the logarithmic correction can be removed in some cases:  — if the point process is isotropic, we have | ¯B  (2)| ≲ λ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  — if the interparticle distance is ℓ := infn̸=m |xn − xm| ≥ 1, we have | ¯B  (2)| ≲ ℓ−2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  The convergence rate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='11) for finite-volume approximations can be improved to  | ¯B(2)  L − ¯B(2)| ≲ L−γ + log L  L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  ♦  EFFECTIVE VISCOSITY OF SEMI-DILUTE SUSPENSIONS  11  The optimality of the logarithmic correction in the bound | ¯B  (2)| ≲ λ2|log λ| is under-  stood as follows: the second term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='14) can be written as  ´  K(y)h2(0, y) dy for some  Calder´on–Zygmund kernel K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' As λ2 is the sup norm of h2, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='5), the validity of the  bound | ¯B  (2)| ≲ λ2 would require to control K∗ in L∞(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' This cannot be possible in  general and we easily construct a correlation function h2 for which it fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  At higher orders, the renormalization of cluster formulas is more problematic as simple  cancellations like (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='13) are no longer sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Using corrector equations, the kth cluster  formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='4) can be written as  E : ¯B(k)E =  ≪ k+1  2  ˆ  (Rd)k−1  � ˆ  ∂B  δ{x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=',xk−1}ψ#  E · σ{0}  E ν  �  × fk(0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' , xk−1) dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' dxk−1 ≫ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='15)  up to other similar terms that are slightly better behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' In order to capture relevant  cancellations, we introduce in [11] some new diagrammatic decomposition of corrector  differences δ{x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=',xk−1}ψ#  E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Instead of using the classical method of reflections, which  would yield an infinite series of terms only involving single-particle solutions, we introduce  a suitable finite decomposition involving multiparticle solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' For instance, we write  ∇δx1,x2,x3ψ(0) =  +  +  +  +  +  ,  where the vertex  stands for evaluation at 0, where other vertices correspond to integration  variables x1, x2, x3, and where an edge between vertices x and y corresponds to a kernel  with pointwise decay ⟨x − y⟩−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' More precisely, edges inside a given loop of the graph  correspond to mutliparticle solutions (associated with the set of particles corresponding  to the different vertices of the loop), while all the other edges correspond to single-particle  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' This provides a crucial separation of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' These diagrammatic expansions  are obtained by iterating Green’s representation formula for corrector differences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' see [11,  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='3] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' We insert this decomposition into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='15) and we further expand  the k-point density function fk in terms of correlation functions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  f3(0, x1, x2) = λ2 + λ  �  h2(0, x1) + h2(0, x2) + h2(x1, x2)  �  + h3(0, x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Assuming algebraic decay of correlations, this leads to additional couplings between the  different integration variables, with some pointwise algebraic decay ⟨·⟩−γ, which we shall  represent by dotted edges in the diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Now the point is that our decomposition of  corrector differences precisely allows to capture relevant cancellations: whenever a graph  can be split into two subgraphs that are only connected by a simple edge, its contribution  can be shown to vanish as a boundary term in the infinite-volume limit, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  ,  ,  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Starting from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='15) and removing all such contributions, we are left with terms that all  correspond to absolutely convergent integrals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  E : ¯B(4)E =  +  +  +  +  +  +  +  +  +  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  The number of terms in such decompositions grows very quickly, which probably makes  them useless in practice, but it leads us to the proof of the following improved version of  Theorem 3, which is our main result in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  12  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' DUERINCKX AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' GLORIA  Theorem 5 (Higher-order cluster expansion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='8 in [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Let the same  assumptions hold as in Theorem 1, and assume (H1) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Given K ≥ 1, further assuming that correlation functions hK+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' , h2K+1 have some  decay C⟨·⟩−γ in each direction for some C, γ > 0 (which is implied by algebraic α-  mixing), the cluster expansion of the effective viscosity holds in form of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='12), where  cluster terms are defined by infinite-volume approximation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  For all k ≥ 1, further assuming that the correlation function hk has some decay C⟨·⟩−γ  in each direction for some C, γ > 0, then  — the kth cluster term is equivalently given by a renormalized formula only involving a  finite number of absolutely convergent integrals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  — it satisfies the bound | ¯B(k)| ≲ λk|log λ|k−1, which is optimal in general;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  — the convergence rate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='11) for finite-volume approximations can be improved to  | ¯B(k)  L − ¯B(k)| ≲ (log L)k−1  Lγ∧1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  ♦  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Remark: a new elliptic regularity result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' In order to prove the above, in [11], we  make repeated use of decay properties of solutions to Stokes problems with finite numbers  of rigid particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' More precisely, we use for instance the following mean-value property,  which seems new and might be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Theorem 6 (Mean-value property;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' see Appendix A in [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Let {In}n∈S be a finite collec-  tion of disjoint subsets of BR, assume that they are uniformly C2 and that for some ρ > 0  we have dist(In, Im) ≥ ρ and dist(In, ∂BR) ≥ ρ for all n ̸= m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Let u ∈ H1(BR)d satisfy  the Stokes problem  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  −△u + ∇p = 0, div(u) = 0,  in BR \\ ∪n∈SIn,  D(u) = 0,  in ∪n∈SIn,  ´  ∂In σ(u, p)ν = 0,  ∀n ∈ S,  ´  ∂In(x − xn) × σ(u, p)ν = 0,  ∀n ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  Then we have  B  |∇u|2 ≤ C(ρ, ♯S)  BR  |∇u|2,  where the multiplicative constant C(ρ, ♯S) only depends on ρ, ♯S, and on dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  ♦  Note that this result is false in general if ♯S = ∞, in link with classical counterexamples  to Lipschitz regularity, but we show in [12] that a corresponding annealed estimate then  holds upon taking a suitable ensemble average with respect to the set of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Further results and extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' In [6], we derive similar results for the effective  viscosity of active suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' In a work in preparation with Pertinand, we further study  Batchelor’s dilute expansion [3] for the mean settling speed of a sedimenting suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  The main difficulty is that, in the sedimentation problem, the long-range nature of hydro-  dynamic interactions is even more drastic: the flow disturbance at x due to a particle at y  then decays like ⟨x−y⟩1−d instead of ⟨x−y⟩−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' For this reason, a suitable renormalization  is needed even to actually define the mean settling speed, which requires a nontrivial decay  of correlations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' see [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' The dilute expansion is then particularly tricky, but the same type  of diagrammatic decompositions proves to be of crucial use in capturing cancellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  EFFECTIVE VISCOSITY OF SEMI-DILUTE SUSPENSIONS  13  Acknowledgements  MD acknowledges financial support from F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='-FNRS, and AG from the European Re-  search Council (ERC) under the European Union’s Horizon 2020 research and innovation  programme (Grant Agreement n◦ 864066).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  References  [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
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+page_content=' Brenner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
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+page_content=' Mourrat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
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+page_content=' Springer, Cham, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' The determination of the bulk stress in suspension of spherical  particles to order c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=', 56:401–427, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='  [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Batchelor and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' The hydrodynamic interaction of two small freely-moving spheres in  a linear flow field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
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+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
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+page_content='duerinckx@ulb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
+page_content='be  (Antoine Gloria) Sorbonne Universit´e, CNRS, Universit´e de Paris, Laboratoire Jacques-  Louis Lions, 75005 Paris, France & Institut Universitaire de France & Universit´e Libre de  Bruxelles, D´epartement de Math´ematique, 1050 Brussels, Belgium  Email address: antoine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfWfeb/content/2301.00165v1.pdf'}
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+ 
+ 
+At the crossroads of epidemiology and biology: bridging the gap between SARS-CoV-2 
+viral strain properties and epidemic wave characteristics. 
+Florian Poydenot1, Alice Lebreton2,3, Jacques Haiech4* and Bruno Andreotti1 
+1) Laboratoire de Physique de l’Ecole Normale Supérieure (LPENS), CNRS UMR 8023, Ecole 
+Normale Supérieure, Université PSL, Sorbonne Université, and Université de Paris, 75005 
+Paris, France 
+2) Institut de Biologie de l’ENS (IBENS), École Normale Supérieure, CNRS, INSERM, Université 
+PSL, 75005 Paris, France 
+ 
+3) INRAE, Micalis Institute, 78350 Jouy-en-Josas, France 
+4) CNRS UMR7242 BSC ESBS, 300 Bd Sébastien Brant, CS 10413, 67412 ILLKIRCH cedex 
+* Corresponding author: haiech@hotmail.fr 
+Declaration of competing interest: The authors declare that they have no known competing financial interests or 
+personal relationships that could have appeared to influence the work reported in this paper. 
+Contributions: F.P, J.H and B.A have collectively designed and written the paper, with input from A.L. 
+KEYWORDS: COVID19; SARS-CoV-2; epidemiology; PFU; GU; viral load 
+ABSTRACT 
+The COVID-19 pandemic has given rise to numerous articles from different scientific fields 
+(epidemiology, virology, immunology, airflow physics...) without any effort to link these 
+different insights. In this review, we aim to establish relationships between epidemiological 
+data and the characteristics of the virus strain responsible for the epidemic wave concerned. 
+We have carried out this study on the Wuhan, Alpha, Delta and Omicron strains allowing us 
+to illustrate the evolution of the relationships we have highlighted according to these different 
+viral strains. 
+We addressed the following questions: 
+1) How can the mean infectious dose (one quantum, by definition in epidemiology) be 
+measured and expressed as an amount of viral RNA molecules (in genome units, GU) 
+or as a number of replicative viral particles (in plaque-forming units, PFU)?  
+2) How many infectious quanta are exhaled by an infected person per unit of time? 
+3) How many infectious quanta are exhaled, on average, integrated over the whole 
+contagious period?  
+4) How do these quantities relate to the epidemic reproduction rate R as measured in 
+epidemiology, and to the viral load, as measured by molecular biological methods?  
+5) How has the infectious dose evolved with the different strains of SARS-CoV-2? 
+
+We make use of state-of-the-art modelling, reviewed and explained in the appendix of the 
+article (Supplemental Information, SI), to answer these questions using data from the 
+literature in both epidemiology and virology. We have considered the modification of these 
+relationships according to the vaccination status of the population.  
+We hope that this work will allow a better integration of data from different fields (virology, 
+epidemiology, and immunology) to anticipate the evolution of the epidemic in the case of 
+COVID-19, but also in respiratory pathologies induced by a virus or a bacterium 
+transmissible in an airborne manner. 
+ 
+INTRODUCTION 
+Evidence of aerosol transmission of SARS-CoV-2, the virus responsible for COVID-19 
+disease, has accumulated over the months (Fennelly, 2020; Lewis, 2020; Morawska and 
+Milton, 2020; Zhang et al., 2020, p. 19) until a consensus was reached, six months after the 
+start of the pandemic. Viral particles, with or without a liquid droplet surrounding them, are 
+dispersed by turbulent air movements. When they are light enough, hydrodynamic 
+fluctuations keep these particles suspended in the air, despite gravity. The mixture of air and 
+particles then constitutes a phase called aerosol. These droplets, which may or may not carry 
+the virus, are produced by atomization in the respiratory tract when an airflow of sufficient 
+velocity causes the fragmentation of a mucus film (Bourouiba, 2021; Johnson et al., 2011; 
+Johnson and Morawska, 2009; Moriarty and Grotberg, 1999). This is the case when large 
+millimeter-sized droplets are emitted by coughing or sneezing, as well as when smaller 
+micron- or submicron-sized droplets are emitted during human exhalation activities (e.g., 
+breathing, speaking or laughing). The water in the droplets then evaporates into the air, 
+concentrating the droplets into virions and mucus proteins, some of which have antiviral 
+properties that help inactivate the virus after a few hours. When airborne viral particles, 
+regardless of their production process, are inhaled, infection can occur (Figure 1). Pathogens 
+responsible for other diseases such as influenza, tuberculosis or measles can also be carried 
+by these small droplets (Blanchard, 1989; Chingin et al., 2018; Du et al., 2020; Zhou et al., 
+2018). 
+Airborne particles are the primary route of transmission of SARS-CoV-2 (Cheng et al., 2021; 
+Johansson et al., 2021). The epithelial cells serving as loci of original infection and reservoirs 
+of dissemination are located in the nasal cavity (for strains prior to Omicron), which points to 
+airborne transmission. The heavier, millimeter-sized droplets emitted specifically in the 
+Covid symptomatic phase have a ballistic trajectory that is relatively insensitive to the 
+presence of air and are stopped by all types of face masks and respirators. The reduction of 
+the risk of transmission of the virus outdoors or in well-ventilated closed environments 
+(Gettings, 2021) that is obtained by wearing respirators designed to filter aerosols (Goldberg 
+et al., 2021; Klompas et al., 2021), but also the long-distance transmission in super-spreading 
+events where a single virus carrier infects a large number of people (Endo et al., 2020; Yang 
+et al., 2021), are evidence of the importance of airborne transmission of SARS-CoV-2. 
+Transmission by contact with fomites on which these drops are deposited is probably 
+insignificant (Chen et al., 2021; Goldman, 2020; Lewis, 2021); however, regardless of its 
+actual weight in SARS-CoV-2 transmission, improved hand hygiene remains a recommended 
+habit to prevent transmission of other pathogens — for instance, viruses causing 
+gastroenteritis have a major handheld transmission route associated with a specific locus of 
+infection: the gastrointestinal epithelium (Green et al., 2020; Robilotti et al., 2015). Finally, 
+
+possible transmission through feces (Nouri-Vaskeh and Alizadeh, 2020) via aerosolization 
+during toilet flushing remains controversial and is probably a minor route, if relevant at all.  
+Two approaches have been proposed in the scientific literature to characterize the infectivity 
+and transmissibility of viral strains. The epidemiological approach, based on contact tracing 
+and population-scale testing, provides precise quantitative information but has a blind spot: 
+the epidemic propagation depends on social practices, the full complexity of which is difficult 
+to delineate, and which themselves depend on age, education, number and duration of social 
+contacts, vaccination status, etc. It also depends on the degree of immunity in the population. 
+The ab initio approach, based on knowledge of virology, immunology, and molecular 
+biology, is complementary; it allows one to characterize viral strains in vivo and in vitro but 
+suffers from the need of large-scale statistics, and can present important biases of 
+parameterization and calibration. This review article aims to bridge the gap between these 
+two approaches and to review methods combining epidemiological and biological 
+measurements performed on viral strains to deduce their intrinsic characteristics. 
+ 
+Figure 1-A illustrates how the viral mist exhaled by an infected person (index case) can infect 
+non-immune individuals (secondary case) at some distance, and after a time delay. The 
+infection risk increases with the intake viral dose d, defined as the amount of inhaled viral 
+particles accumulated over time. d increases with the time of exposure to the virus and with 
+the concentration of infectious viral particles in the inhaled air. The dilution factor between 
+exhaled and inhaled air is controlled at short distance by turbulent dispersion and at long 
+distance by ventilation, which is the process of introducing fresh air into indoor spaces while 
+removing stale air (Poydenot et al., 2022). The dose-response curve expresses the ratio of 
+infected individuals as a function of the intake dose d (Figure 1-A). We hypothesize that a 
+single replicating virion among the numerous particles inhaled can initiate infection. In tissue 
+culture assays, the number of infected cells is proportional to the number of viral cells, which 
+shows that there is no cooperativity between viral particles, in vitro(Houng et al., 2004). 
+Then, each inhaled viral particle can be considered as an independent attempt to contaminate 
+the individual. Statistically, more than one is required, as the probability of a single virus 
+successfully overcoming the host immune defenses is low, and since a large fraction of 
+inhaled viral particles, being damaged or defective, are intrinsically unable to infect cells and 
+replicate therein. Infection occurs when a single flawless virion enters a vulnerable location 
+where conditions are permissive to cellular colonization and viral replication. The dose-
+response therefore follows a cumulative Poisson distribution. By definition, an infectious 
+quantum is the dose inhaled by the individuals of a cohort which leads to the infection of 
+63% of the cohort. In the field of epidemiology, infectious quanta are used to express a 
+quantity of virus needed to induce contamination or a given symptom (fever, mortality for 
+example). 
+ 
+The dose-response curve has not been directly measured on humans. Infectious challenge 
+trials during which healthy young volunteers are deliberately infected are rare since they are 
+ethically controversial (Adams-Phipps et al., 2022); in the single one led to study COVID-19 
+(Killingley et al., 2022), a large viral dose (10 TCID50) is introduced via intranasal drops. 
+How can the mean infectious dose (one quantum, by definition) be measured and expressed 
+as an amount of viral RNA molecules (in genome units, GU, Figure 1-B) or as a number of 
+replicative viral particles (in plaque-forming units, PFU, Figure 1-C)? How many infectious 
+quanta are exhaled by an infected person per unit of time? How many infectious quanta are 
+exhaled, on average, integrated over the whole contagious period? How do these quantities 
+relate to the epidemic reproduction rate R as measured in epidemiology, and to the viral load, 
+
+as measured by molecular biological methods? How has the infectious dose evolved with the 
+different strains of SARS-CoV-2?  
+ 
+In this manuscript, we review concepts and methods providing preliminary answers to these 
+questions. We first describe the mechanism of viral infection and the antiviral response. 
+Then, we detail the viral kinetics and the subsequent time evolution of the viral exhalation 
+flux. The determination of the infectious dose by combining methods from molecular biology 
+and epidemiology is then reviewed. Finally, the evolution of the characteristics of successive 
+viral strains is presented and discussed. 
+To generate the figures of this review, we have used standard epidemiological modelling. The 
+specific theoretical frame chosen has been published in a previous paper (Poydenot et al., 
+2022). In the supplementary material section, we provide a detailed model accessible with 
+basic knowledge of physics and mathematics, focusing on the parameter’s calibration. 
+          Mechanism of viral infection 
+SARS-CoV-2 is a virus enveloped by a lipid bilayer in which E, M, and S proteins are 
+inserted (Figure 1-B). The lipid membrane originates from the cell in which the virus 
+replicated before being released. The virus contains a copy of the viral genomic RNA 
+protected by a capsid, structured by the assembly of the nucleocapsid protein N. Viral 
+particles measure 80-90 nm in diameter, and are decorated with an average of 48 spike (S) 
+proteins anchored in their envelope. The RNA genome encodes 29 proteins, including the 
+envelope (E, M, and S) and capsid (N) proteins, as well as non-structural proteins required for 
+replication and assembly of the virus within the host cell (Bar-On et al., 2020; Yao et al., 
+2020). To colonize a cell, the virus interacts via the S protein which is cleaved by a host cell 
+protease (mainly the TMPRSS2 protease for the wild strain, Wuhan-1) with a host cell 
+membrane protein (mainly the ACE2 receptor). This interaction leads to the formation of a 
+virus-ACE2 complex (via the virus spike, a trimer of the S (Yin et al., 2022)) which triggers 
+the internalization of the virus into the cell (Figure 2-A). A series of cellular events leads to 
+the disassembly of the virion and the undressing of the RNA molecule. The released viral 
+RNA is taken over by the host cell ribosomes, which read the information it encodes and 
+produce the viral proteins needed for virus production. New viral particles are then assembled 
+by hijacking the host cell mechanisms, and then released, leading to the colonization of 
+neighboring cells (Snijder et al., 2020). From the nasal cavity or throat, which are probably 
+the first tissues to be infected, the virus, embedded in the mucus secreted by goblet cells of 
+the nasal epithelium, is transported to the trachea, then to the lungs or esophagus, and finally 
+to deeper organs (Figure 2-B and 2-C). The severity and variety of disease symptoms depend 
+on the likelihood of the viral infection overcoming host defenses and reaching multiple sites, 
+as well as on the damage caused to the host by the potent inflammatory and interferon 
+responses launched against the viral assault. In contrast, the spread of the virus depends 
+primarily on its ability to colonize the host airways, and thus the viral load cannot be 
+correlated with symptom severity (Le Borgne et al., 2021). From the nasal cavity, the virus 
+has the ability to travel to the brain and colonize cells of olfactory bulb, which could explain 
+the changes in taste and smell and, in the long term, some of the neurological symptoms 
+associated with long COVID (Fodoulian et al., 2020; Pereira, 2020). More rarely, the virus 
+cpean be found in the blood or lymph, reach different organs in the body and colonize 
+specific cell types in various organs (liver, kidney, heart, prostate, etc.) (Dong et al., 2020) 
+(Figure 2-C). 
+
+When the virus is concentrated in the nasal cavity or in the throat, it is spread by a mist of 
+fine droplets of mucus or saliva which can be dispersed by breathing, speaking or singing. A 
+sneeze or cough produces larger droplets containing viral particles. As these droplets are 
+formed, their viral particle content increases linearly with the viral load, in the nasal cavity 
+for mucus droplets, or in the throat for saliva droplets (Buonanno et al., 2020b). An organism 
+can be infected if enough viral particles interact with cells expressing both the TMPRSS2 
+protease and the ACE2 receptor and if the virus is able to hijack cellular mechanisms to 
+produce and disseminate new virions. 
+For the Omicron viral strain, the TMPRSS2 protease appears to be less essential, in favor of 
+an alternative pathway of entry into epithelial cells via the endosomal route (Peacock et al., 
+2022b). At each cycle, the virus must enter a novel host cell, replicate its RNA molecule, 
+produce the proteins necessary for its self-assembly and then be released as virions. 
+Omics databases such as the Human Cell Atlas (https://www.humancellatlas.org/) provide 
+insight into the tissues and cell types that express the TMPRSS2 protease and ACE2 receptor 
+in different tissues of the human body. It is commonly assumed that these cells are the 
+primary target cells of SARS-CoV-2 (Hoffmann et al., 2020). The nasal epithelium is most 
+often the first infected tissue, which is amongst the strongest proofs in favor of a transmission 
+of SARS-CoV-2 primarily by inhalation of virus-laden aerosols. Accordingly, there exists a 
+cluster of nasal cells in the ACE2/TMPRSS2 expression map (Singh et al., 2020; Sungnak et 
+al., 2020). However, due to its lower dependence on TMPRSS2, the Omicron viral strain 
+illustrates the changes of cellular tropism that can happen as a result of evolution of the virus 
+(Gupta, 2022).  
+Antiviral response 
+The viral load of an infected person is the result of viral replication and of the antiviral 
+response. The arrival of the virus in the nasal cavity or in the throat induces an antiviral 
+response, due to the recognition of viral components considered as danger signals by relevant 
+cellular receptors (Silva-Lagos et al., 2021). This interaction triggers a non-specific antiviral 
+defensive response, which relies to a large extent on the production of a class of molecules 
+called interferons. By binding to their receptors on target cells, interferons induce the 
+expression of a wide range of interferon-stimulated genes (ISGs) that, through various 
+mechanisms, help limit viral replication (Gallo et al., 2021). 
+Interferons are produced by infected cells but also by sentinel immune cells such as 
+macrophages and dendritic cells. After their secretion, they diffuse and bind to their receptors 
+on surrounding cells without discriminating whether they are infected or not, stopping 
+cellular functions. Since the concentration is higher around the site of infection, diffusion 
+leads to an efficient stochastic control of infected cells. If the interferon response is initiated 
+early, in a localized and circumscribed manner, viral spread through the tissue can be 
+controlled (Katze et al., 2002; Perry et al., 2005). However, a too strong interferon response 
+is not only antiviral but also destructive to host cells. Moreover, the outcome of the virus-host 
+crosstalk is further complicated by the fact that some viral proteins counteract host interferon 
+responses and, conversely, host interferon responses can sometimes amplify viral infectivity 
+(Ziegler et al., 2020). Others mechanisms may also modulate the infectivity of specific 
+strains, for instance the modulation of splicing of ACE2 in response to interferon responses 
+(Blume et al., 2021).Therefore, the nasal cavity is a site of ongoing conflict between SARS-
+CoV-2 replication and the body's efforts to inhibit the virus. The efficiency of the viral 
+replication and secretion processes, as well as the body's interferon response, can vary across 
+
+time and space within the nasal cavity. These above described and antagonists processes can 
+influence the ability of the virus to establish an infection in the body. 
+Because viral transmission depends directly on viral loads, the kinetics of the replication and 
+secretion of the virus characterizes the capacity of an individual to transmit the infection. 
+Moreover, the kinetics of the interferon response correlates with an individual’s susceptibility 
+to infection (Hadjadj et al., 2020). Although it is often forgotten, mucus also modulates 
+significantly that kinetics in the nasal cavity and participates in the immune response. Mucus 
+is composed mainly of water (95%), lipids and glycoproteins (mucins) (Bansil and Turner, 
+2018; Fahy and Dickey, 2010). It is secreted by specialized cells named goblet cells which 
+are also among the cells infected by the virus. Mucus forms two layers at the surface of the 
+nasal epithelium: a gel constituted by a network of polymerized mucins anchored to the 
+membrane of epithelial cells, and a solution that moves under the action of the cilia of the 
+epithelial cells. The viscoelastic properties of mucus depend on the type of mucins (Moniaux 
+et al., 2001) and on the physicochemical environment (humidity, calcium concentration and 
+pH). Inflammation modifies the expression and the glycosylation/fucosylation of MUC5B 
+and MUC5AC (Amini et al., 2019; Chatterjee et al., 2020), the major airway gel-forming 
+mucins. 
+Mucosal immunity results from a synergy between the mucus and the immune system (innate 
+immunity, secreted IgA and IgA for the acquired immunity, and cellular immunity), as the 
+network created by polymerized mucins traps particles down to a few hundred nanometers in 
+size (which is the size of viruses). Mucus also contains enzymes that can neutralize viruses 
+and bacteria in a non-specific manner. It should be noted that nasal mucus and saliva, which 
+is also a specific mucus, have different compositions and physicochemical characteristics. 
+Lastly, the nasal microbiota, a community of bacteria that live within the mucus, also plays a 
+role in defense against infection by viruses or pathogenic bacteria (Kolhe et al., 2021). 
+Viral load 
+The viral load in the oropharyngeal cavities, which determines the exhaled particle flux, is 
+highly time-dependent (Figure 3-A). It can be measured in different ways. The concentration 
+of viral genomes in a sample can be measured using RT-qPCR (Reverse transcription 
+followed by quantitative Polymerase Chain Reaction), using as targets one or more nucleic 
+acid sequences in the viral RNA genome (Figure 3-B). The sample is collected with a swab, 
+and then its RNA contents are extracted in a given volume of an RNA extracting and 
+stabilizing solution. This volume differs depending on the test used. The RT-qPCR result is 
+expressed in threshold polymerase reaction cycles (Ct) which is transformed into a number of 
+copies of viral RNA molecules (genome units, GU), after calibration using a solution of viral 
+RNA molecules of known concentration. The concentration of viral genomes is thus 
+expressed in viral RNA molecules per unit of volume of extraction solution (GU/mL in 
+practice). Considering that the swab collects at most 100 microliters of nasal extract and the 
+volume of extracting solution varies between 1 and 3 mL depending on the kit used, the 
+concentration of viral RNA is probably an order of magnitude higher in the oropharyngeal 
+cavity than the concentration in the extraction solution. Alternatively, the ability of the virus 
+to infect a cell monolayer and cause cell lysis can be measured. Using serial dilutions of a 
+virus-containing sample, the number of lysis plaques (Plate Forming Unit, PFU) produced 
+within a well monolayer for a given volume of sample is measured (Figure 3-C). The 
+concentration of replicative virus expressed in PFU/mL depends on the type of cells used, as 
+susceptibility to infection varies depending on cell types. 
+
+For a given sample and a given cell type, the number of cell lysis plaques obtained is 
+proportional to the concentration of viral genomes in solution. This experimental linearity 
+indicates that cell infection events can be considered as independent from each other. The 
+number of lysis plaques induced on the average by one viral genome unit is the viral 
+infectivity and is expressed in PFU/GU (Dabisch et al., 2021; Wang et al., 2021). As this 
+quantity reflects the ability of the virus to enter cells and replicate, it primarily depends on the 
+viral strain considered and on the type of cells. In practice, the viral infectivity is a highly 
+variable quantity as it reflects viral integrity for a given sample. It therefore depends on the 
+time and site of sampling, as well as on the protocol used for sample collection and 
+preservation. A viral infectivity equal to one would imply that each virion is able to colonize 
+a cell, replicate, be released to colonize neighboring cells and lyse the colonized cells. This 
+value is never reached as it implies a perfect replication and assembly of all virus particles, 
+the absence of damaged or inactivated viruses, as well as perfect sample preservation. In 
+other words, low values of the viral infectivity indicate viral solutions that contain high loads 
+of mis-assembled, non-functional or degraded viral particles. Part of this degradation may be 
+due to the antiviral response or the immune response of the body. Conversely, a null viral 
+infectivity means that no virion can form lysis plaques on a cell monolayer. For SARS-CoV-
+1, the viral infectivity was between 6 × 10-4 and 8 × 10-4 PFU/GU on VERO cells (Houng et 
+al., 2004) : one viral particle over 1200 to 1600 was able to form a cell lysis plaque. For the 
+Wuhan-1 strain of SARS-CoV-2, the literature provides different measurements of infectivity 
+for high preservation quality samples (Bao et al., 2020; Yu et al., 2020), which lead to an 
+estimate of  7 × 10-4 PFU/GU (1 cell lysis plaque for 1400 virions) for the initial, 
+asymptomatic phase of the pathology. As the disease progresses, the fraction of replicating 
+virions decreases and the infectivity decreases. 
+The viral kinetics has standardly been described in the literature by an exponential growth of 
+the viral load (replication and shedding) followed by an exponential decay (immunity 
+response). The SARS- CoV-2 “human challenge” trial (Killingley et al., 2022) has provided 
+unprecedented time-resolved data showing a more rounded viral load curve (Figure 3-A). The 
+viral strain used in this study was close to the wild-type strain, Wuhan-1. In the nose, the 
+maximum viral load, expressed in genome units, was reached after Tm=7.0 days, around the 
+onset of symptoms. This time lapse provides an estimate of the incubation period. This period 
+was slightly smaller, around Tm=5.9 days, when considering the maximum replicable viral 
+load, measured in PFU. This difference may be ascribed to the effect of the antiviral 
+responses on infectivity. As an alternative hypothesis, we cannot exclude the possibility of 
+incorrect assembly of the virus by the colonized cell leading to a decrease of infectivity (in 
+PFU/GU), although a potential inhibition of the virus assembly could also be a consequence 
+of antiviral responses. The infectious period T, defined as the average time between 
+infections of the index and secondary cases, is estimated to be 7.2 days (Figure 3-A). The 
+exponential growth rate just after infection is approximately 15.0 days-1 (growth time 8.5 
+hours). The exponential decay rate long after this maximum is 3.3 days-1 (decay time 39 
+hours). The infectivity is around 10-5 PFU/GU at maximum, which is two orders of 
+magnitude weaker than the value found when using virus replicated on cultured cells. 
+The viral emission rate, defined as the quantity of viral particles exhaled per unit of time, has 
+been measured in several pioneering studies. In the study by Ma et al. (Ma et al., 2020), 
+patients were asked to exhale into a cooled hydrophobic film through a long straw to collect a 
+sample of exhaled breath condensate. The measured concentration of SARS-CoV-2 viral 
+particles was between 105 and 2 × 107 GU/m3. The authors noted that the exhalation rate was 
+correlated with the viral load in the nose and in the throat but not in the lungs. In the study by 
+
+Malik et al. (2021),	the mean viral load per swab was 7.8 × 106 GU whereas exhaled breath 
+samples displayed 2.47 × 103 GU per 20 times exhaling, which corresponds to 2.5 × 105 
+GU/m3. In the scientific literature, it is standardly considered that the viral emission rate is 
+proportional to the viral load. This is reasonable if the infected epithelium surface is 
+reasonably constant and if there is no viral enrichment at the interface between muco-salivary 
+fluid and air. In the following sections, we will adopt this hypothesis and use the 
+multiplicative factor 0.4 between the viral load, in GU/mL, and the viral emission rate, in 
+GU/ day found in Malik et al. (2021). 
+The total viral emission of an infected person is the quantity of viral particles exhaled during 
+its infection. Considering the viral kinetic displayed in Figure 3-A, the total viral emission, 
+obtained by integrating over time the viral emission rate, would be around 9 × 107 GU, or 
+6 × 104 PFU, within a half-decade uncertainty. 
+Molecular vs epidemiologic determination of the infectious dose 
+As mentioned in the introduction, the determination of the infectious dose a priori requires 
+measuring a dose-response relationship. Importantly, infection can be defined from different 
+observables (amount of virus in the nasal cavity, presence of specific symptoms — like 
+rhinitis, pneumonia or acute respiratory distress syndrome — or mortality). The infectious 
+dose therefore depends on the viral strain, on the capacity of the upper airway mucosal 
+immune system and of the systemic immune system for the lung or other organs to neutralize 
+the virus, and on the type of observables used to monitor infection. The inhaled dose only 
+dictates the primary infection in the upper airways. The severe acute respiratory syndrome in 
+the lungs typically results from a secondary infection, after the virus has colonized the nasal 
+cavity (Wölfel et al., 2020). As a consequence, the relevant viral dose is the one that is 
+transferred from upper to lower airways. The viral dose in the inoculum is therefore not 
+directly related to disease severity, as it is negligible compared to the production of virus by 
+colonized host cells in the upper airways. 
+ 
+The dose response relationship allows expressing the mean infectious dose (one quantum, by 
+definition) as an amount of viral RNA molecules or as a number of replicative viral particles. 
+However, it can only be measured on animal models. No such study exists for SARS-CoV-2, 
+to the best of our knowledge, but it has been measured on a non-human primate model for 
+SARS-CoV-1 (Watanabe et al., 2010). The ID50 (median infectious dose) describes the 
+amount of replicable virus (expressed in PFU) needed to infect 50% of the population (Figure 
+1-A). Note that the same quantity, when measured on a population of cells in tissue culture 
+assays, is called TCID50 (median tissue culture infectious dose). For SARS-CoV-1, the 
+reported measurement is ID50=280 TCID50. We recall that an infectious quantum is the dose 
+inhaled by the individuals of a cohort that leads to the infection of 63% of the cohort, and that 
+it is used in epidemiology as a unit. This convention leads to a multiplicative factor 
+1/log(2)=1.44 between the epidemic quantum and the ID50 quantity. The mean infectious 
+dose therefore relates an epidemiological quantity to a characteristic quantity of virus, 
+defined using molecular biology experiments (Figure 1): the reported measurement for 
+SARS-CoV-1 leads to 400 PFU/quantum. Using these estimates, the infectious dose for the 
+raw Wuhan-1 strain on humans is in the range of 5.6 × 105 GU, within a factor 2. Considering 
+the viral kinetic displayed in Figure 3-A, the total viral emission ħ would be around 150 
+quanta, within a factor 6. This corresponds to a typical viral emission rate of 1 quantum/hour, 
+and a maximal viral emission rate of 2 quanta/hour.  
+
+From the epidemiological point of view, reference points are provided by closed micro-
+societies inside which the virus propagates. They provide estimates of the total viral emission 
+ħ expressed in infectious quanta. In other words, the exhaled dose of an infected person as the 
+potential to infect ħ other people, on the average. However, thanks to ventilation, only a small 
+fraction of this exhaled dose is actually inhaled. The exponentially growing epidemics 
+onboard the ship Diamond Princess (Almilaji and Thomas, 2020) and onboard the French 
+aircraft carrier Charles de Gaulle are the most important events for the Wuhan-1 strain 
+(Figure 3-A). Both events were characterized by a low rate of replacement of stale air by 
+fresh air, and by an air conditioning system lacking HEPA filters to remove pathogens. Using 
+the viral load curve of Figure 3, the growth rate can be converted into an epidemic 
+reproduction rate (R=4.8 and R=3.2, respectively). Then, using a model dilution factor 
+between exhaled air and inhaled air (Poydenot et al., 2022), one deduces the total viral 
+emission ħ=490 quanta (retired people) and ħ=460 quanta (young adults), respectively. This 
+corresponds to a typical viral emission rate of 3 quanta/hour, and a maximal viral emission 
+rate of 6.3 quanta/hour. Schools constitute the best documented social sub-system (Bazant 
+and Bush, 2021; Vouriot et al., 2021), although not isolated from the rest of society as ships 
+were. Figure S4 shows the epidemic evolution for secondary school pupils, between 
+lockdown and holidays, in UK, together with the typical CO2 concentration measurement 
+(Poydenot et al., 2022) that gives a total viral emission ħ = 270 quanta for pupils from age 10 
+to age 14. It corresponds to a typical viral emission rate of 1.6 quanta/hour and a maximal 
+emission rate in the range of 3.3 quanta/hour.  
+In conclusion, although calibrations are lacking large-scale statistics, the molecular and 
+epidemiologic determinations of the mean infectious dose are consistent with each other, 
+within error bars. It is important to note that the infectious dose is approximately 6 × 105 GU 
+for the wild type strain Wuhan-1 and not between 10 and 100 as mentioned in a series of 
+recent articles (Bazant and Bush, 2021; Buonanno et al., 2020a; Lelieveld et al., 2020; 
+Pöhlker et al., 2021; Vouriot et al., 2021; Vuorinen et al., 2020). A large part of the results on 
+the airborne transmission risk, based on the volume of exhaled drops, is quantitatively flawed 
+by the omission of the infectivity ratio between plaque forming units (PFU) and genomic 
+units (GU). The estimated typical emission rate is between 15 and 30 quanta/hour in Bazant 
+and Bush (2021) and Miller et al. (2021) found 1,000 quanta/hour for singing, to be compared 
+to 1-3 quanta/hour estimated here. Accordingly, the total viral emission ħ, which ranges 
+between 450 and 500 quanta on average for adults, is almost 10-fold smaller than previous 
+estimates. 
+Evolution of viral strain characteristics      
+In the previous sections, we have discussed at length the absolute characteristics of the wild 
+strain, Wuhan-1. Comparing the characteristics of the variants presents different difficulties. 
+On the one hand, relative measurements are much more precise; on the other hand, the 
+immunity induced by infection or by vaccination induces a strong heterogeneity in the 
+population. The evolution of the parameters presented here for the wild type strain, Wuhan-1, 
+and for variants Alpha (B1.1.7), Delta (B.1.617.2), Omicron BA.1 and Omicron BA.2, is 
+based on relative measurements, the wild type strain serving as a reference. We have selected 
+the variants that have led to an epidemic wave and we have retained the date of the first case 
+in France to display the results. Although this choice is somehow arbitrary, it is justified by 
+the fact that most measurements are performed using French epidemic data. 
+
+Figure 4-D shows that the incubation period Tm and the infectious period T, measured in vivo 
+using the viral load kinetics, are almost equal. They are slightly decreased for the variants 
+Alpha and Delta, which leads to a higher growth rate and therefore to an evolutionary 
+advantage. By contrast, the advantage of Omicron is not due to a change of incubation and 
+infectious period. Figure 4-D compares these two characteristic times, Tm and T, to the 
+replication time, measured in vitro using human nasal epithelial cells (hNECs). To get 
+comparable orders of magnitude, we define the latter as the time needed in vitro to multiply 
+the number of viral particles by 1 billion. Interestingly, the replication time of the virus in a 
+human lung cell line that expresses abundant ACE2 and TMPRSS2 (Calu-3) has strongly 
+decreased for the Delta variant, in direct relation with a higher risk of pneumonia. Omicron is 
+milder than Delta (but more severe than the wild-type strain) for lung symptoms, which is 
+coherent with its increased replication time in Calu-3 cells. 
+In Figure 5, we propose a meta-analysis of various characteristics for the successive variants 
+of interest. It combines epidemiologic and biological data, following standard methods in 
+both domains. We emphasize that this meta-analysis is deduced from the definitions and 
+measurement methods defined previously in a straightforward and robust way.  
+Figure 5-A shows the evolution of the inverse of infectivity, measured as the average number 
+of viral particles (GU) needed to induce one lysis plaque on a generic Vero cell culture (PFU) 
+(data from Bao et al., 2020; Despres et al., 2022; Ghezzi et al., 2020; Houng et al., 2004; 
+Killingley et al., 2022; Paton et al., 2021; Peacock et al., 2022; Peña-Hernández et al., 2022; 
+Puhach et al., 2022); measurements performed on hNECs and Calu-3 cells would be more 
+relevant but are lacking. The Delta variant is more infectious (240 GU/PFU) than the Alpha 
+variant (1,000 GU/PFU), itself being more infectious than the wild-type strain (1,400 
+GU/PFU). Omicron is slightly less infectious than Delta (300 GU/PFU). This confirms the 
+evolution of the replication time. The evolutionary advantage of the Delta variant is due to its 
+higher binding affinity with ACE2, which increases the probability of cell entry and 
+replication. 
+ 
+Figure 5-B shows the total viral emission ħ (in quanta), deduced from epidemiologic 
+measurements. The reference value of ħ is deduced from the infections onboard the Diamond 
+Princess and Charles de Gaulle boats (Figure 3-A).  ħ is the key biological characteristic that 
+determines the epidemic growth rate, for given social practices. The ratio of ħ for two 
+successive strains is deduced from the period of time during which these two strains coexist, 
+as the ratio of the epidemic reproduction rates R. The two values of R are themselves deduced 
+using the Euler-Lotka equation from the epidemic growth rates. Figure 5-B shows that ħ has 
+increased from one variant to the next. For Alpha and Delta, this increased epidemic growth 
+rate is probably a direct consequence of the evolutionary increase of binding affinity for the 
+ACE2 receptor. For Omicron, a reasonable hypothesis is that the increased epidemic growth 
+rate results from a displacement of the first entry point from the nasal cavity to the throat. A 
+possibility would be that lower density of lymphid follicles in the throat than in the 
+nasopharynx renders this mucosa less potent in launching an effective immune response 
+against infection (Ogra, 2000). The total viral emission ħ (in quanta) is also displayed when 
+both the index and secondary cases are fully vaccinated. For this, we used data of the relative 
+transmissibility and susceptibility deduced from household transmission. Although the 
+vaccines were optimized to induce circulating antibodies and systemic T and B cell responses 
+to prevent viral diseases, they were very effective to prevent transmission for the Alpha and 
+Delta strain. On these variants, the vaccines were reducing both the transmissibility, which is 
+by definition the replicable viral emission rate, and the susceptibility of contact cases to be 
+
+infected. However, this prevention of viral spreading by the vaccine has almost disappeared 
+with Omicron. Incidentally, by maintaining a broad viral pool in circulation, increased 
+immune escape is amongst the reasons for Omicron increased transmissibility (Paz et al., 
+2022). 
+ 
+Figure 5-C shows the evolution of the infectious dose (1 quantum) in the upper airways, in 
+genome units (GU). This measurement combines the number ħ of infectious quanta exhaled 
+during the infectious period (Figure 5-B) and the evolution of the maximum viral load. The 
+ratio between the number of genome units in a swab and in one breath is around 1.6 105 
+(Adenaiye et al., 2021; Leung et al., 2020; Ma et al., 2021; Malik et al., 2021; Ryan et al., 
+2021). The result is multiplied by the mean number of breathes per day (2 × 104) and by the 
+integral over time of the viral load in the upper airways to obtain the number of genome units 
+(GU) exhaled on the average during the whole infectious period. The number of genome 
+units in the infectious dose is obtained by dividing the number of genome units (GU) exhaled 
+in total by the number of quanta exhaled in total, ħ. Figure 5-C shows that the reason for the 
+increase of the integral viral exhalation ħ is not primarily a larger viral load but a strong 
+decrease of the number of virions statistically needed to induce the infection. 
+Figure 5-D shows the evolution of the infectious dose (1 quantum) in the lower airways, 
+measured in genome units (GU), for the successive variants of interest. The mean dose 
+required to infect the lower airway is measured as the ratio of the number of genome units 
+(GU) transferred by inhalation from the upper airways to the lungs, divided by the probability 
+to develop a lung pathology. In first approximation, inhalation moves the same quantity of 
+viral particles towards the lungs as exhalation does towards the outside. The probability that 
+the lungs get infected by SARS-CoV-2 when the upper airways are, is approximated by the 
+hospitalization hazard ratio. The infectious dose is much larger for the lungs than for the 
+upper airways due to the fact that viral particles are not diluted when transported to the lower 
+airways (Heyder, 2004; Hinds, 1999). The rather low probability of infection shows that the 
+immune system is more effective in the lungs than in the nose or throat. 
+Figures 5-E and S8 shows the maximum likelihood phylogenies inferred from spike 
+nucleotide sequences for major SARS-CoV-2 lineages. SARS-CoV-2 does not currently 
+show the unbalanced, unidirectional phylogenetic tree that is a hallmark of immune escape 
+for viruses under a strong immune selection pressure, and where each new variant emerges 
+from the last dominant variant (Volz et al., 2013). Indeed, the Omicron strain is not a 
+mutation of the Delta strain, nor was the Delta strain a mutation of the Alpha strain, but each 
+one of them emerged from very different branches of the phylogenetic tree. So far, each new 
+variant of concern has arisen from very different branches of the phylogenetic tree by novel 
+mutations that have remained undetected over long periods of time, resulting in a relatively 
+balanced tree. By contrast, endemic viruses such as influenza or seasonal coronaviruses can 
+be recognized by the shape of their unbalanced, unidirectional phylogenetic tree (or ladder-
+like evolutionary tree, when represented as a function of time): they circulate and mutate; in 
+parallel, immunity develops against them, leading to the gradual extinction of the ancestral 
+branches. Some of the mutations lead to new variants that escape immunity; their branches 
+expand until immunity develops against these new variants, and so on. At this endemic stage, 
+each new variant thus derives by mutation from one of the last hegemonic strains, usually by 
+a limited evolutionary jump that allows it to escape immunity, which is partially 
+predictable(Carabelli et al., 2023). Meanwhile, at the present stage of SARS-CoV-2 
+evolution, new viral strains do not present such a phylogenetic pattern: they may emerge 
+
+from all branches as well as from the root of the tree, after having spread silently for a time. It 
+is then impossible to predict how close the next hegemonic variant will be to the previous 
+one. 
+Discussion 
+In this article, we have first reviewed a series of epidemiological and molecular biology 
+methods that can be combined to characterize the airborne transmission of respiratory 
+viruses: 
+(i) 
+replication kinetics of viral strains in tissue culture assays, using RT-qPCR and 
+cell lysis plaques measurements, provides access to the viral replication time in 
+the absence of immunity response and to the viral infectivity 
+(ii) 
+the dose response curve, determined using model animals close enough to 
+humans, provides the expression of the epidemic quantum in GU and PFU 
+(iii) 
+the viral exhalation rate can be measured as a function of time t after infection, 
+directly, by collecting virus in a mask, or via viral load curves. This measurement 
+gives access to the total viral exhalation in GU and in PFU, to the infection time T 
+and to the rescaled viral transmissibility ψ(t) 
+(iv) 
+the epidemic reproduction rate R can be deduced from the epidemic growth rate σ, 
+using the rescaled viral transmissibility ψ(t). This measurement gives access to the 
+total viral exhalation in quanta, if the epidemic growth rate σ is measured on an 
+isolated micro-society 
+This toolbox can be used to characterize viral strain and calibrate quantitatively the models 
+for airborne transmission risk. 
+Some limitations to the present conclusions of this study should be highlighted. The severity 
+of the symptoms and the transmissibility depend on the personal status (age, weight, 
+immunodeficiency conditions, comorbidity factors, etc.). Epidemic databases annotated with 
+such information do not exist as open data for obvious ethical reasons: it would be necessary 
+to chain different databases. Such an investigation would be extremely interesting but is 
+outside the scope of this review. 
+A better molecular determination of infectious quanta requires measuring the number of viral 
+particles per unit of time (or per unit of volume of exhaled air) during expiratory human 
+activities such as breathing or speaking. This would require the design and calibration of face 
+masks that allow patients to breathe normally and collect all viral particles in a filter. We 
+have used here epidemiological data to estimate the infectious quantum, expressed in viral 
+RNA (GU), for the successive SARS-CoV-2 variants. The systematic use of respiratory 
+aerosol samplers (Li et al., 2021) is essential to characterize quantitatively SARS-CoV-2 
+strains, which is necessary for risk assessment and subsequent risk reduction policies. 
+Similarly, standard quantitative molecular biology techniques, such as plaque assays and RT-
+qPCR could be used directly to measure face mask efficacy. 
+Second, it would be useful to measure systematically the replication kinetics of SARS-CoV-2 
+strains in tissue culture assays with large enough statistics and controls to infer replication 
+properties in the absence of immune response. Although viral multiplicative curves are 
+
+regularly measured and published, the necessary scientific coordination is missing as well as 
+systematic comparison with epidemiologic data, for which the immune response is present. 
+Third, another limitation of this study is the lack of knowledge about the generation of virus-
+containing aerosols in the upper respiratory tract, particularly in the nasal cavity. We have 
+shown that indirectly determining the exhalation rate of viral particles using the aerosol 
+droplet emission rate underestimates the result by three orders of magnitude (Duan et al., 
+2021). 
+Fourth, several issues regarding evolution during desiccation of mucus droplets carrying 
+virions remain open: 
+- The evolution of the physicochemical characteristics of mucus as a function of time, 
+temperature, humidity, pH, ionic concentration (especially calcium, which condenses mucin 
+polymers at high concentrations), and pathology (Ma et al., 2018). 
+- The role of mucus in the formation of droplets and their contaminating character: 
+concentration of the virus, size of the droplets, mixing of the mucus of two different 
+individuals—mucus of the transmitter and mucus of the receiver (Edwards et al., 2021). 
+- The respective contributions of mucus and interferon responses to the clearance of SARS-
+CoV-2 (Persson, 2021). 
+Finally, the mechanisms of SARS-CoV-2 inactivation remain poorly understood. In order to 
+assess the effectiveness of alternative risk reduction techniques, it is necessary to know how 
+environmental conditions (temperature, humidity, chemical concentrations, ultraviolet 
+irradiation) affect the viability of SARS-CoV-2 (Duan et al., 2003; Fears et al., 2020; 
+Lednicky et al., 2020). 
+The evolution of the virus takes place under a double selection pressure, an increase in 
+transmissibility and an escape from neutralizing antibodies. The increase in transmissibility is 
+due to an optimization of the virus ability to replicate in the epithelial cells of the throat and 
+nose, which release more and more replicable virions. This optimization occurs 
+independently of the symptoms that the different mutants may cause in the contaminated 
+organisms. The existence of neutralizing antibodies affects the quality of the exhaled virions. 
+Thus, immune escape of new variants participates in a temporary increase of transmissibility 
+by increasing the quantity of replicable exhaled viruses. 
+The severity of symptoms induced by a given variant is not correlated with the 
+transmissibility and therefore, does not seem to be important in the evolution of the virus 
+(Alizon and Sofonea, 2021). Thus, new variants appear randomly, which may either induce 
+more severe, or milder, symptoms. However, the speed and diversity of virus evolution is 
+correlated with its circulation flow and its ability to remain present in an organism for long 
+periods. Thus, the ability of new variants to infect animals or persist in immunocompromised 
+individuals accelerates virus circulation and the appearance of new variants with enhanced 
+transmissibility. The greater the diversity of these new variants, the greater the possibility of 
+variants inducing severe symptoms, even in younger individuals. Following a precautionary 
+principle, it is important to decrease as much as possible the circulation of the virus by 
+improving the air quality in closed areas and to monitor the circulation of the virus and the 
+appearance of new variants in a given spatial territory. This requires appropriate means 
+
+allowing a timely monitoring of viral circulation in the environment and not only at the level 
+of individuals (Rios et al., 2021). 
+Figure captions 
+Figure 1: How to relate epidemiologic characteristics to measurements performed in 
+molecular biology? (A) The dose-response curve relates the probability of infection to the 
+amount of inhaled viral particles accumulated over time, called the intake dose. The mean 
+infectious dose is defined as ID50 for animals (50 % probability), as TCID50 for cells (50 % 
+probability) and as the infectious quantum for humans (dose-response curve approximated by 
+1-exp(-d), where d is the dose expressed in quanta; red curve). (B) The mean infectious dose 
+can be expressed as a quantity of viral genome copies, expressed in genome units (GU). (C) 
+Alternatively, it can be expressed as a quantity of viral particles able to replicate on a certain 
+type of cell, expressed in plate forming units (PFU). 
+Biological characteristics
+Epidemiological characteristics
+Viral particles
+genome units (GU)
+Replicative virus
+with respect to a cell type
+plaque-forming units (PFU)
+Mean infectious dose
+infectious quantum
+1
+0
+0.5
+102
+103
+104
+101
+102
+101
+10-1
+10-2
+100
+Index
+case
+Exhaled
+dose
+Probability of infection
+dose (PFU)
+dose (quanta)
+Inhaled
+dose
+Secondary
+case
+ID50
+B
+A
+C
+Spike Glycoprotein (S)
+M-Protein
+E-Protein
+RNA and N-Protein
+
+ 
+Figure 2: Mechanism of viral infection. (A) Pathway used by SARS coronavirus 2 (SARS-
+CoV-2) to enter and infect the cell by intermolecular interactions between the spike protein of 
+the virus and its host cellular receptor angiotensin converting enzyme 2 (ACE2). Effective 
+binding is dependent upon spike protein activation by transmembrane protease or furin. (B) 
+The nasal cavity and the throat are usually the first tissues to be infected, after inhalation of 
+viral particles. The infection of other organs where the ACE2 receptor is expressed is induced 
+in a second stage, after the virus has colonized the upper airways. (C) Viral particles issued 
+by replication in the nasal cavity can be transported by the air (to the lungs), in the nerves (to 
+the brain) and possibly through the blood or lymph, by a Trojan horse mechanism (to deeper 
+organs). 
+Inhalation
+ACE2
+receptor
+Viral RNA
+release
+Endocytosis
+Protease
+Upper
+airways
+infection
+Viral
+replication
+Membrane
+fusion
+Blood and
+lymphatic
+transport
+Axonal
+transport
+Respiratory
+transport
+Lungs secondary infection
+through lower
+respiratory tract
+Brain secondary
+infection through
+olphactory nerve
+Multiple organ secondary
+infection through blood,
+lymph or nerves
+A
+B
+C
+
+Figure 3: Viral kinetics of SARS-CoV-2. (A) Viral kinetics of SARS-CoV-2. Average viral 
+load in the nasal cavity and the throat as a function of time since infection. Data from 
+Killingley et al. (2022), using a strain close to Wuhan-1. (B) Viral load expressed in genome 
+units (GU) is obtained by reverse transcription followed by quantitative PCR (RT-qPCR), 
+which measures the quantity of viral genome copies in a viral solution. (C) Viral load 
+expressed in plaque forming units (PFU) is measured using plaque assays, which consist 
+counting lysis plaques in a cell monolayer in contact with a viral solution. Data of panel A are 
+obtained for Vero Cells, using a 3 mL solution per swab.  
+Figure 4: Calibration of the epidemic quantum. (A) Curve of the proportion P of infected 
+individuals aboard the cruise ship Diamond Princess (blue) and aboard the French aircraft 
+carrier Charles de Gaulle (red) as a function of time t. The solid lines correspond to the best 
+fit by exponential growth, resulting in a growth rate σ = 0.23 day−1 and σ = 0.17 day−1, 
+respectively. The dashed horizontal line is the theoretical collective immunity limit P → 1/R. 
+The dotted line is the numerical integration of the infection equation SI-(6). On both boats, 
+the persistence of viral particles is limited by their deactivation timescale rather than by the 
+ventilation. The fast transmission in the Charles de Gaulle French aircraft carrier (Laval et al., 
+15
+10
+5
+0
+104
+102
+100
+106
+108
+Time since infection (days) 
+Viral load (per mL)
+GU
+PFU
+PFU
+GU
+Nasal
+swab
+Nose
+Throat
+Successive dilutions
+Dilution
+Amplification
+Reverse transcription
+Pharyngeal
+swab
+PFU
+100
+0
+10-2
+10-4
+100
+RNA
+DNA
+Primer
+A
+C
+B
+Cycle
+Fluorescence
+0
+10 20
+30 40
+Time (days) 
+date
+100
+10-1
+10-2
+10-3
+10-4
+50
+40
+30
+20
+10
+0
+Epidemic prevalence P
+Charles de Gaulle
+aircraft carrier
+Diamond Princess
+cruise ship
+8
+6
+4
+2
+0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+Time since infection (days) 
+growth rate σ (day-1)
+Infectious period T
+Incubation period Tm
+Viral timescales (day)
+Rescaled
+transmissibility
+Reproduction number R
+8
+6
+4
+2
+0
+4
+5
+6
+3
+2
+1
+0
+-0.25
+0
+0.25
+2022
+2021
+2020
+Wuhan-1
+alpha
+delta
+omicron
+BA.1
+BA.2
+T
+HNEP
+Calu-3
+Tm
+A
+B
+C
+D
+
+2022) is due to the lack of filtration of the recycled air. Over 1,767 people onboard, crew and 
+commandos together, 1,288 were infected in a short period of time. The indoor relevant 
+volume is estimated around 150 × 103 m3. The fast transmission on the Diamond Princess 
+boat (Bazant and Bush, 2021) is due to a mostly recycling (at 70%, no HEPA filters), air 
+conditioning, due to the cold weather conditions (−5°C). The surface accessible to passengers 
+is 78 × 103 m2 and the ceiling height is 2.4 m. The indoor volume, 187 × 103 m3, is rather 
+large compared to the number of people, N = 3,711, crew and passengers together. (B) Model 
+dimensionless transmissibility as a function of time t, in days, after infection. The viral 
+transmissibility of a person (index case) is defined as the rate of emission of replicable viral 
+particles. In first approximation, transmissibility is proportional to the viral load in the upper 
+respiratory tract. The incubation period Tm is defined as the time between infection and 
+maximum transmissibility, and the infectious period T as the average time between infections 
+of the index and secondary cases. (C) Relation between the epidemic reproduction rate R, 
+defined as the mean number of secondary cases per index case, and the epidemic growth rate 
+σ, predicted by the Euler-Lotka equation SI-(7), for a given transmissibility curve. It gives R 
+= 4.8 for the cruise ship and R = 3.2 for the aircraft carrier. The red line is the approximation 
+for small growth rate: σ = (R − 1)/T. (D) Incubation period Tm and infectious period T for the 
+wild-type strain, Wuhan-1, and of variants Alpha (B1.1.7), Delta (B.1.617.2), Omicron BA.1 
+and Omicron BA.2, represented at the date of the first case in France. Viral replication time-
+scale is measured using primary cultures of human nasal epithelial cells (hNECs), human 
+lung cell that expresses abundant ACE2 and TMPRSS2 (Calu-3). It is defined as the time 
+needed to multiply the viral particles by 1 billion (109).	 
+Figure 5: Epidemiologic and biological characteristics of the wild-type strain, Wuhan-1, and 
+of variants Alpha (B1.1.7), Delta (B.1.617.2), Omicron BA.1 and Omicron BA.2. (A) Viral 
+infectivity, defined as the average number of lysis induced by a viral particle on a Vero cell 
+culture. It is expressed in plaque forming units per genome units (PFU/GU). (B) Mean viral 
+2022
+2021
+2020
+date
+2022
+2021
+2020
+date
+104
+103
+102
+101
+full vaccination
+no vaccination
+109
+1010
+full vaccination
+no vaccination
+full vaccination
+no vaccination
+10-3
+10-2
+Wuhan-1
+Wuhan-1
+Alpha
+Delta
+Omicron
+Alpha
+Delta
+Delta (21J)
+Delta (21I)
+Delta (21A)
+Beta
+Gamma
+Omicron
+BA.1
+BA.2
+2022
+2021
+2020
+date
+2022
+2021
+2020
+date
+Wuhan-1
+Alpha
+Delta
+Omicron
+BA.1
+BA.2
+BA.1 (21K)
+BA.2 (21L)
+Integral viral exhalation ħ (quanta)
+Lower airway infectious dose (GU)
+Upper airway infectious dose (GU)
+Infectivity (PFU/GU)
+106
+105
+104
+A
+B
+E
+C
+D
+
+exhalation during the whole contagious time, expressed in quanta. The measurement is 
+deduced from the variant frequency curves. (C) Infectious dose in upper airways, measured in 
+genome units (GU). (D) Infectious dose in lower airways, measured in genome units (GU). 
+(E) Maximum likelihood phylogenies inferred from spike nucleotide sequence; Distance 
+corresponds to number of mutations. Source: Gisaid/ Nextstrain. 
+ 
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+ 
+ 
+
+At the crossroads of epidemiology and biology: bridging the
+gap between SARS-CoV-2 viral strain properties and
+epidemic wave characteristics.
+— Supplementary Information —
+Florian Poydenot, Jacques Haiech, Alice Lebreton and B. Andreotti
+January 30, 2023
+Abstract
+In this supplementary information document, we provide a self-contained review of the foundations
+of standard epidemiological models, aimed at being accessible with basic knowledge of physics and
+mathematics. The particular formulation of epidemiological equations used to construct the figures of the
+review has been published in a companion paper [1]. We review the assumptions and the parametrization
+of the model using epidemiological data. We furthermore provide the table of published measurements
+used in the review paper.
+1
+Epidemic growth rate and reproduction rate
+Sustained airborne transmission of SARS-CoV-2 from an index case to contacts depends on biological
+and immune characteristics of the index case (transmissibility) and the contacts (susceptibility), but also
+on social and physical characteristics such as the number of available contacts, the duration of exposure,
+and the ventilation and mask wear during the contact (see Figure S1). We have developed the social and
+physical aspects of transmission in a companion paper [1], and we discuss here the calibration of biological
+quantities, both from a molecular and epidemiologic point of view.
+Starting from the molecular description of the index case, viral kinetics determine both the transmis-
+sibility through the viral shedding rate, and the dynamics of transmission through the course of evolution
+of the illness. The viral kinetics is standardly described in the literature by an exponential growth of the
+viral load (replication and shedding) followed by an exponential decay (immunity response). However, the
+SARS-CoV-2 human challenge trial [2] has provided unprecedented time resolved data showing a more
+rounded viral load curve (Fig. 3). Here, we parametrize the viral load V by the law
+V = Vψ(t)
+(1)
+where V is a characteristic concentration for a particular infected person and ψ(t) the rescaled transmissibil-
+ity at a time t after infection (Fig. 4-B). We therefore assume that infected people with different maximum
+viral load present on the average the same viral kinetics up to a constant. The infectious period T is defined
+as the average time from infection, weighted by the transmissibility:
+T =
+� ∞
+0
+τψ(τ)dτ
+� ∞
+0
+ψ(τ)dτ
+(2)
+It is worth noting that the integrals run from t = 0, which is the infection time. More precisely, we choose
+a convenient normalization of ψ(t) to ensure that it is dimensionless:
+T =
+� ∞
+0
+ψ(τ)dτ
+and
+T 2 =
+� ∞
+0
+τψ(τ)dτ
+(3)
+1
+
+index case
+secondary
+case
+N-1 contacts
+transmissibility
+susceptibility
+ventilation
+dilution
+facial mask
+N-1 contacts
+duration Δt
+work
+restaurant
+public
+transportation
+index case
+home
+non-infectible fraction
+of the population
+infection rate
+Figure S1: (a) Transmission of SARS-CoV-2 in a certain space depends on the duration ∆t, on the number
+N of people present, on the the viral emission rate (transmissibility), on the susceptibility of contacts to
+be infected and on the viral dilution ϵ due to ventilation and to face masks. (b) The mean number of
+people infected by an index case is the average over possible situations weighted by the duration ∆t. Four
+examples are schematized. From left to right: high number N but short time ∆t and moderate dilution
+factor ϵ (public transportation); Long time ∆t but moderate number N and low dilution factor ϵ (work);
+Long time ∆t and high dilution factor ϵ (restaurant); Very long time ∆t with a high dilution factor ϵ but
+small N (household transmission). (c) Transmissibility is by definition the viral emission rate, which is
+proportional to the replicable viral load. The infection rate I is the number of new infected people per unit
+time. It is related to the infection rate over the past period, to the epidemic reproduction rate, and to the
+fraction of non-infectible people.
+Here, we have introduced a phenomenological equation of the form
+log ψ = log ψm − a(t − Tm)2
+1 + bt
+(4)
+2
+
+to fit these data with the same number of parameters. V is a characteristic viral load. Tm is the incubation
+period, defined as the time from infection to the maximum viral load. The parameters a and b determine the
+initial growth rate and final decay rate. The constant ψm is fixed by the normalization condition. We have
+performed a mapping of the double exponential model onto equation (4) in order to deduce the parameters
+that best describe the average kinetics for different variants. The resulting curves are shown in Figure S2.
+8
+6
+4
+2
+0
+20
+15
+10
+5
+0
+15
+10
+5
+0
+104
+102
+106
+108
+104
+102
+100
+106
+108
+Delta
+ Wuhan
+ Alpha
+ Delta
+ Omicron BA.1
+ Omicron BA.2
+Figure S2: (a) Modelled average viral kinetics for younger and older patients, vaccinated or not, obtained
+by mapping the double exponential kinetics to equation (4). The characteristics are those of the Delta strain.
+From N´eant et al. [3] (b) Modelled average viral kinetics for the successive variants of interest, obtained
+using the data reported in table 1. The parameters of equation (4) are reported in table 2.
+The viral transmissibility of a particular infected person (the index case) is defined here as the rate
+of emission of replicable viral particles (Fig. S1). Under the simplifying assumption that emission is
+proportional to the viral load in the upper respiratory tract, the transmissibility is proportional to the viral
+load in the upper respiratory tract. We therefore parametrize the concentration of viral particles in the
+exhaled air by the law:
+C = Cψ(t)
+(5)
+where C is a characteristic concentration for a particular infected person and ψ(t) the rescaled transmissi-
+bility at a time t after infection (Fig. 4-B).
+The infection rate I is defined as the mean number of infected people per unit time in a given population.
+At time t, a secondary case infection is induced by an index case contaminated before, at a time t−t′, with
+a probability proportional to ψ(t′). Introducing R, the epidemic reproduction rate, defined as the mean
+number of secondary cases per index case, I obeys an integral equation (see Grassly and Fraser [4] for a
+review and derivation):
+I(t) = A(t)R
+T
+� ∞
+0
+I(t − t′)ψ(t′)dt′
+(6)
+where A(t) is the fraction of the population susceptible to be infected (Fig. S1). A(t)R(t)ψ(t′)/T is
+classically known as the infectiousness at time t of the index case that has been infected for a duration t′.
+It is the product of the contact rate AR/T at the time of contact and the biological factor of infectiousness
+since the onset of infection ψ(t′). Importantly, equation (6) remains valid even when the transmission time
+is comparable to the epidemic growth timescale as it takes into account the viral kinetics.I(t) and A(t) are
+unknowns, and must be solved together. When a small fraction of the population is immune, A is close to 1,
+and the equation admits an exact exponential solution to I(t), I(t) = I0e−σt. Plugging this into equation
+(6), we can relate the growth rate σ to the epidemic reproduction rate R by the Euler-Lotka equation:
+R =
+�
+ψ(t)dt
+�
+ψ(t) exp(−σt)dt
+(7)
+3
+
+This relation is plotted in figure 4-C for the rescaled viral load shown in figure 4-B. A discussion of the
+Euler-Lotka renewal equation in a more general context can be found in chap. 13 of Martcheva [5].
+The epidemic prevalence P is defined as the fraction of the population that has been infected in the
+past:
+P(t) = N −1
+� t
+−∞
+I(t′)dt′
+(8)
+Then, assuming that each infection leads to long term immunization, a simple approximation of the sus-
+ceptible fraction is A(t) = 1 − P(t). However, A may be very different from 1 − P due to vaccination, or
+to the gradual loss of immunity.
+2
+Reference point for the evaluation of the epidemic reproduction
+rate R0
+The epidemic reproduction rate R is the average number of secondary infections per index case. It com-
+bines biological factors that determine infection susceptibility and viral transmissibility, and social factors
+(Fig. S1). To define a reference reproduction rate R0 that would characterize the transmissibility of a given
+viral strain, it is necessary to choose a reference state of the social behaviors. The simplest choice is to
+define R0 as the epidemic reproduction rate when society ignores the virus and behaves ”normally”. This
+is, by definition, only possible when the epidemic starts. Figure S3 shows the initial stage of SARS-CoV-2
+epidemics in different French departments and different European countries, before the first 2020 lock-
+down. It can be safely assumed that the number of deaths per unit time was proportional to the number of
+cases, in this initial stage. The curves show the multiplicative nature of the epidemic and is direct evidence
+of the effect of the lockdown. The epidemic reproduction number is around the same value R0 = 6.9 in
+both departments/countries where the epidemic arrived earlier or later. For small values of the epidemic
+growth rate σ, the Euler-Lotka equation can be linearized:
+R ≃
+�
+ψ(t)dt
+�
+ψ(t)(1 − σt)dt ≃ 1 + σT
+(9)
+This popular approximation leads to a much lower value of the epidemic reproduction number, around
+R0 = 3. This difference can be ascribed to the non-linearity in the relation between σ and R, at large σT,
+when the growth time σ−1 is comparable or larger than the infectious period T.
+3
+Relation between the reproduction rate and the infectious quan-
+tum
+The model described in this section is discussed in a companion paper [1] in the context of the social
+and physical aspects of transmission. We recall here its underlying assumptions and the calibration of the
+biological factors in the transmission risk. We assume here that a single viral particle initiates the infection
+when it penetrates a vulnerable locus where conditions are favorable. The probability that at least one viral
+particle manages to enter a cell and replicates is independent of the presence of others viral particles. It
+depends on factors such as the type of cells or the density of ACE2 receptors. Wells [6] introduced the
+notion of dose and quantum to describe this probability. For a person having inhaled an intake dose d, the
+probability law of infection p(d) takes the form p(d) = 1 − e−ad. a−1 is the infection dose of the person
+considered. Its average over the population, ¯a−1, is by definition the quantum of infection. The product
+ad is therefore the dose, expressed in infectious quanta. At small ad, the probability of infection can be
+linearized: p(d) ≃ ad. This excludes super-spreading events, which occur when an infected person with
+a large exhaled viral concentration C attends an under-ventilated place, leading to multiple simultaneous
+infections. The total number of exhaled viral particles is, on average, equal to
+�
+¯qCψ(t)dt = ¯qCT, where
+¯q is the mean exhalation flow rate. This number can be expressed in infectious quanta to define the mean
+integrated quantum emission ¯h:
+¯h = ¯qC¯aT
+(10)
+4
+
+50
+40
+30
+20
+10
+10-2
+10-1
+100
+101
+102
+0
+-10
+-20
+ Italy
+France
+Germany
+Sweden
+60
+50
+40
+30
+20
+10
+0
+103
+100
+101
+102
+Bouches du Rhone
+ Loiret
+ Seine Saint-Denis
+ Paris
+Côtes d’Armor
+ Alpes-de-Haute
+-Provence
+Figure S3: (a) Curve of the cumulative number of deaths D as a function of time, in days, in different
+French departments. The time axis is shifted so as to superimpose the curves in the first phase of the
+epidemic on a jointly fitted exponential (σ = 0.28 day−1). For each curve, the best fit gives the time
+at which, statistically, the first death would have occurred on average, given the epidemic history. The
+further away departments are from major cities, the later the epidemic occurs and the more lockdown,
+imposed at the same date everywhere, has limited the number of deaths. (b) Curve of the cumulative
+number of COVID-19 deaths normalized by the same number, on the day of lockdown, as a function of
+time, in days, relative to the date of lockdown. Although the epidemic arrived at very different dates in the
+countries represented, the curves superimpose on a master curve, which shows the multiplicative nature
+of the epidemic. The best fit of the first phase of the epidemic by an exponential gives the growth rate
+σ = 0.28 ± 0.03 day−1, which corresponds to R0 = 6.9 ± 2. This is much higher than the common value
+deduced from the linearization of the Euler-Lotka equation (9), which underestimates R0 to 3.
+The mean integrated quantum emission ¯h measures the transmissibility and encodes all the biological
+part of the risk (Fig. S1). It is defined as an average over the sub-population attending the public space
+considered of the number of quanta that would have been exhaled if an infected person were there. It may
+depend on the particular activity taking place in the public space through the mean inhalation rate ¯q.
+We consider a virtual situation in which N people would stay in a certain place during their entire
+infectious period. Consider that an infected person amongst the N people. It would exhale a dose d = ¯h/¯a
+or equivalently a number of infectious quanta ¯ad = ¯h. Introducing the dilution factor ϵ between exhalation
+and inhalation (Fig. S1), which characterizes the ventilation and dispersion efficiency, as well as the effect
+of face masks, the inhaled dose (in quanta) is ¯hϵ. The average secondary case number is therefore:
+r = (N − 1)¯ad = (N − 1)ϵ¯h
+(11)
+It is proportional to the total number of exhaled infectious quanta, to the number of infectible people
+(Fig. S1). The epidemic reproduction rate R deduces by averaging over the population, weighting the
+different places in which they live according to the time they spent inside (Fig. S1):
+R = ⟨ϵ(N − 1)⟩ ¯h
+(12)
+R is the product of three terms (Fig. S1): ⟨ϵ(N − 1)⟩ characterizes the social behaviour, including the
+effect of ventilation and face masks; ¯h characterizes the biological factors. The mean integrated quantum
+emission ¯h can be determined using equation (12), if social behaviours are known. Figure S4 shows the
+calibration of the mean integrated quantum emission using the epidemic evolution in secondary schools in
+the United Kingdom. We discuss the effect of masks on this determination in a companion paper [1].
+5
+
+1200
+1000
+800
+600
+400
+200
+0
+8
+6
+4
+2
+0
+0.3
+0.2
+0.1
+0.0
+20
+15
+10
+5
+0
+70
+60
+50
+40
+30
+20
+10
+0
+Figure S4: The epidemic evolution for secondary school pupils, between lockdown and holidays. (a) Cases
+per million people in the United Kingdom, from 1 February to 12 April. Blue: pupils from age 10 to age
+14, with no mandatory mask. The best fit by an exponential provides the reproduction number: R = 1.45
+(σ = 0.052 day−1) during school period vs R = 0.76 before (σ = −0.037 day−1) and R = 0.53
+(σ = −0.087 day−1) after. Red: pupils from age 15 to age 19, with mandatory masks. The best fit by
+an exponential provides the reproduction number: R = 1.10 during school period vs R = 0.78 before
+and R = 0.63 after. Uncertainties are typically 5%. The contribution of schools to the epidemic rate,
+in the absence of mandatory masks, is estimated to R = 0.8 in March 2021. (b) Typical ventilation in
+British schools in March, as deduced from Vouriot et al. [7], in a classroom of volume per pupil 10 m3.
+Left axis: CO2 concentration as a function of time. The pupils are not present in the classroom during
+the periods of time shown in gray. The average concentration is C = 1070 ppm. Right axis: deduced
+transmission risk r. Considering that the average school time for secondary schools is 27.5 hours per
+week, a total viral emission ¯h = 270 quanta is deduced. It corresponds to a typical viral emission rate
+¯q¯aC = 1.6 quanta/hour and an emission rate at maximum on the order of 3.3 quanta/hour.
+4
+Relative transmissibility and infectivity of successive variants
+Variants only interact through immunization. At low prevalence, each variant epidemic can be considered
+as independent from the others. The results of massive RT-PCR tests can then be used to determine the in-
+fectious quantum of successive variants of concern. Let us consider the simplest case were a variant noted
++ replaces a variant noted −. The local epidemic growth rates σ− and σ+ are measured during the replace-
+ment period. The infection rate of the strains are written I− = I− exp(σ−t) and I+ = I+ exp(σ+t). The
+relative prevalence of the new variant therefore obeys the logistic equation:
+I+
+I+ + I−
+=
+1
+1 + I−/I+ exp ((σ− − σ+) t)
+(13)
+The best fit of the new variant relative prevalence by equation 13 gives the difference σ− − σ+ within a
+few percent uncertainty. The growth rate σ+ of the new variant is determined by fitting the evolution of
+the number of new cases during the same period of time. Using the Euler-Lotka equation, the epidemic
+reproduction rates R− and R+ are deduced. Taking the ratio R+/R−, the social component of the repro-
+duction rate ⟨ϵ(N − 1)⟩ is eliminated, leaving the ratio of total viral emission ¯h+/¯h−. For simplicity we
+have ignored possible small differences of immunity between the strains Wuhan-1, Alpha and Delta.
+Figure S5 shows the replacement of the Wuhan strain by the Alpha strain during the winter 2021 and the
+replacement of the Alpha strain by the Delta strain during the summer 2021, in France. The transmissibility
+of the strain Alpha (resp. Delta), as measured by ¯h, is 1.7 (resp. 3.4) times larger than the wild strain. The
+description of the transition from the strain Delta to the strain Omicron is described in the next section.
+6
+
+100
+80
+60
+40
+20
+0
+10-2
+10-1
+100
+100
+80
+60
+40
+20
+0
+10-2
+10-1
+100
+105
+104
+103
+104
+103
+102
+Jan 1, 2021
+Feb 1, 2021
+Mar 1, 2021
+Apr 1, 2021
+Jun 1, 2021
+Jul 1, 2021
+Aug 1, 2021
+Sep 1, 2021
+60
+50
+40
+30
+20
+10
+0
+100
+80
+60
+40
+20
+0
+Figure S5: (a) Frequency of the variant Alpha in RT-PCR tests performed in France, as a function of time.
+The time origin t = 0 corresponds to January 1, 2021. The best fit by the logistic equation (12) gives
+σ+ − σ− = 0.077 day−1. Insert: number of Alpha cases identified by positive RT-PCR tests in France.
+The growth rate at the emergence of the alpha epidemic wave is σ+ = 0.070 day−1. (b) Frequency of the
+variant Delta in RT-PCR tests performed in France, as a function of time. The time origin t = 0 corresponds
+to January 1, 2021. The best fit by the logistic equation (12) gives σ+−σ− = 0.135 day−1. Insert: number
+of Alpha cases identified by positive RT-PCR tests in France. The growth rate at the emergence of the delta
+epidemic wave is σ+ = 0.147 day−1.
+5
+Relative transmissibility and susceptibility of vaccinated people
+The relative susceptibility S is by definition the ratio of the probability that vaccinated people get infected
+to the probability that unvaccinated, never infected people, with the same immunological characteristics,
+get infected. It depends on age and on the vaccination status (type of vaccine, vaccination date). The
+relative susceptibility can be measured from the transmission rate of sub-populations. Such measurements
+are well converged statistically but suffers from social biases (mask wearing, attendance of restaurants and
+bars, attendance of public spaces). Alternatively, it can be measured from household transmission, which
+removes an important bias: the vaccination status of the index case is known. On the other hand, social
+biases persist and the statistics is in general much lower. Other biases like age can be adjusted. However,
+as age, vaccination status and intrinsic susceptibility to infection (quality of the immunity) are strongly
+correlated, it becomes problematic to exhibit a single quantity characterizing vaccination efficiency. From
+the molecular biology point of view, relative susceptibility characterizes the neutralization of virus by
+the antigenic response. There currently exists no calibration relating molecular aspects to epidemiologic
+aspects.
+The relative transmissibility T is by definition the ratio of the probability that vaccinated people with
+the virus (index case) infect other people (secondary cases) to the same probability for unvaccinated people.
+From the molecular biology point of view, the relative transmissibility can be measured as the ratio of the
+integral viral emission between vaccinated and unvaccinated people. From the epidemiological point of
+view, the relative transmissibility can only be measured through contact tracing and in particular household
+transmission statistics. The crude measurement is the ratio of the secondary attack rates, conditioned by
+the index vaccination status. The measurement suffers from social biases and a lack of statistics. Age,
+vaccination status and intrinsic transmissibility (quality of the immunity) are strongly correlated. It is
+therefore problematic to exhibit a single quantity characterizing vaccination efficiency against transmission,
+after an adjustment.
+Figure S6 (d) shows a compilation of measurements of S and T for an up to date vaccination status.
+Although dispersed, the data show a clear common decrease of relative susceptibility S and relative trans-
+7
+
+25
+20
+15
+10
+5
+0
+25
+20
+15
+10
+5
+0
+Unvaccinated
+Alpha
+Delta
+Omicron BA.1
+Omicron BA.2
+Dose 2, 12-24 weeks
+Dose 3, <12 weeks
+Dose 3, >24 weeks
+Dose 2, <12 weeks
+Dose 1, <2 weeks
+Dose 2, >24, weeks
+Dose 3, 12-24 weeks
+Dose 1, >2 weeks
+10-5
+10-4
+10-3
+10-5
+10-4
+10-3
+2.
+1.5
+1
+0.5
+0
+1
+0.8
+0.6
+0.4
+0.2
+0
+1
+0.8
+0.6
+0.4
+0.2
+0
+Dec 15, 2021
+Dec 15, 2021
+Jan 1, 2022
+Figure S6: Incidence of SARS-CoV-2 infection for all sub-populations with different vaccination status, as
+a function of time, in France. (a) Raw incidence data for the Delta variant. (b) Raw incidence data for the
+Omicron BA.1 variant. The time origin t = 0 corresponds to December 7th, 2021. The best fit by expo-
+nentials with the same rate for all vaccination status is superimposed. The good fit confirms that vaccinated
+and unvaccinated people infect each other sufficiently to share the same overall dynamics. (c) Relative sus-
+ceptibility between fully vaccinated people and unvaccinated people, estimated from the relative incidence
+Ij/Nj, as a function of relative susceptibility between fully vaccinated people and unvaccinated people.
+(d) Relative transmissibility T as a function of relative susceptibility S for up to date vaccination status. T
+and S both should tend to 1 (no effect of vaccination on transmission) and to 0 (effective barrier immunity)
+together. Solid line: phenomenological fit T = 1 − (1 − S)2.
+missibility T . We know that they both should tend to 1 (no effect of vaccination on transmission) and to 0
+(effective barrier immunity) together. A good phenomenological fit to the data is provided by the relation
+T = 1 − (1 − S)2 and is shown in solid line in figure S6 (d).
+As a simplifying assumption, we consider that contacts of vaccinated and unvaccinated people are
+similarly composed: then, each person is statistically in contact with vaccinated and unvaccinated people,
+8
+
+100
+101
+102
+103
+104
+105
+Jan 1, 2022
+Feb 1, 2022
+Dec 1, 2021
+60
+40
+20
+0
+BA.1
+BA.2
+Figure S7: (a) Number of new cases of the variants BA.1 and BA.2 in France. The black and green solid
+lines are the best exponential fit in two different periods of time.
+following the fractions of the whole society. We consider a division of society into J classes (according
+to age, vaccination status, etc.) whose size is denoted Nj (such that N = �
+j Nj). We denote by Sj and
+Tj, the relative susceptibility and relative onward transmissibility associated with the class j. The infection
+rate Ij, defined as the mean number of infected people per unit time in the class j, obeys the equation:
+Ij(t) = Sj(1 − Pj(t))R
+T
+Nj
+N
+� ∞
+0
+�
+k
+TkIk(t − t′)ψ(τ)dt′
+with
+T =
+� ∞
+0
+ψ(τ)dt′
+(14)
+The epidemic reproduction rate R is here defined for a virtual society of unvaccinated, never infected
+people. At small P, these equations admit an exact exponential solution of the form:
+Ij ∝ Sj(1 − Pj(t)) Nj
+N exp(σt)
+(15)
+whose growth rate σ is related to R by the generalised Euler-Lotka equation:
+R =
+�
+ψ(t)dt (�
+j Nj)
+�
+ψ(t) exp(−σt)dt (�
+j SjTjNj)
+(16)
+Figure S6 (a) and (b) shows that incidences for all sub-populations share the same growth rate, meaning
+that vaccinated and unvaccinated people infect each other enough to obey the same dynamics. Under this
+assumption, the relative incidence Ij/Nj provides an estimate of the relative susceptibility Sj for different
+vaccination schemes, displayed in Figure S6 (c). For the Delta strain, the susceptibility is ordered from 0.15
+for a complete vaccination scheme in 3 doses to 1, the unvaccinated reference. For Omicron, on the other
+hand, the susceptibility is mostly 1 within noise except for people vaccinated with two doses but delaying
+or refusing the third one. This may point to biases introduced by different social behaviors correlated with
+the vaccination status.
+During three weeks, the growth rate of Omicron was σ+ = 0.23 day−1 vs σ− = 0.0 day−1 for Delta.
+The transmissibility of the strain Omicron BA.1, as measured by ¯h, is 1.8 times larger than the Delta strain,
+6 times larger than the wild strain. Figure S7 (a) shows the number of cases of Omicron BA.1 and BA.2 in
+France during the replacement period. An exponential phase is observed during two short periods of time.
+The transmissibility of the strain Omicron BA.2 is 1.7 larger for BA.2 than BA.1, consistently between the
+two estimates.
+9
+
+Dec 1, 2019
+Apr 1, 2020
+Aug 1, 2020
+Dec 1,2020
+Apr 1, 2021
+Aug 1, 2021
+Dec 1, 2021
+date
+Wuhan-1
+Alpha
+(21J)
+(21I)
+Delta 
+Beta
+Gamma
+Omicron
+BA.1 (21K)
+BA.2 (21L)
+(21A)
+Alpha
+Delta (21I)
+Delta (21A)
+Delta (21J) 
+Beta
+Gamma
+Omicron BA.1 (21K)
+Omicron BA.2 (21L)
+Figure S8: Phylogenetic tree up for the strains included in the article: Alpha, Beta, Gamma, Delta, Omicron
+BA.1 and Omicron BA.2. Source: GISAID/Nextstrain [8].
+10
+
+Dec 1, 2019
+May 1, 2020
+Oct 1, 2020
+Mar 1,2021
+Aug 1, 2021
+Jan 1, 2022
+Nov 1, 2022
+date
+Wuhan-1
+Alpha
+Delta 
+Beta
+Gamma
+Omicron
+BA.2 (21L)
+BQ.1.1
+(22E)
+BA.2.75 (22D)
+XBB (22F)
+BA.4 (22A)
+BA.5
+(22B)
+BA.1 (21K)
+Alpha
+Delta (21I)
+Delta (21A)
+Delta (21J) 
+Beta
+Gamma
+Omicron BA.1 (21K)
+Omicron BA.2 (21L)
+Omicron BA.4 (22A)
+Omicron BA.5 (22B)
+Omicron BA.2.75 (22D)
+Omicron BQ.1.1 (22E)
+Omicron XBB (22F)
+Figure S9: Phylogenetic tree including the Omicron variants: BA.1, BA.2, BA.4, BA.5, BA.2.75 and XBB.
+Source: GISAID/Nextstrain [8].
+11
+
+6
+Supplementary Tables
+Table 1: References used for viral kinetics
+Strain
+Maximum
+viral
+load
+(log10 copies/mL)
+Growth
+time
+(days)
+Decay time (days)
+Reference;
+com-
+ments
+Wuhan
+7.2
+0.35
+0.62
+[2]
+Wuhan
+8.1
+0.22
+2.7
+[9]
+Wuhan
+8.2
+0.30
+0.53
+[10]
+Wuhan
+9.7
+0.19
+1.2
+[3]; old people
+Wuhan
+9.6
+0.22
+0.91
+[3]; young people
+Wuhan
+7.6
+1.4
+[11]
+Wuhan
+9.2
+[12]
+Alpha
+8.0
+0.26
+0.47
+[10]
+Alpha
+7.8
+1.3
+[11]
+Delta
+7.6
+1.1
+[11]
+Delta
+8.3
+0.21
+0.44
+[10]
+Delta
+7.4
+[13]
+Omicron
+6.9
+[13]
+Omicron
+7.1
+[13]
+Omicron
+7.0
+1.2
+0.70
+[14]
+Unvaccinated
+8.1
+0.26
+0.56
+[10]
+Vaccinated
+8.1
+0.24
+0.41
+[10]
+Delta
+8.1
+0.35
+0.47
+[10]
+Delta
+8.0
+[12]
+Delta
+unvacci-
+nated
+8.6
+[12]
+Delta vaccinated
+8.2
+[12]
+Delta vaccinated
+7.7
+[15]
+Delta
+unvacci-
+nated
+7.6
+[15]
+Delta
+symp-
+tomatic
+vacci-
+nated
+8.0
+[15]
+Delta
+symp-
+tomatic
+unvacci-
+nated
+7.9
+[15]
+Delta
+asymp-
+tomatic
+vacci-
+nated
+7.2
+[15]
+Delta
+asymp-
+tomatic
+unvacci-
+nated
+7.4
+[15]
+Delta
+unvacci-
+nated
+7.2
+[16]
+Delta vaccinated
+7.4
+[16]
+Delta vaccinated
+7.7
+[17]
+Delta
+unvacci-
+nated
+7.3
+[17]
+Delta
+7.3
+0.25
+0.89
+[18]
+Omicron BA.1
+5.7
+0.70
+0.76
+[18]
+12
+
+Omicron BA.1
+7.3
+0.40
+0.47
+[10]
+Omicron BA.1
+7.8
+[12]
+Omicron
+BA.1
+boosted
+6.9
+[19]
+Omicron
+BA.1
+unvaccinated
+7.1
+[19]
+Omicron
+BA.2
+boosted
+7.0
+[19]
+Omicron
+BA.2
+vaccinated
+7.2
+[19]
+Omicron
+BA.2
+unvaccinated
+7.4
+[19]
+Table 2: Parameters of the viral kinetics model
+Strain
+log ψm
+(log10 copies/mL)
+a
+(log10 copies/mL)
+b (days−1)
+Tm (days)
+Wuhan
+7.8
+0.28
+0.19
+7.0
+Alpha
+7.8
+0.40
+0.33
+5.8
+Delta
+8.0
+0.58
+0.40
+4.9
+BA.1
+7.5
+0.51
+0.30
+5.1
+BA.2
+7.8
+0.51
+0.30
+5.1
+Table 3: References used for the PFU/GU ratio
+Strain
+PFU/GU
+Cells
+Replication
+Reference
+Wuhan
+85000
+Vero
+In vivo
+[2]
+Wuhan throat
+94000
+Vero
+In vivo
+[2]
+Wuhan
+13000000
+Vero
+In vivo
+[12]
+Alpha
+430000
+Vero
+In vivo
+[20]
+Delta
+43000
+Vero
+In vivo
+[20]
+Delta
+590000
+Vero
+In vivo
+[12]
+Omicron BA.1
+1500000
+Vero
+In vivo
+[12]
+Delta
+unvacci-
+nated
+1300000
+Vero
+In vivo
+[12]
+Delta vaccinated
+2500000
+Vero
+In vivo
+[12]
+Delta
+unvacci-
+nated
+1400000
+Vero
+In vivo
+[12]
+Delta vaccinated
+63000
+Vero
+In vivo
+[16]
+Delta
+unvacci-
+nated
+70000
+Vero
+In vivo
+[16]
+Delta
+200
+hNEC
+In vitro
+[21]
+Delta
+2100
+Vero
+In vitro
+[21]
+Delta
+600
+Calu
+In vitro
+[21]
+Omicron
+4400
+hNEC
+In vitro
+[21]
+Omicron
+5600
+Vero
+In vitro
+[21]
+Delta
+22000
+hNEC
+In vitro
+[22]
+Omicron
+17000
+hNEC
+In vitro
+[22]
+Alpha
+360
+Vero-TMPRSS2
+In vitro
+[20]
+Delta
+90
+Vero-TMPRSS2
+In vitro
+[20]
+13
+
+Epsilon
+1400
+Vero-TMPRSS2
+In vitro
+[20]
+Alpha
+210
+Calu-3
+In vitro
+[20]
+Delta
+60
+Calu-3
+In vitro
+[20]
+Beta
+1700 − 2800
+Vero
+In vitro
+[23]
+SARS-CoV-1 Ur-
+bani
+2300
+Vero
+In vitro
+[24]
+Alpha
+700
+Vero
+In vitro
+[25]
+Wuhan
+6400
+Vero
+In vivo
+[26]
+Table 4: References used for the susceptibility and transmissibility
+Strain
+Susceptibility
+Transmissibility
+Odds ratio type
+Reference
+Alpha
+0.15
+0.32
+adjusted
+[27]
+Alpha
+0.34
+0.29
+adjusted
+[28]
+Wuhan and Alpha
+0.53
+adjusted
+[29]
+Delta
+0.15
+raw
+This study
+Delta
+0.16
+adjusted
+[30]
+Delta
+0.4
+raw
+[30]
+Delta
+0.19
+0.5
+adjusted
+[27]
+Delta
+0.34
+0.45
+raw
+[31]
+Delta
+0.53
+0.48
+adjusted
+[32]
+Delta
+0.61
+1.1
+raw
+[33]
+Delta
+0.36
+0.91
+raw
+[34]
+Delta
+0.33
+0.71
+adjusted
+[34]
+Delta
+0.29
+adjusted
+[35]
+BA.1
+0.77
+adjusted
+[35]
+BA.1
+0.53
+adjusted
+[30]
+BA.1
+0.83
+raw
+[30]
+BA.1
+0.53
+0.83
+adjusted
+[19]
+BA.1
+0.67
+raw
+[19]
+BA.1
+0.83
+raw
+This study
+BA.2
+0.71
+0.67
+adjusted
+[19]
+BA.2
+0.77
+raw
+[19]
+Table 5: References used for the prevention of hospitalization
+Strain
+Hospitalization hazard ratio
+Reference; comments
+Alpha
+1.62
+[36]
+Alpha
+2
+[37] (from [38])
+Alpha
+1.7
+[39] (from [38])
+Alpha
+1.42
+[40]
+Alpha
+1.52
+[41] (from [38])
+Alpha
+1.62
+[36] (from [38])
+Alpha
+1.89
+[42] (from [38])
+Alpha
+1.47
+[43] (from [38])
+Alpha
+1.6
+[44] (from [38])
+Alpha
+1.52
+[45] (from [38])
+Delta
+2.08 w.r.t. Wuhan
+[41]
+Delta
+0.97 w.r.t. Alpha
+[46]
+Delta
+2.3 w.r.t. Alpha
+[47]
+14
+
+Delta
+2.8 w.r.t. Alpha
+[48]
+Delta
+1.9 w.r.t. Alpha
+[49]
+Omicron
+0.4 w.r.t. Delta
+[50]
+Omicron
+0.56 w.r.t. Delta
+[51]
+Table 6: References used for the vaccine efficiency against hospitalization
+Strain
+Vaccine efficiency against hos-
+pitalization
+Reference
+Alpha
+4.3
+[46]
+Delta
+3.6
+[46]
+Omicron
+4.5
+[50]
+Omicron
+10
+[52]
+15
+
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+Aranday-Cortes, Natasha Johnson, Igor Starinskij, Ana da Silva Filipe, David Robertson, Richard
+Orton, Joseph Hughes, Sreenu Vattipally, Joshua Singer, Seema Nickbakhsh, Antony Hale, Louissa
+Macfarlane-Smith, Katherine Harper, Holli Carden, Yusri Taha, Brendan Payne, Shirelle Burton-
+Fanning, Sheila Waugh, Jennifer Collins, Gary Eltringham, Steven Rushton, Sarah O’Brien, Amanda
+Bradley, Alasdair Maclean, Guy Mollett, Rachel Blacow, Kate Templeton, Martin McHugh, Rebecca
+Dewar, Elizabeth Wastenge, Samir Dervisevic, Rachael Stanley, Emma Meader, Lindsay Coupland,
+Louise Smith, Clive Graham, Edward Barton, Debra Padgett, Garren Scott, Emma Swindells, Jane
+Greenaway, Andrew Nelson, Clare McCann, Wen Yew, Monique Andersson, Timothy Peto, Anita
+Justice, David Eyre, Derrick Crook, Tim Sloan, Nichola Duckworth, Sarah Walsh, Anoop Chauhan,
+Sharon Glaysher, Kelly Bicknell, Sarah Wyllie, Scott Elliott, Allyson Lloyd, Robert Impey, Nick Lev-
+ene, Lynn Monaghan, Declan Bradley, Tim Wyatt, Elias Allara, Clare Pearson, Husam Osman, An-
+drew Bosworth, Esther Robinson, Peter Muir, Ian Vipond, Richard Hopes, Hannah Pymont, Stephanie
+Hutchings, Martin Curran, Surendra Parmar, Angie Lackenby, Tamyo Mbisa, Steven Platt, Shahjahan
+Miah, David Bibby, Carmen Manso, Jonathan Hubb, Meera Chand, Gavin Dabrera, Mary Ramsay,
+Daniel Bradshaw, Alicia Thornton, Richard Myers, Ulf Schaefer, Natalie Groves, Eileen Gallagher,
+David Lee, David Williams, Nicholas Ellaby, Ian Harrison, Hassan Hartman, Nikos Manesis, Vi-
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+neet Patel, Chloe Bishop, Vicki Chalker, Juan Ledesma, Katherine Twohig, Matthew Holden, Sharif
+Shaaban, Alec Birchley, Alexander Adams, Alisha Davies, Amy Gaskin, Amy Plimmer, Bree Gatica-
+Wilcox, Caoimhe McKerr, Catherine Moore, Chris Williams, David Heyburn, Elen De Lacy, Ember
+Hilvers, Fatima Downing, Giri Shankar, Hannah Jones, Hibo Asad, Jason Coombes, Joanne Watkins,
+Johnathan Evans, Laia Fina, Laura Gifford, Lauren Gilbert, Lee Graham, Malorie Perry, Mari Mor-
+gan, Matthew Bull, Michelle Cronin, Nicole Pacchiarini, Noel Craine, Rachel Jones, Robin Howe,
+Sally Corden, Sara Rey, Sara Kumziene-SummerhaYes, Sarah Taylor, Simon Cottrell, Sophie Jones,
+Sue Edwards, Justin O’Grady, Andrew Page, Alison Mather, David Baker, Steven Rudder, Alp Aydin,
+Gemma Kay, Alexander Trotter, Nabil-Fareed Alikhan, Leonardo de Oliveira Martins, Thanh Le-Viet,
+Lizzie Meadows, Anna Casey, Liz Ratcliffe, David Simpson, Zoltan Molnar, Thomas Thompson,
+Erwan Acheson, Jane Masoli, Bridget Knight, Sian Ellard, Cressida Auckland, Christopher Jones,
+Tabitha Mahungu, Dianne Irish-Tavares, Tanzina Haque, Jennifer Hart, Eric Witele, Melisa Fenton,
+Ashok Dadrah, Amanda Symmonds, Tranprit Saluja, Yann Bourgeois, Garry Scarlett, Katie Love-
+son, Salman Goudarzi, Christopher Fearn, Kate Cook, Hannah Dent, Hannah Paul, David Partridge,
+Mohammad Raza, Cariad Evans, Kate Johnson, Steven Liggett, Paul Baker, Stephen Bonner, Sarah
+Essex, Ronan Lyons, Kordo Saeed, Adhyana Mahanama, Buddhini Samaraweera, Siona Silveira,
+Emanuela Pelosi, Eleri Wilson-Davies, Rachel Williams, Mark Kristiansen, Sunando Roy, Charlotte
+Williams, Marius Cotic, Nadua Bayzid, Adam Westhorpe, John Hartley, Riaz Jannoo, Helen Lowe,
+Angeliki Karamani, Leah Ensell, Jacqui Prieto, Sarah Jeremiah, Dimitris Grammatopoulos, Saro-
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+
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new file mode 100644
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+page_content='At the crossroads of epidemiology and biology: bridging the gap between SARS-CoV-2  viral strain properties and epidemic wave characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Florian Poydenot1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Alice Lebreton2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Jacques Haiech4* and Bruno Andreotti1  1) Laboratoire de Physique de l’Ecole Normale Supérieure (LPENS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' CNRS UMR 8023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Ecole  Normale Supérieure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Université PSL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Sorbonne Université,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' and Université de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' 75005  Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content=' Université  PSL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' 75005 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content=' Micalis Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' 78350 Jouy-en-Josas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' France  4) CNRS UMR7242 BSC ESBS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' 300 Bd Sébastien Brant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' CS 10413,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' 67412 ILLKIRCH cedex  Corresponding author: haiech@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='fr  Declaration of competing interest: The authors declare that they have no known competing financial interests or  personal relationships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Contributions: F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='P, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='H and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='A have collectively designed and written the paper, with input from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  KEYWORDS: COVID19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' SARS-CoV-2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' epidemiology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' PFU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' GU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' viral load  ABSTRACT  The COVID-19 pandemic has given rise to numerous articles from different scientific fields  (epidemiology, virology, immunology, airflow physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=') without any effort to link these  different insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' In this review, we aim to establish relationships between epidemiological  data and the characteristics of the virus strain responsible for the epidemic wave concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  We have carried out this study on the Wuhan, Alpha, Delta and Omicron strains allowing us  to illustrate the evolution of the relationships we have highlighted according to these different  viral strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  We addressed the following questions:  1) How can the mean infectious dose (one quantum, by definition in epidemiology) be  measured and expressed as an amount of viral RNA molecules (in genome units, GU)  or as a number of replicative viral particles (in plaque-forming units, PFU)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  2) How many infectious quanta are exhaled by an infected person per unit of time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  3) How many infectious quanta are exhaled, on average, integrated over the whole  contagious period?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  4) How do these quantities relate to the epidemic reproduction rate R as measured in  epidemiology, and to the viral load, as measured by molecular biological methods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  5) How has the infectious dose evolved with the different strains of SARS-CoV-2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  We make use of state-of-the-art modelling, reviewed and explained in the appendix of the  article (Supplemental Information, SI), to answer these questions using data from the  literature in both epidemiology and virology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We have considered the modification of these  relationships according to the vaccination status of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  We hope that this work will allow a better integration of data from different fields (virology,  epidemiology, and immunology) to anticipate the evolution of the epidemic in the case of  COVID-19, but also in respiratory pathologies induced by a virus or a bacterium  transmissible in an airborne manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  INTRODUCTION  Evidence of aerosol transmission of SARS-CoV-2, the virus responsible for COVID-19  disease, has accumulated over the months (Fennelly, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Lewis, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Morawska and  Milton, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' 19) until a consensus was reached, six months after the  start of the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Viral particles, with or without a liquid droplet surrounding them, are  dispersed by turbulent air movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' When they are light enough, hydrodynamic  fluctuations keep these particles suspended in the air, despite gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The mixture of air and  particles then constitutes a phase called aerosol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' These droplets, which may or may not carry  the virus, are produced by atomization in the respiratory tract when an airflow of sufficient  velocity causes the fragmentation of a mucus film (Bourouiba, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Johnson and Morawska, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Moriarty and Grotberg, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This is the case when large  millimeter-sized droplets are emitted by coughing or sneezing, as well as when smaller  micron- or submicron-sized droplets are emitted during human exhalation activities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=',  breathing, speaking or laughing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The water in the droplets then evaporates into the air,  concentrating the droplets into virions and mucus proteins, some of which have antiviral  properties that help inactivate the virus after a few hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' When airborne viral particles,  regardless of their production process, are inhaled, infection can occur (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Pathogens  responsible for other diseases such as influenza, tuberculosis or measles can also be carried  by these small droplets (Blanchard, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Chingin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=',  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Airborne particles are the primary route of transmission of SARS-CoV-2 (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Johansson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The epithelial cells serving as loci of original infection and reservoirs  of dissemination are located in the nasal cavity (for strains prior to Omicron), which points to  airborne transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The heavier, millimeter-sized droplets emitted specifically in the  Covid symptomatic phase have a ballistic trajectory that is relatively insensitive to the  presence of air and are stopped by all types of face masks and respirators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The reduction of  the risk of transmission of the virus outdoors or in well-ventilated closed environments  (Gettings, 2021) that is obtained by wearing respirators designed to filter aerosols (Goldberg  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Klompas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021), but also the long-distance transmission in super-spreading  events where a single virus carrier infects a large number of people (Endo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Yang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021), are evidence of the importance of airborne transmission of SARS-CoV-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Transmission by contact with fomites on which these drops are deposited is probably  insignificant (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Goldman, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Lewis, 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' however, regardless of its  actual weight in SARS-CoV-2 transmission, improved hand hygiene remains a recommended  habit to prevent transmission of other pathogens — for instance, viruses causing  gastroenteritis have a major handheld transmission route associated with a specific locus of  infection: the gastrointestinal epithelium (Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Robilotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Finally,  possible transmission through feces (Nouri-Vaskeh and Alizadeh, 2020) via aerosolization  during toilet flushing remains controversial and is probably a minor route, if relevant at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Two approaches have been proposed in the scientific literature to characterize the infectivity  and transmissibility of viral strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The epidemiological approach, based on contact tracing  and population-scale testing, provides precise quantitative information but has a blind spot:  the epidemic propagation depends on social practices, the full complexity of which is difficult  to delineate, and which themselves depend on age, education, number and duration of social  contacts, vaccination status, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It also depends on the degree of immunity in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The ab initio approach, based on knowledge of virology, immunology, and molecular  biology, is complementary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' it allows one to characterize viral strains in vivo and in vitro but  suffers from the need of large-scale statistics, and can present important biases of  parameterization and calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This review article aims to bridge the gap between these  two approaches and to review methods combining epidemiological and biological  measurements performed on viral strains to deduce their intrinsic characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Figure 1-A illustrates how the viral mist exhaled by an infected person (index case) can infect  non-immune individuals (secondary case) at some distance, and after a time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The  infection risk increases with the intake viral dose d, defined as the amount of inhaled viral  particles accumulated over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' d increases with the time of exposure to the virus and with  the concentration of infectious viral particles in the inhaled air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The dilution factor between  exhaled and inhaled air is controlled at short distance by turbulent dispersion and at long  distance by ventilation, which is the process of introducing fresh air into indoor spaces while  removing stale air (Poydenot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The dose-response curve expresses the ratio of  infected individuals as a function of the intake dose d (Figure 1-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We hypothesize that a  single replicating virion among the numerous particles inhaled can initiate infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' In tissue  culture assays, the number of infected cells is proportional to the number of viral cells, which  shows that there is no cooperativity between viral particles, in vitro(Houng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Then, each inhaled viral particle can be considered as an independent attempt to contaminate  the individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Statistically, more than one is required, as the probability of a single virus  successfully overcoming the host immune defenses is low, and since a large fraction of  inhaled viral particles, being damaged or defective, are intrinsically unable to infect cells and  replicate therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Infection occurs when a single flawless virion enters a vulnerable location  where conditions are permissive to cellular colonization and viral replication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The dose-  response therefore follows a cumulative Poisson distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' By definition, an infectious  quantum is the dose inhaled by the individuals of a cohort which leads to the infection of  63% of the cohort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' In the field of epidemiology, infectious quanta are used to express a  quantity of virus needed to induce contamination or a given symptom (fever, mortality for  example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The dose-response curve has not been directly measured on humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Infectious challenge  trials during which healthy young volunteers are deliberately infected are rare since they are  ethically controversial (Adams-Phipps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' in the single one led to study COVID-19  (Killingley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2022), a large viral dose (10 TCID50) is introduced via intranasal drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  How can the mean infectious dose (one quantum, by definition) be measured and expressed  as an amount of viral RNA molecules (in genome units, GU, Figure 1-B) or as a number of  replicative viral particles (in plaque-forming units, PFU, Figure 1-C)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' How many infectious  quanta are exhaled by an infected person per unit of time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' How many infectious quanta are  exhaled, on average, integrated over the whole contagious period?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' How do these quantities  relate to the epidemic reproduction rate R as measured in epidemiology, and to the viral load,  as measured by molecular biological methods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' How has the infectious dose evolved with the  different strains of SARS-CoV-2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  In this manuscript, we review concepts and methods providing preliminary answers to these  questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We first describe the mechanism of viral infection and the antiviral response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Then, we detail the viral kinetics and the subsequent time evolution of the viral exhalation  flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The determination of the infectious dose by combining methods from molecular biology  and epidemiology is then reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Finally, the evolution of the characteristics of successive  viral strains is presented and discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  To generate the figures of this review, we have used standard epidemiological modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The  specific theoretical frame chosen has been published in a previous paper (Poydenot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=',  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' In the supplementary material section, we provide a detailed model accessible with  basic knowledge of physics and mathematics, focusing on the parameter’s calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Mechanism of viral infection  SARS-CoV-2 is a virus enveloped by a lipid bilayer in which E, M, and S proteins are  inserted (Figure 1-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The lipid membrane originates from the cell in which the virus  replicated before being released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The virus contains a copy of the viral genomic RNA  protected by a capsid, structured by the assembly of the nucleocapsid protein N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Viral  particles measure 80-90 nm in diameter, and are decorated with an average of 48 spike (S)  proteins anchored in their envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The RNA genome encodes 29 proteins, including the  envelope (E, M, and S) and capsid (N) proteins, as well as non-structural proteins required for  replication and assembly of the virus within the host cell (Bar-On et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=',  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' To colonize a cell, the virus interacts via the S protein which is cleaved by a host cell  protease (mainly the TMPRSS2 protease for the wild strain, Wuhan-1) with a host cell  membrane protein (mainly the ACE2 receptor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This interaction leads to the formation of a  virus-ACE2 complex (via the virus spike, a trimer of the S (Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2022)) which triggers  the internalization of the virus into the cell (Figure 2-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' A series of cellular events leads to  the disassembly of the virion and the undressing of the RNA molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The released viral  RNA is taken over by the host cell ribosomes, which read the information it encodes and  produce the viral proteins needed for virus production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' New viral particles are then assembled  by hijacking the host cell mechanisms, and then released, leading to the colonization of  neighboring cells (Snijder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' From the nasal cavity or throat, which are probably  the first tissues to be infected, the virus, embedded in the mucus secreted by goblet cells of  the nasal epithelium, is transported to the trachea, then to the lungs or esophagus, and finally  to deeper organs (Figure 2-B and 2-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The severity and variety of disease symptoms depend  on the likelihood of the viral infection overcoming host defenses and reaching multiple sites,  as well as on the damage caused to the host by the potent inflammatory and interferon  responses launched against the viral assault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' In contrast, the spread of the virus depends  primarily on its ability to colonize the host airways, and thus the viral load cannot be  correlated with symptom severity (Le Borgne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' From the nasal cavity, the virus  has the ability to travel to the brain and colonize cells of olfactory bulb, which could explain  the changes in taste and smell and, in the long term, some of the neurological symptoms  associated with long COVID (Fodoulian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Pereira, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' More rarely, the virus  cpean be found in the blood or lymph, reach different organs in the body and colonize  specific cell types in various organs (liver, kidney, heart, prostate, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=') (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020)  (Figure 2-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  When the virus is concentrated in the nasal cavity or in the throat, it is spread by a mist of  fine droplets of mucus or saliva which can be dispersed by breathing, speaking or singing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' A  sneeze or cough produces larger droplets containing viral particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' As these droplets are  formed, their viral particle content increases linearly with the viral load, in the nasal cavity  for mucus droplets, or in the throat for saliva droplets (Buonanno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' An organism  can be infected if enough viral particles interact with cells expressing both the TMPRSS2  protease and the ACE2 receptor and if the virus is able to hijack cellular mechanisms to  produce and disseminate new virions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  For the Omicron viral strain, the TMPRSS2 protease appears to be less essential, in favor of  an alternative pathway of entry into epithelial cells via the endosomal route (Peacock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=',  2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' At each cycle, the virus must enter a novel host cell, replicate its RNA molecule,  produce the proteins necessary for its self-assembly and then be released as virions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Omics databases such as the Human Cell Atlas (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='humancellatlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='org/) provide  insight into the tissues and cell types that express the TMPRSS2 protease and ACE2 receptor  in different tissues of the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It is commonly assumed that these cells are the  primary target cells of SARS-CoV-2 (Hoffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The nasal epithelium is most  often the first infected tissue, which is amongst the strongest proofs in favor of a transmission  of SARS-CoV-2 primarily by inhalation of virus-laden aerosols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Accordingly, there exists a  cluster of nasal cells in the ACE2/TMPRSS2 expression map (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Sungnak et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' However, due to its lower dependence on TMPRSS2, the Omicron viral strain  illustrates the changes of cellular tropism that can happen as a result of evolution of the virus  (Gupta, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Antiviral response  The viral load of an infected person is the result of viral replication and of the antiviral  response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The arrival of the virus in the nasal cavity or in the throat induces an antiviral  response, due to the recognition of viral components considered as danger signals by relevant  cellular receptors (Silva-Lagos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This interaction triggers a non-specific antiviral  defensive response, which relies to a large extent on the production of a class of molecules  called interferons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' By binding to their receptors on target cells, interferons induce the  expression of a wide range of interferon-stimulated genes (ISGs) that, through various  mechanisms, help limit viral replication (Gallo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Interferons are produced by infected cells but also by sentinel immune cells such as  macrophages and dendritic cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' After their secretion, they diffuse and bind to their receptors  on surrounding cells without discriminating whether they are infected or not, stopping  cellular functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Since the concentration is higher around the site of infection, diffusion  leads to an efficient stochastic control of infected cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' If the interferon response is initiated  early, in a localized and circumscribed manner, viral spread through the tissue can be  controlled (Katze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Perry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' However, a too strong interferon response  is not only antiviral but also destructive to host cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Moreover, the outcome of the virus-host  crosstalk is further complicated by the fact that some viral proteins counteract host interferon  responses and, conversely, host interferon responses can sometimes amplify viral infectivity  (Ziegler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Others mechanisms may also modulate the infectivity of specific  strains, for instance the modulation of splicing of ACE2 in response to interferon responses  (Blume et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content="Therefore, the nasal cavity is a site of ongoing conflict between SARS-  CoV-2 replication and the body's efforts to inhibit the virus." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=" The efficiency of the viral  replication and secretion processes, as well as the body's interferon response, can vary across  time and space within the nasal cavity." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' These above described and antagonists processes can  influence the ability of the virus to establish an infection in the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Because viral transmission depends directly on viral loads, the kinetics of the replication and  secretion of the virus characterizes the capacity of an individual to transmit the infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Moreover, the kinetics of the interferon response correlates with an individual’s susceptibility  to infection (Hadjadj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Although it is often forgotten, mucus also modulates  significantly that kinetics in the nasal cavity and participates in the immune response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Mucus  is composed mainly of water (95%), lipids and glycoproteins (mucins) (Bansil and Turner,  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Fahy and Dickey, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It is secreted by specialized cells named goblet cells which  are also among the cells infected by the virus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Mucus forms two layers at the surface of the  nasal epithelium: a gel constituted by a network of polymerized mucins anchored to the  membrane of epithelial cells, and a solution that moves under the action of the cilia of the  epithelial cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The viscoelastic properties of mucus depend on the type of mucins (Moniaux  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2001) and on the physicochemical environment (humidity, calcium concentration and  pH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Inflammation modifies the expression and the glycosylation/fucosylation of MUC5B  and MUC5AC (Amini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Chatterjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020), the major airway gel-forming  mucins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Mucosal immunity results from a synergy between the mucus and the immune system (innate  immunity, secreted IgA and IgA for the acquired immunity, and cellular immunity), as the  network created by polymerized mucins traps particles down to a few hundred nanometers in  size (which is the size of viruses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Mucus also contains enzymes that can neutralize viruses  and bacteria in a non-specific manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It should be noted that nasal mucus and saliva, which  is also a specific mucus, have different compositions and physicochemical characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Lastly, the nasal microbiota, a community of bacteria that live within the mucus, also plays a  role in defense against infection by viruses or pathogenic bacteria (Kolhe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Viral load  The viral load in the oropharyngeal cavities, which determines the exhaled particle flux, is  highly time-dependent (Figure 3-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It can be measured in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The concentration  of viral genomes in a sample can be measured using RT-qPCR (Reverse transcription  followed by quantitative Polymerase Chain Reaction), using as targets one or more nucleic  acid sequences in the viral RNA genome (Figure 3-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The sample is collected with a swab,  and then its RNA contents are extracted in a given volume of an RNA extracting and  stabilizing solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This volume differs depending on the test used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The RT-qPCR result is  expressed in threshold polymerase reaction cycles (Ct) which is transformed into a number of  copies of viral RNA molecules (genome units, GU), after calibration using a solution of viral  RNA molecules of known concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The concentration of viral genomes is thus  expressed in viral RNA molecules per unit of volume of extraction solution (GU/mL in  practice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Considering that the swab collects at most 100 microliters of nasal extract and the  volume of extracting solution varies between 1 and 3 mL depending on the kit used, the  concentration of viral RNA is probably an order of magnitude higher in the oropharyngeal  cavity than the concentration in the extraction solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Alternatively, the ability of the virus  to infect a cell monolayer and cause cell lysis can be measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Using serial dilutions of a  virus-containing sample, the number of lysis plaques (Plate Forming Unit, PFU) produced  within a well monolayer for a given volume of sample is measured (Figure 3-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The  concentration of replicative virus expressed in PFU/mL depends on the type of cells used, as  susceptibility to infection varies depending on cell types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  For a given sample and a given cell type, the number of cell lysis plaques obtained is  proportional to the concentration of viral genomes in solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This experimental linearity  indicates that cell infection events can be considered as independent from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The  number of lysis plaques induced on the average by one viral genome unit is the viral  infectivity and is expressed in PFU/GU (Dabisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' As this  quantity reflects the ability of the virus to enter cells and replicate, it primarily depends on the  viral strain considered and on the type of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' In practice, the viral infectivity is a highly  variable quantity as it reflects viral integrity for a given sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It therefore depends on the  time and site of sampling, as well as on the protocol used for sample collection and  preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' A viral infectivity equal to one would imply that each virion is able to colonize  a cell, replicate, be released to colonize neighboring cells and lyse the colonized cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This  value is never reached as it implies a perfect replication and assembly of all virus particles,  the absence of damaged or inactivated viruses, as well as perfect sample preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' In  other words, low values of the viral infectivity indicate viral solutions that contain high loads  of mis-assembled, non-functional or degraded viral particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Part of this degradation may be  due to the antiviral response or the immune response of the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Conversely, a null viral  infectivity means that no virion can form lysis plaques on a cell monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' For SARS-CoV-  1, the viral infectivity was between 6 × 10-4 and 8 × 10-4 PFU/GU on VERO cells (Houng et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2004) : one viral particle over 1200 to 1600 was able to form a cell lysis plaque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' For the  Wuhan-1 strain of SARS-CoV-2, the literature provides different measurements of infectivity  for high preservation quality samples (Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020), which lead to an  estimate of  7 × 10-4 PFU/GU (1 cell lysis plaque for 1400 virions) for the initial,  asymptomatic phase of the pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' As the disease progresses, the fraction of replicating  virions decreases and the infectivity decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The viral kinetics has standardly been described in the literature by an exponential growth of  the viral load (replication and shedding) followed by an exponential decay (immunity  response).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The SARS- CoV-2 “human challenge” trial (Killingley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2022) has provided  unprecedented time-resolved data showing a more rounded viral load curve (Figure 3-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The  viral strain used in this study was close to the wild-type strain, Wuhan-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' In the nose, the  maximum viral load, expressed in genome units, was reached after Tm=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='0 days, around the  onset of symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This time lapse provides an estimate of the incubation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This period  was slightly smaller, around Tm=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='9 days, when considering the maximum replicable viral  load, measured in PFU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This difference may be ascribed to the effect of the antiviral  responses on infectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' As an alternative hypothesis, we cannot exclude the possibility of  incorrect assembly of the virus by the colonized cell leading to a decrease of infectivity (in  PFU/GU), although a potential inhibition of the virus assembly could also be a consequence  of antiviral responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The infectious period T, defined as the average time between  infections of the index and secondary cases, is estimated to be 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2 days (Figure 3-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The  exponential growth rate just after infection is approximately 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='0 days-1 (growth time 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='5  hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The exponential decay rate long after this maximum is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='3 days-1 (decay time 39  hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The infectivity is around 10-5 PFU/GU at maximum, which is two orders of  magnitude weaker than the value found when using virus replicated on cultured cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The viral emission rate, defined as the quantity of viral particles exhaled per unit of time, has  been measured in several pioneering studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' In the study by Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020),  patients were asked to exhale into a cooled hydrophobic film through a long straw to collect a  sample of exhaled breath condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The measured concentration of SARS-CoV-2 viral  particles was between 105 and 2 × 107 GU/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The authors noted that the exhalation rate was  correlated with the viral load in the nose and in the throat but not in the lungs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' In the study by  Malik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (2021), the mean viral load per swab was 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='8 × 106 GU whereas exhaled breath  samples displayed 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='47 × 103 GU per 20 times exhaling, which corresponds to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='5 × 105  GU/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' In the scientific literature, it is standardly considered that the viral emission rate is  proportional to the viral load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This is reasonable if the infected epithelium surface is  reasonably constant and if there is no viral enrichment at the interface between muco-salivary  fluid and air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' In the following sections, we will adopt this hypothesis and use the  multiplicative factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='4 between the viral load, in GU/mL, and the viral emission rate, in  GU/ day found in Malik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The total viral emission of an infected person is the quantity of viral particles exhaled during  its infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Considering the viral kinetic displayed in Figure 3-A, the total viral emission,  obtained by integrating over time the viral emission rate, would be around 9 × 107 GU, or  6 × 104 PFU, within a half-decade uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Molecular vs epidemiologic determination of the infectious dose  As mentioned in the introduction, the determination of the infectious dose a priori requires  measuring a dose-response relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Importantly, infection can be defined from different  observables (amount of virus in the nasal cavity, presence of specific symptoms — like  rhinitis, pneumonia or acute respiratory distress syndrome — or mortality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The infectious  dose therefore depends on the viral strain, on the capacity of the upper airway mucosal  immune system and of the systemic immune system for the lung or other organs to neutralize  the virus, and on the type of observables used to monitor infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The inhaled dose only  dictates the primary infection in the upper airways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The severe acute respiratory syndrome in  the lungs typically results from a secondary infection, after the virus has colonized the nasal  cavity (Wölfel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' As a consequence, the relevant viral dose is the one that is  transferred from upper to lower airways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The viral dose in the inoculum is therefore not  directly related to disease severity, as it is negligible compared to the production of virus by  colonized host cells in the upper airways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The dose response relationship allows expressing the mean infectious dose (one quantum, by  definition) as an amount of viral RNA molecules or as a number of replicative viral particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  However, it can only be measured on animal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' No such study exists for SARS-CoV-2,  to the best of our knowledge, but it has been measured on a non-human primate model for  SARS-CoV-1 (Watanabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The ID50 (median infectious dose) describes the  amount of replicable virus (expressed in PFU) needed to infect 50% of the population (Figure  1-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Note that the same quantity, when measured on a population of cells in tissue culture  assays, is called TCID50 (median tissue culture infectious dose).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' For SARS-CoV-1, the  reported measurement is ID50=280 TCID50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We recall that an infectious quantum is the dose  inhaled by the individuals of a cohort that leads to the infection of 63% of the cohort, and that  it is used in epidemiology as a unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This convention leads to a multiplicative factor  1/log(2)=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='44 between the epidemic quantum and the ID50 quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The mean infectious  dose therefore relates an epidemiological quantity to a characteristic quantity of virus,  defined using molecular biology experiments (Figure 1): the reported measurement for  SARS-CoV-1 leads to 400 PFU/quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Using these estimates, the infectious dose for the  raw Wuhan-1 strain on humans is in the range of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='6 × 105 GU, within a factor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Considering  the viral kinetic displayed in Figure 3-A, the total viral emission ħ would be around 150  quanta, within a factor 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This corresponds to a typical viral emission rate of 1 quantum/hour,  and a maximal viral emission rate of 2 quanta/hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  From the epidemiological point of view, reference points are provided by closed micro-  societies inside which the virus propagates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' They provide estimates of the total viral emission  ħ expressed in infectious quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' In other words, the exhaled dose of an infected person as the  potential to infect ħ other people, on the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' However, thanks to ventilation, only a small  fraction of this exhaled dose is actually inhaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The exponentially growing epidemics  onboard the ship Diamond Princess (Almilaji and Thomas, 2020) and onboard the French  aircraft carrier Charles de Gaulle are the most important events for the Wuhan-1 strain  (Figure 3-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Both events were characterized by a low rate of replacement of stale air by  fresh air, and by an air conditioning system lacking HEPA filters to remove pathogens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Using  the viral load curve of Figure 3, the growth rate can be converted into an epidemic  reproduction rate (R=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='8 and R=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Then, using a model dilution factor  between exhaled air and inhaled air (Poydenot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2022), one deduces the total viral  emission ħ=490 quanta (retired people) and ħ=460 quanta (young adults), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This  corresponds to a typical viral emission rate of 3 quanta/hour, and a maximal viral emission  rate of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='3 quanta/hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Schools constitute the best documented social sub-system (Bazant  and Bush, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Vouriot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021), although not isolated from the rest of society as ships  were.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Figure S4 shows the epidemic evolution for secondary school pupils, between  lockdown and holidays, in UK, together with the typical CO2 concentration measurement  (Poydenot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2022) that gives a total viral emission ħ = 270 quanta for pupils from age 10  to age 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It corresponds to a typical viral emission rate of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='6 quanta/hour and a maximal  emission rate in the range of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='3 quanta/hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  In conclusion, although calibrations are lacking large-scale statistics, the molecular and  epidemiologic determinations of the mean infectious dose are consistent with each other,  within error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It is important to note that the infectious dose is approximately 6 × 105 GU  for the wild type strain Wuhan-1 and not between 10 and 100 as mentioned in a series of  recent articles (Bazant and Bush, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Buonanno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Lelieveld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Pöhlker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Vouriot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Vuorinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' A large part of the results on  the airborne transmission risk, based on the volume of exhaled drops, is quantitatively flawed  by the omission of the infectivity ratio between plaque forming units (PFU) and genomic  units (GU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The estimated typical emission rate is between 15 and 30 quanta/hour in Bazant  and Bush (2021) and Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (2021) found 1,000 quanta/hour for singing, to be compared  to 1-3 quanta/hour estimated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Accordingly, the total viral emission ħ, which ranges  between 450 and 500 quanta on average for adults, is almost 10-fold smaller than previous  estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Evolution of viral strain characteristics  In the previous sections, we have discussed at length the absolute characteristics of the wild  strain, Wuhan-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Comparing the characteristics of the variants presents different difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  On the one hand, relative measurements are much more precise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' on the other hand, the  immunity induced by infection or by vaccination induces a strong heterogeneity in the  population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The evolution of the parameters presented here for the wild type strain, Wuhan-1,  and for variants Alpha (B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='7), Delta (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2), Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1 and Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2, is  based on relative measurements, the wild type strain serving as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We have selected  the variants that have led to an epidemic wave and we have retained the date of the first case  in France to display the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Although this choice is somehow arbitrary, it is justified by  the fact that most measurements are performed using French epidemic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Figure 4-D shows that the incubation period Tm and the infectious period T, measured in vivo  using the viral load kinetics, are almost equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' They are slightly decreased for the variants  Alpha and Delta, which leads to a higher growth rate and therefore to an evolutionary  advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' By contrast, the advantage of Omicron is not due to a change of incubation and  infectious period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Figure 4-D compares these two characteristic times, Tm and T, to the  replication time, measured in vitro using human nasal epithelial cells (hNECs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' To get  comparable orders of magnitude, we define the latter as the time needed in vitro to multiply  the number of viral particles by 1 billion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Interestingly, the replication time of the virus in a  human lung cell line that expresses abundant ACE2 and TMPRSS2 (Calu-3) has strongly  decreased for the Delta variant, in direct relation with a higher risk of pneumonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Omicron is  milder than Delta (but more severe than the wild-type strain) for lung symptoms, which is  coherent with its increased replication time in Calu-3 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  In Figure 5, we propose a meta-analysis of various characteristics for the successive variants  of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It combines epidemiologic and biological data, following standard methods in  both domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We emphasize that this meta-analysis is deduced from the definitions and  measurement methods defined previously in a straightforward and robust way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Figure 5-A shows the evolution of the inverse of infectivity, measured as the average number  of viral particles (GU) needed to induce one lysis plaque on a generic Vero cell culture (PFU)  (data from Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Despres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Ghezzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Houng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Killingley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Paton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Peacock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Peña-Hernández et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Puhach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' measurements performed on hNECs and Calu-3 cells would be more  relevant but are lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The Delta variant is more infectious (240 GU/PFU) than the Alpha  variant (1,000 GU/PFU), itself being more infectious than the wild-type strain (1,400  GU/PFU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Omicron is slightly less infectious than Delta (300 GU/PFU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This confirms the  evolution of the replication time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The evolutionary advantage of the Delta variant is due to its  higher binding affinity with ACE2, which increases the probability of cell entry and  replication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Figure 5-B shows the total viral emission ħ (in quanta), deduced from epidemiologic  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The reference value of ħ is deduced from the infections onboard the Diamond  Princess and Charles de Gaulle boats (Figure 3-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' ħ is the key biological characteristic that  determines the epidemic growth rate, for given social practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The ratio of ħ for two  successive strains is deduced from the period of time during which these two strains coexist,  as the ratio of the epidemic reproduction rates R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The two values of R are themselves deduced  using the Euler-Lotka equation from the epidemic growth rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Figure 5-B shows that ħ has  increased from one variant to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' For Alpha and Delta, this increased epidemic growth  rate is probably a direct consequence of the evolutionary increase of binding affinity for the  ACE2 receptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' For Omicron, a reasonable hypothesis is that the increased epidemic growth  rate results from a displacement of the first entry point from the nasal cavity to the throat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' A  possibility would be that lower density of lymphid follicles in the throat than in the  nasopharynx renders this mucosa less potent in launching an effective immune response  against infection (Ogra, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The total viral emission ħ (in quanta) is also displayed when  both the index and secondary cases are fully vaccinated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' For this, we used data of the relative  transmissibility and susceptibility deduced from household transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Although the  vaccines were optimized to induce circulating antibodies and systemic T and B cell responses  to prevent viral diseases, they were very effective to prevent transmission for the Alpha and  Delta strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' On these variants, the vaccines were reducing both the transmissibility, which is  by definition the replicable viral emission rate, and the susceptibility of contact cases to be  infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' However, this prevention of viral spreading by the vaccine has almost disappeared  with Omicron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Incidentally, by maintaining a broad viral pool in circulation, increased  immune escape is amongst the reasons for Omicron increased transmissibility (Paz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=',  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Figure 5-C shows the evolution of the infectious dose (1 quantum) in the upper airways, in  genome units (GU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This measurement combines the number ħ of infectious quanta exhaled  during the infectious period (Figure 5-B) and the evolution of the maximum viral load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The  ratio between the number of genome units in a swab and in one breath is around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='6 105  (Adenaiye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Malik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Ryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=',  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The result is multiplied by the mean number of breathes per day (2 × 104) and by the  integral over time of the viral load in the upper airways to obtain the number of genome units  (GU) exhaled on the average during the whole infectious period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The number of genome  units in the infectious dose is obtained by dividing the number of genome units (GU) exhaled  in total by the number of quanta exhaled in total, ħ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Figure 5-C shows that the reason for the  increase of the integral viral exhalation ħ is not primarily a larger viral load but a strong  decrease of the number of virions statistically needed to induce the infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Figure 5-D shows the evolution of the infectious dose (1 quantum) in the lower airways,  measured in genome units (GU), for the successive variants of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The mean dose  required to infect the lower airway is measured as the ratio of the number of genome units  (GU) transferred by inhalation from the upper airways to the lungs, divided by the probability  to develop a lung pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' In first approximation, inhalation moves the same quantity of  viral particles towards the lungs as exhalation does towards the outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The probability that  the lungs get infected by SARS-CoV-2 when the upper airways are, is approximated by the  hospitalization hazard ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The infectious dose is much larger for the lungs than for the  upper airways due to the fact that viral particles are not diluted when transported to the lower  airways (Heyder, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Hinds, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The rather low probability of infection shows that the  immune system is more effective in the lungs than in the nose or throat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Figures 5-E and S8 shows the maximum likelihood phylogenies inferred from spike  nucleotide sequences for major SARS-CoV-2 lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' SARS-CoV-2 does not currently  show the unbalanced, unidirectional phylogenetic tree that is a hallmark of immune escape  for viruses under a strong immune selection pressure, and where each new variant emerges  from the last dominant variant (Volz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Indeed, the Omicron strain is not a  mutation of the Delta strain, nor was the Delta strain a mutation of the Alpha strain, but each  one of them emerged from very different branches of the phylogenetic tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' So far, each new  variant of concern has arisen from very different branches of the phylogenetic tree by novel  mutations that have remained undetected over long periods of time, resulting in a relatively  balanced tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' By contrast, endemic viruses such as influenza or seasonal coronaviruses can  be recognized by the shape of their unbalanced, unidirectional phylogenetic tree (or ladder-  like evolutionary tree, when represented as a function of time): they circulate and mutate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' in  parallel, immunity develops against them, leading to the gradual extinction of the ancestral  branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Some of the mutations lead to new variants that escape immunity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' their branches  expand until immunity develops against these new variants, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' At this endemic stage,  each new variant thus derives by mutation from one of the last hegemonic strains, usually by  a limited evolutionary jump that allows it to escape immunity, which is partially  predictable(Carabelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Meanwhile, at the present stage of SARS-CoV-2  evolution, new viral strains do not present such a phylogenetic pattern: they may emerge  from all branches as well as from the root of the tree, after having spread silently for a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It  is then impossible to predict how close the next hegemonic variant will be to the previous  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Discussion  In this article,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' we have first reviewed a series of epidemiological and molecular biology  methods that can be combined to characterize the airborne transmission of respiratory  viruses:  (i)  replication kinetics of viral strains in tissue culture assays,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' using RT-qPCR and  cell lysis plaques measurements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' provides access to the viral replication time in  the absence of immunity response and to the viral infectivity  (ii)  the dose response curve,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' determined using model animals close enough to  humans,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' provides the expression of the epidemic quantum in GU and PFU  (iii)  the viral exhalation rate can be measured as a function of time t after infection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  directly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' by collecting virus in a mask,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' or via viral load curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This measurement  gives access to the total viral exhalation in GU and in PFU, to the infection time T  and to the rescaled viral transmissibility ψ(t)  (iv)  the epidemic reproduction rate R can be deduced from the epidemic growth rate σ,  using the rescaled viral transmissibility ψ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This measurement gives access to the  total viral exhalation in quanta, if the epidemic growth rate σ is measured on an  isolated micro-society  This toolbox can be used to characterize viral strain and calibrate quantitatively the models  for airborne transmission risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Some limitations to the present conclusions of this study should be highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The severity  of the symptoms and the transmissibility depend on the personal status (age, weight,  immunodeficiency conditions, comorbidity factors, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Epidemic databases annotated with  such information do not exist as open data for obvious ethical reasons: it would be necessary  to chain different databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Such an investigation would be extremely interesting but is  outside the scope of this review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  A better molecular determination of infectious quanta requires measuring the number of viral  particles per unit of time (or per unit of volume of exhaled air) during expiratory human  activities such as breathing or speaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This would require the design and calibration of face  masks that allow patients to breathe normally and collect all viral particles in a filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We  have used here epidemiological data to estimate the infectious quantum, expressed in viral  RNA (GU), for the successive SARS-CoV-2 variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The systematic use of respiratory  aerosol samplers (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021) is essential to characterize quantitatively SARS-CoV-2  strains, which is necessary for risk assessment and subsequent risk reduction policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Similarly, standard quantitative molecular biology techniques, such as plaque assays and RT-  qPCR could be used directly to measure face mask efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Second, it would be useful to measure systematically the replication kinetics of SARS-CoV-2  strains in tissue culture assays with large enough statistics and controls to infer replication  properties in the absence of immune response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Although viral multiplicative curves are  regularly measured and published, the necessary scientific coordination is missing as well as  systematic comparison with epidemiologic data, for which the immune response is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Third, another limitation of this study is the lack of knowledge about the generation of virus-  containing aerosols in the upper respiratory tract, particularly in the nasal cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We have  shown that indirectly determining the exhalation rate of viral particles using the aerosol  droplet emission rate underestimates the result by three orders of magnitude (Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=',  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Fourth, several issues regarding evolution during desiccation of mucus droplets carrying  virions remain open:  The evolution of the physicochemical characteristics of mucus as a function of time,  temperature, humidity, pH, ionic concentration (especially calcium, which condenses mucin  polymers at high concentrations), and pathology (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The role of mucus in the formation of droplets and their contaminating character:  concentration of the virus, size of the droplets, mixing of the mucus of two different  individuals—mucus of the transmitter and mucus of the receiver (Edwards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The respective contributions of mucus and interferon responses to the clearance of SARS-  CoV-2 (Persson, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Finally, the mechanisms of SARS-CoV-2 inactivation remain poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' In order to  assess the effectiveness of alternative risk reduction techniques, it is necessary to know how  environmental conditions (temperature, humidity, chemical concentrations, ultraviolet  irradiation) affect the viability of SARS-CoV-2 (Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Fears et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Lednicky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The evolution of the virus takes place under a double selection pressure, an increase in  transmissibility and an escape from neutralizing antibodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The increase in transmissibility is  due to an optimization of the virus ability to replicate in the epithelial cells of the throat and  nose, which release more and more replicable virions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This optimization occurs  independently of the symptoms that the different mutants may cause in the contaminated  organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The existence of neutralizing antibodies affects the quality of the exhaled virions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Thus, immune escape of new variants participates in a temporary increase of transmissibility  by increasing the quantity of replicable exhaled viruses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The severity of symptoms induced by a given variant is not correlated with the  transmissibility and therefore, does not seem to be important in the evolution of the virus  (Alizon and Sofonea, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Thus, new variants appear randomly, which may either induce  more severe, or milder, symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' However, the speed and diversity of virus evolution is  correlated with its circulation flow and its ability to remain present in an organism for long  periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Thus, the ability of new variants to infect animals or persist in immunocompromised  individuals accelerates virus circulation and the appearance of new variants with enhanced  transmissibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The greater the diversity of these new variants, the greater the possibility of  variants inducing severe symptoms, even in younger individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Following a precautionary  principle, it is important to decrease as much as possible the circulation of the virus by  improving the air quality in closed areas and to monitor the circulation of the virus and the  appearance of new variants in a given spatial territory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This requires appropriate means  allowing a timely monitoring of viral circulation in the environment and not only at the level  of individuals (Rios et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Figure captions  Figure 1: How to relate epidemiologic characteristics to measurements performed in  molecular biology?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (A) The dose-response curve relates the probability of infection to the  amount of inhaled viral particles accumulated over time, called the intake dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The mean  infectious dose is defined as ID50 for animals (50 % probability), as TCID50 for cells (50 %  probability) and as the infectious quantum for humans (dose-response curve approximated by  1-exp(-d), where d is the dose expressed in quanta;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (B) The mean infectious dose  can be expressed as a quantity of viral genome copies, expressed in genome units (GU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (C)  Alternatively, it can be expressed as a quantity of viral particles able to replicate on a certain  type of cell, expressed in plate forming units (PFU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Biological characteristics  Epidemiological characteristics  Viral particles  genome units (GU)  Replicative virus  with respect to a cell type  plaque-forming units (PFU)  Mean infectious dose  infectious quantum  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='5  102  103  104  101  102  101  10-1  10-2  100  Index  case  Exhaled  dose  Probability of infection  dose (PFU)  dose (quanta)  Inhaled  dose  Secondary  case  ID50  B  A  C  Spike Glycoprotein (S)  M-Protein  E-Protein  RNA and N-Protein  Figure 2: Mechanism of viral infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (A) Pathway used by SARS coronavirus 2 (SARS-  CoV-2) to enter and infect the cell by intermolecular interactions between the spike protein of  the virus and its host cellular receptor angiotensin converting enzyme 2 (ACE2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Effective  binding is dependent upon spike protein activation by transmembrane protease or furin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (B)  The nasal cavity and the throat are usually the first tissues to be infected, after inhalation of  viral particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The infection of other organs where the ACE2 receptor is expressed is induced  in a second stage, after the virus has colonized the upper airways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (C) Viral particles issued  by replication in the nasal cavity can be transported by the air (to the lungs), in the nerves (to  the brain) and possibly through the blood or lymph, by a Trojan horse mechanism (to deeper  organs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Inhalation  ACE2  receptor  Viral RNA  release  Endocytosis  Protease  Upper  airways  infection  Viral  replication  Membrane  fusion  Blood and  lymphatic  transport  Axonal  transport  Respiratory  transport  Lungs secondary infection  through lower  respiratory tract  Brain secondary  infection through  olphactory nerve  Multiple organ secondary  infection through blood,  lymph or nerves  A  B  C  Figure 3: Viral kinetics of SARS-CoV-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (A) Viral kinetics of SARS-CoV-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Average viral  load in the nasal cavity and the throat as a function of time since infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Data from  Killingley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (2022), using a strain close to Wuhan-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (B) Viral load expressed in genome  units (GU) is obtained by reverse transcription followed by quantitative PCR (RT-qPCR),  which measures the quantity of viral genome copies in a viral solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (C) Viral load  expressed in plaque forming units (PFU) is measured using plaque assays, which consist  counting lysis plaques in a cell monolayer in contact with a viral solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Data of panel A are  obtained for Vero Cells, using a 3 mL solution per swab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Figure 4: Calibration of the epidemic quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (A) Curve of the proportion P of infected  individuals aboard the cruise ship Diamond Princess (blue) and aboard the French aircraft  carrier Charles de Gaulle (red) as a function of time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The solid lines correspond to the best  fit by exponential growth, resulting in a growth rate σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='23 day−1 and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='17 day−1,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The dashed horizontal line is the theoretical collective immunity limit P → 1/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The dotted line is the numerical integration of the infection equation SI-(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' On both boats,  the persistence of viral particles is limited by their deactivation timescale rather than by the  ventilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The fast transmission in the Charles de Gaulle French aircraft carrier (Laval et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=',  15  10  5  0  104  102  100  106  108  Time since infection (days)  Viral load (per mL)  GU  PFU  PFU  GU  Nasal  swab  Nose  Throat  Successive dilutions  Dilution  Amplification  Reverse transcription  Pharyngeal  swab  PFU  100  0  10-2  10-4  100  RNA  DNA  Primer  A  C  B  Cycle  Fluorescence  0  10 20  30 40  Time (days)  date  100  10-1  10-2  10-3  10-4  50  40  30  20  10  0  Epidemic prevalence P  Charles de Gaulle  aircraft carrier  Diamond Princess  cruise ship  8  6  4  2  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='0  Time since infection (days)  growth rate σ (day-1)  Infectious period T  Incubation period Tm  Viral timescales (day)  Rescaled  transmissibility  Reproduction number R  8  6  4  2  0  4  5  6  3  2  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='25  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='25  2022  2021  2020  Wuhan-1  alpha  delta  omicron  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2  T  HNEP  Calu-3  Tm  A  B  C  D  2022) is due to the lack of filtration of the recycled air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Over 1,767 people onboard, crew and  commandos together, 1,288 were infected in a short period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The indoor relevant  volume is estimated around 150 × 103 m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The fast transmission on the Diamond Princess  boat (Bazant and Bush, 2021) is due to a mostly recycling (at 70%, no HEPA filters), air  conditioning, due to the cold weather conditions (−5°C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The surface accessible to passengers  is 78 × 103 m2 and the ceiling height is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='4 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The indoor volume, 187 × 103 m3, is rather  large compared to the number of people, N = 3,711, crew and passengers together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (B) Model  dimensionless transmissibility as a function of time t, in days, after infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The viral  transmissibility of a person (index case) is defined as the rate of emission of replicable viral  particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' In first approximation, transmissibility is proportional to the viral load in the upper  respiratory tract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The incubation period Tm is defined as the time between infection and  maximum transmissibility, and the infectious period T as the average time between infections  of the index and secondary cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (C) Relation between the epidemic reproduction rate R,  defined as the mean number of secondary cases per index case, and the epidemic growth rate  σ, predicted by the Euler-Lotka equation SI-(7), for a given transmissibility curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It gives R  = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='8 for the cruise ship and R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2 for the aircraft carrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The red line is the approximation  for small growth rate: σ = (R − 1)/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (D) Incubation period Tm and infectious period T for the  wild-type strain, Wuhan-1, and of variants Alpha (B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='7), Delta (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2), Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1  and Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2, represented at the date of the first case in France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Viral replication time-  scale is measured using primary cultures of human nasal epithelial cells (hNECs), human  lung cell that expresses abundant ACE2 and TMPRSS2 (Calu-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It is defined as the time  needed to multiply the viral particles by 1 billion (109).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Figure 5: Epidemiologic and biological characteristics of the wild-type strain, Wuhan-1, and  of variants Alpha (B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='7), Delta (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2), Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1 and Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (A) Viral  infectivity, defined as the average number of lysis induced by a viral particle on a Vero cell  culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It is expressed in plaque forming units per genome units (PFU/GU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (B) Mean viral  2022  2021  2020  date  2022  2021  2020  date  104  103  102  101  full vaccination  no vaccination  109  1010  full vaccination  no vaccination  full vaccination  no vaccination  10-3  10-2  Wuhan-1  Wuhan-1  Alpha  Delta  Omicron  Alpha  Delta  Delta (21J)  Delta (21I)  Delta (21A)  Beta  Gamma  Omicron  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2  2022  2021  2020  date  2022  2021  2020  date  Wuhan-1  Alpha  Delta  Omicron  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1 (21K)  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2 (21L)  Integral viral exhalation ħ (quanta)  Lower airway infectious dose (GU)  Upper airway infectious dose (GU)  Infectivity (PFU/GU)  106  105  104  A  B  E  C  D  exhalation during the whole contagious time, expressed in quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The measurement is  deduced from the variant frequency curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (C) Infectious dose in upper airways, measured in  genome units (GU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (D) Infectious dose in lower airways, measured in genome units (GU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  (E) Maximum likelihood phylogenies inferred from spike nucleotide sequence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Distance  corresponds to number of mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Source: Gisaid/ Nextstrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Bibliography  Adams-Phipps, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Toomey, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Więcek, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Schmit, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Wilkinson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Scholl, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Jamrozik,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Osowicki, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Roestenberg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Manheim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' A Systematic Review of  Human Challenge Trials, Designs, and Safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Clin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Infect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Dis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' ciac820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1093/cid/ciac820  Adenaiye, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Lai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', de Mesquita, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Hong, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Youssefi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', German, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Tai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=',  Albert, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Schanz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Weston, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Hang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Fung, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Chung, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Coleman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=',  Sapoval, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Treangen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Berry, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Mullins, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Frieman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Ma, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Milton,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', University of Maryland StopCOVID Research Group, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Infectious SARS-  CoV-2 in Exhaled Aerosols and Efficacy of Masks During Early Mild Infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Clin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Infect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Dis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Infect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Dis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' ciab797.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content=',  Schiller, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content=', Leslie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Kiem, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=',  Fortune, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Berger, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Finberg, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Garber, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Schmidt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=',  Lingwood, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Shalek, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', Ordovas-Montanes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=', HCA Lung Biological Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Electronic address: lung-network@humancellatlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='org, HCA Lung Biological  Network, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' SARS-CoV-2 Receptor ACE2 Is an Interferon-Stimulated Gene in  Human Airway Epithelial Cells and Is Detected in Specific Cell Subsets across  Tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Cell 181, 1016-1035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='e19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='035  At the crossroads of epidemiology and biology: bridging the  gap between SARS-CoV-2 viral strain properties and  epidemic wave characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  — Supplementary Information —  Florian Poydenot, Jacques Haiech, Alice Lebreton and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Andreotti  January 30, 2023  Abstract  In this supplementary information document, we provide a self-contained review of the foundations  of standard epidemiological models, aimed at being accessible with basic knowledge of physics and  mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The particular formulation of epidemiological equations used to construct the figures of the  review has been published in a companion paper [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We review the assumptions and the parametrization  of the model using epidemiological data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We furthermore provide the table of published measurements  used in the review paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  1  Epidemic growth rate and reproduction rate  Sustained airborne transmission of SARS-CoV-2 from an index case to contacts depends on biological  and immune characteristics of the index case (transmissibility) and the contacts (susceptibility), but also  on social and physical characteristics such as the number of available contacts, the duration of exposure,  and the ventilation and mask wear during the contact (see Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We have developed the social and  physical aspects of transmission in a companion paper [1], and we discuss here the calibration of biological  quantities, both from a molecular and epidemiologic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Starting from the molecular description of the index case, viral kinetics determine both the transmis-  sibility through the viral shedding rate, and the dynamics of transmission through the course of evolution  of the illness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The viral kinetics is standardly described in the literature by an exponential growth of the  viral load (replication and shedding) followed by an exponential decay (immunity response).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' However, the  SARS-CoV-2 human challenge trial [2] has provided unprecedented time resolved data showing a more  rounded viral load curve (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Here, we parametrize the viral load V by the law  V = Vψ(t)  (1)  where V is a characteristic concentration for a particular infected person and ψ(t) the rescaled transmissibil-  ity at a time t after infection (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' 4-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We therefore assume that infected people with different maximum  viral load present on the average the same viral kinetics up to a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The infectious period T is defined  as the average time from infection, weighted by the transmissibility:  T =  � ∞  0  τψ(τ)dτ  � ∞  0  ψ(τ)dτ  (2)  It is worth noting that the integrals run from t = 0, which is the infection time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' More precisely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' we choose  a convenient normalization of ψ(t) to ensure that it is dimensionless:  T =  � ∞  0  ψ(τ)dτ  and  T 2 =  � ∞  0  τψ(τ)dτ  (3)  1  index case  secondary  case  N-1 contacts  transmissibility  susceptibility  ventilation  dilution  facial mask  N-1 contacts  duration Δt  work  restaurant  public  transportation  index case  home  non-infectible fraction  of the population  infection rate  Figure S1: (a) Transmission of SARS-CoV-2 in a certain space depends on the duration ∆t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' on the number  N of people present,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' on the the viral emission rate (transmissibility),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' on the susceptibility of contacts to  be infected and on the viral dilution ϵ due to ventilation and to face masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (b) The mean number of  people infected by an index case is the average over possible situations weighted by the duration ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Four  examples are schematized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' From left to right: high number N but short time ∆t and moderate dilution  factor ϵ (public transportation);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Long time ∆t but moderate number N and low dilution factor ϵ (work);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Long time ∆t and high dilution factor ϵ (restaurant);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Very long time ∆t with a high dilution factor ϵ but  small N (household transmission).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (c) Transmissibility is by definition the viral emission rate, which is  proportional to the replicable viral load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The infection rate I is the number of new infected people per unit  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It is related to the infection rate over the past period, to the epidemic reproduction rate, and to the  fraction of non-infectible people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Here, we have introduced a phenomenological equation of the form  log ψ = log ψm − a(t − Tm)2  1 + bt  (4)  2  to fit these data with the same number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' V is a characteristic viral load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Tm is the incubation  period, defined as the time from infection to the maximum viral load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The parameters a and b determine the  initial growth rate and final decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The constant ψm is fixed by the normalization condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We have  performed a mapping of the double exponential model onto equation (4) in order to deduce the parameters  that best describe the average kinetics for different variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The resulting curves are shown in Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  8  6  4  2  0  20  15  10  5  0  15  10  5  0  104  102  106  108  104  102  100  106  108  Delta  Wuhan  Alpha  Delta  Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1  Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2  Figure S2: (a) Modelled average viral kinetics for younger and older patients, vaccinated or not, obtained  by mapping the double exponential kinetics to equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The characteristics are those of the Delta strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  From N´eant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' [3] (b) Modelled average viral kinetics for the successive variants of interest, obtained  using the data reported in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The parameters of equation (4) are reported in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The viral transmissibility of a particular infected person (the index case) is defined here as the rate  of emission of replicable viral particles (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Under the simplifying assumption that emission is  proportional to the viral load in the upper respiratory tract, the transmissibility is proportional to the viral  load in the upper respiratory tract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We therefore parametrize the concentration of viral particles in the  exhaled air by the law:  C = Cψ(t)  (5)  where C is a characteristic concentration for a particular infected person and ψ(t) the rescaled transmissi-  bility at a time t after infection (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' 4-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The infection rate I is defined as the mean number of infected people per unit time in a given population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  At time t, a secondary case infection is induced by an index case contaminated before, at a time t−t′, with  a probability proportional to ψ(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Introducing R, the epidemic reproduction rate, defined as the mean  number of secondary cases per index case, I obeys an integral equation (see Grassly and Fraser [4] for a  review and derivation):  I(t) = A(t)R  T  � ∞  0  I(t − t′)ψ(t′)dt′  (6)  where A(t) is the fraction of the population susceptible to be infected (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' A(t)R(t)ψ(t′)/T is  classically known as the infectiousness at time t of the index case that has been infected for a duration t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  It is the product of the contact rate AR/T at the time of contact and the biological factor of infectiousness  since the onset of infection ψ(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Importantly, equation (6) remains valid even when the transmission time  is comparable to the epidemic growth timescale as it takes into account the viral kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='I(t) and A(t) are  unknowns, and must be solved together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' When a small fraction of the population is immune, A is close to 1,  and the equation admits an exact exponential solution to I(t), I(t) = I0e−σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Plugging this into equation  (6), we can relate the growth rate σ to the epidemic reproduction rate R by the Euler-Lotka equation:  R =  �  ψ(t)dt  �  ψ(t) exp(−σt)dt  (7)  3  This relation is plotted in figure 4-C for the rescaled viral load shown in figure 4-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' A discussion of the  Euler-Lotka renewal equation in a more general context can be found in chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' 13 of Martcheva [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The epidemic prevalence P is defined as the fraction of the population that has been infected in the  past:  P(t) = N −1  � t  −∞  I(t′)dt′  (8)  Then, assuming that each infection leads to long term immunization, a simple approximation of the sus-  ceptible fraction is A(t) = 1 − P(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' However, A may be very different from 1 − P due to vaccination, or  to the gradual loss of immunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  2  Reference point for the evaluation of the epidemic reproduction  rate R0  The epidemic reproduction rate R is the average number of secondary infections per index case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It com-  bines biological factors that determine infection susceptibility and viral transmissibility, and social factors  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' To define a reference reproduction rate R0 that would characterize the transmissibility of a given  viral strain, it is necessary to choose a reference state of the social behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The simplest choice is to  define R0 as the epidemic reproduction rate when society ignores the virus and behaves ”normally”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This  is, by definition, only possible when the epidemic starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Figure S3 shows the initial stage of SARS-CoV-2  epidemics in different French departments and different European countries, before the first 2020 lock-  down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It can be safely assumed that the number of deaths per unit time was proportional to the number of  cases, in this initial stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The curves show the multiplicative nature of the epidemic and is direct evidence  of the effect of the lockdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The epidemic reproduction number is around the same value R0 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='9 in  both departments/countries where the epidemic arrived earlier or later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' For small values of the epidemic  growth rate σ, the Euler-Lotka equation can be linearized:  R ≃  �  ψ(t)dt  �  ψ(t)(1 − σt)dt ≃ 1 + σT  (9)  This popular approximation leads to a much lower value of the epidemic reproduction number, around  R0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This difference can be ascribed to the non-linearity in the relation between σ and R, at large σT,  when the growth time σ−1 is comparable or larger than the infectious period T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  3  Relation between the reproduction rate and the infectious quan-  tum  The model described in this section is discussed in a companion paper [1] in the context of the social  and physical aspects of transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We recall here its underlying assumptions and the calibration of the  biological factors in the transmission risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We assume here that a single viral particle initiates the infection  when it penetrates a vulnerable locus where conditions are favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The probability that at least one viral  particle manages to enter a cell and replicates is independent of the presence of others viral particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It  depends on factors such as the type of cells or the density of ACE2 receptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Wells [6] introduced the  notion of dose and quantum to describe this probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' For a person having inhaled an intake dose d, the  probability law of infection p(d) takes the form p(d) = 1 − e−ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' a−1 is the infection dose of the person  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Its average over the population, ¯a−1, is by definition the quantum of infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The product  ad is therefore the dose, expressed in infectious quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' At small ad, the probability of infection can be  linearized: p(d) ≃ ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This excludes super-spreading events, which occur when an infected person with  a large exhaled viral concentration C attends an under-ventilated place, leading to multiple simultaneous  infections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The total number of exhaled viral particles is, on average, equal to  �  ¯qCψ(t)dt = ¯qCT, where  ¯q is the mean exhalation flow rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This number can be expressed in infectious quanta to define the mean  integrated quantum emission ¯h:  ¯h = ¯qC¯aT  (10)  4  50  40  30  20  10  10-2  10-1  100  101  102  0  10  20  Italy  France  Germany  Sweden  60  50  40  30  20  10  0  103  100  101  102  Bouches du Rhone  Loiret  Seine Saint-Denis  Paris  Côtes d’Armor  Alpes-de-Haute  Provence  Figure S3: (a) Curve of the cumulative number of deaths D as a function of time, in days, in different  French departments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The time axis is shifted so as to superimpose the curves in the first phase of the  epidemic on a jointly fitted exponential (σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='28 day−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' For each curve, the best fit gives the time  at which, statistically, the first death would have occurred on average, given the epidemic history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The  further away departments are from major cities, the later the epidemic occurs and the more lockdown,  imposed at the same date everywhere, has limited the number of deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (b) Curve of the cumulative  number of COVID-19 deaths normalized by the same number, on the day of lockdown, as a function of  time, in days, relative to the date of lockdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Although the epidemic arrived at very different dates in the  countries represented, the curves superimpose on a master curve, which shows the multiplicative nature  of the epidemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The best fit of the first phase of the epidemic by an exponential gives the growth rate  σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='03 day−1, which corresponds to R0 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This is much higher than the common value  deduced from the linearization of the Euler-Lotka equation (9), which underestimates R0 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The mean integrated quantum emission ¯h measures the transmissibility and encodes all the biological  part of the risk (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It is defined as an average over the sub-population attending the public space  considered of the number of quanta that would have been exhaled if an infected person were there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It may  depend on the particular activity taking place in the public space through the mean inhalation rate ¯q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  We consider a virtual situation in which N people would stay in a certain place during their entire  infectious period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Consider that an infected person amongst the N people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It would exhale a dose d = ¯h/¯a  or equivalently a number of infectious quanta ¯ad = ¯h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Introducing the dilution factor ϵ between exhalation  and inhalation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' S1), which characterizes the ventilation and dispersion efficiency, as well as the effect  of face masks, the inhaled dose (in quanta) is ¯hϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The average secondary case number is therefore:  r = (N − 1)¯ad = (N − 1)ϵ¯h  (11)  It is proportional to the total number of exhaled infectious quanta, to the number of infectible people  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The epidemic reproduction rate R deduces by averaging over the population, weighting the  different places in which they live according to the time they spent inside (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' S1):  R = ⟨ϵ(N − 1)⟩ ¯h  (12)  R is the product of three terms (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' S1): ⟨ϵ(N − 1)⟩ characterizes the social behaviour, including the  effect of ventilation and face masks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' ¯h characterizes the biological factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The mean integrated quantum  emission ¯h can be determined using equation (12), if social behaviours are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Figure S4 shows the  calibration of the mean integrated quantum emission using the epidemic evolution in secondary schools in  the United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We discuss the effect of masks on this determination in a companion paper [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  5  1200  1000  800  600  400  200  0  8  6  4  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='0  20  15  10  5  0  70  60  50  40  30  20  10  0  Figure S4: The epidemic evolution for secondary school pupils, between lockdown and holidays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (a) Cases  per million people in the United Kingdom, from 1 February to 12 April.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Blue: pupils from age 10 to age  14, with no mandatory mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The best fit by an exponential provides the reproduction number: R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='45  (σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='052 day−1) during school period vs R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='76 before (σ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='037 day−1) and R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='53  (σ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='087 day−1) after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Red: pupils from age 15 to age 19, with mandatory masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The best fit by  an exponential provides the reproduction number: R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='10 during school period vs R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='78 before  and R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='63 after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Uncertainties are typically 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The contribution of schools to the epidemic rate,  in the absence of mandatory masks, is estimated to R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='8 in March 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (b) Typical ventilation in  British schools in March, as deduced from Vouriot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' [7], in a classroom of volume per pupil 10 m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Left axis: CO2 concentration as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The pupils are not present in the classroom during  the periods of time shown in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The average concentration is C = 1070 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Right axis: deduced  transmission risk r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Considering that the average school time for secondary schools is 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='5 hours per  week, a total viral emission ¯h = 270 quanta is deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It corresponds to a typical viral emission rate  ¯q¯aC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='6 quanta/hour and an emission rate at maximum on the order of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='3 quanta/hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  4  Relative transmissibility and infectivity of successive variants  Variants only interact through immunization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' At low prevalence, each variant epidemic can be considered  as independent from the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The results of massive RT-PCR tests can then be used to determine the in-  fectious quantum of successive variants of concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Let us consider the simplest case were a variant noted  + replaces a variant noted −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The local epidemic growth rates σ− and σ+ are measured during the replace-  ment period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The infection rate of the strains are written I− = I− exp(σ−t) and I+ = I+ exp(σ+t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The  relative prevalence of the new variant therefore obeys the logistic equation:  I+  I+ + I−  =  1  1 + I−/I+ exp ((σ− − σ+) t)  (13)  The best fit of the new variant relative prevalence by equation 13 gives the difference σ− − σ+ within a  few percent uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The growth rate σ+ of the new variant is determined by fitting the evolution of  the number of new cases during the same period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Using the Euler-Lotka equation, the epidemic  reproduction rates R− and R+ are deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Taking the ratio R+/R−, the social component of the repro-  duction rate ⟨ϵ(N − 1)⟩ is eliminated, leaving the ratio of total viral emission ¯h+/¯h−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' For simplicity we  have ignored possible small differences of immunity between the strains Wuhan-1, Alpha and Delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Figure S5 shows the replacement of the Wuhan strain by the Alpha strain during the winter 2021 and the  replacement of the Alpha strain by the Delta strain during the summer 2021, in France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The transmissibility  of the strain Alpha (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Delta), as measured by ¯h, is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='7 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='4) times larger than the wild strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The  description of the transition from the strain Delta to the strain Omicron is described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  6  100  80  60  40  20  0  10-2  10-1  100  100  80  60  40  20  0  10-2  10-1  100  105  104  103  104  103  102  Jan 1, 2021  Feb 1, 2021  Mar 1, 2021  Apr 1, 2021  Jun 1, 2021  Jul 1, 2021  Aug 1, 2021  Sep 1, 2021  60  50  40  30  20  10  0  100  80  60  40  20  0  Figure S5: (a) Frequency of the variant Alpha in RT-PCR tests performed in France, as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The time origin t = 0 corresponds to January 1, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The best fit by the logistic equation (12) gives  σ+ − σ− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='077 day−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Insert: number of Alpha cases identified by positive RT-PCR tests in France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The growth rate at the emergence of the alpha epidemic wave is σ+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='070 day−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (b) Frequency of the  variant Delta in RT-PCR tests performed in France, as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The time origin t = 0 corresponds  to January 1, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The best fit by the logistic equation (12) gives σ+−σ− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='135 day−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Insert: number  of Alpha cases identified by positive RT-PCR tests in France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The growth rate at the emergence of the delta  epidemic wave is σ+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='147 day−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  5  Relative transmissibility and susceptibility of vaccinated people  The relative susceptibility S is by definition the ratio of the probability that vaccinated people get infected  to the probability that unvaccinated, never infected people, with the same immunological characteristics,  get infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It depends on age and on the vaccination status (type of vaccine, vaccination date).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The  relative susceptibility can be measured from the transmission rate of sub-populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Such measurements  are well converged statistically but suffers from social biases (mask wearing, attendance of restaurants and  bars, attendance of public spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Alternatively, it can be measured from household transmission, which  removes an important bias: the vaccination status of the index case is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' On the other hand, social  biases persist and the statistics is in general much lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Other biases like age can be adjusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' However,  as age, vaccination status and intrinsic susceptibility to infection (quality of the immunity) are strongly  correlated, it becomes problematic to exhibit a single quantity characterizing vaccination efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' From  the molecular biology point of view, relative susceptibility characterizes the neutralization of virus by  the antigenic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' There currently exists no calibration relating molecular aspects to epidemiologic  aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The relative transmissibility T is by definition the ratio of the probability that vaccinated people with  the virus (index case) infect other people (secondary cases) to the same probability for unvaccinated people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  From the molecular biology point of view, the relative transmissibility can be measured as the ratio of the  integral viral emission between vaccinated and unvaccinated people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' From the epidemiological point of  view, the relative transmissibility can only be measured through contact tracing and in particular household  transmission statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The crude measurement is the ratio of the secondary attack rates, conditioned by  the index vaccination status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The measurement suffers from social biases and a lack of statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Age,  vaccination status and intrinsic transmissibility (quality of the immunity) are strongly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' It is  therefore problematic to exhibit a single quantity characterizing vaccination efficiency against transmission,  after an adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Figure S6 (d) shows a compilation of measurements of S and T for an up to date vaccination status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Although dispersed, the data show a clear common decrease of relative susceptibility S and relative trans-  7  25  20  15  10  5  0  25  20  15  10  5  0  Unvaccinated  Alpha  Delta  Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1  Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2  Dose 2, 12-24 weeks  Dose 3, <12 weeks  Dose 3, >24 weeks  Dose 2, <12 weeks  Dose 1, <2 weeks  Dose 2, >24, weeks  Dose 3, 12-24 weeks  Dose 1, >2 weeks  10-5  10-4  10-3  10-5  10-4  10-3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='5  0  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2  0  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2  0  Dec 15, 2021  Dec 15, 2021  Jan 1, 2022  Figure S6: Incidence of SARS-CoV-2 infection for all sub-populations with different vaccination status, as  a function of time, in France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (a) Raw incidence data for the Delta variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (b) Raw incidence data for the  Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1 variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The time origin t = 0 corresponds to December 7th, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The best fit by expo-  nentials with the same rate for all vaccination status is superimposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The good fit confirms that vaccinated  and unvaccinated people infect each other sufficiently to share the same overall dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' (c) Relative sus-  ceptibility between fully vaccinated people and unvaccinated people, estimated from the relative incidence  Ij/Nj, as a function of relative susceptibility between fully vaccinated people and unvaccinated people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  (d) Relative transmissibility T as a function of relative susceptibility S for up to date vaccination status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' T  and S both should tend to 1 (no effect of vaccination on transmission) and to 0 (effective barrier immunity)  together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Solid line: phenomenological fit T = 1 − (1 − S)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  missibility T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We know that they both should tend to 1 (no effect of vaccination on transmission) and to 0  (effective barrier immunity) together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' A good phenomenological fit to the data is provided by the relation  T = 1 − (1 − S)2 and is shown in solid line in figure S6 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  As a simplifying assumption, we consider that contacts of vaccinated and unvaccinated people are  similarly composed: then, each person is statistically in contact with vaccinated and unvaccinated people,  8  100  101  102  103  104  105  Jan 1, 2022  Feb 1, 2022  Dec 1, 2021  60  40  20  0  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2  Figure S7: (a) Number of new cases of the variants BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1 and BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2 in France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The black and green solid  lines are the best exponential fit in two different periods of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  following the fractions of the whole society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We consider a division of society into J classes (according  to age, vaccination status, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=') whose size is denoted Nj (such that N = �  j Nj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' We denote by Sj and  Tj, the relative susceptibility and relative onward transmissibility associated with the class j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The infection  rate Ij, defined as the mean number of infected people per unit time in the class j, obeys the equation:  Ij(t) = Sj(1 − Pj(t))R  T  Nj  N  � ∞  0  �  k  TkIk(t − t′)ψ(τ)dt′  with  T =  � ∞  0  ψ(τ)dt′  (14)  The epidemic reproduction rate R is here defined for a virtual society of unvaccinated, never infected  people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' At small P, these equations admit an exact exponential solution of the form:  Ij ∝ Sj(1 − Pj(t)) Nj  N exp(σt)  (15)  whose growth rate σ is related to R by the generalised Euler-Lotka equation:  R =  �  ψ(t)dt (�  j Nj)  �  ψ(t) exp(−σt)dt (�  j SjTjNj)  (16)  Figure S6 (a) and (b) shows that incidences for all sub-populations share the same growth rate, meaning  that vaccinated and unvaccinated people infect each other enough to obey the same dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Under this  assumption, the relative incidence Ij/Nj provides an estimate of the relative susceptibility Sj for different  vaccination schemes, displayed in Figure S6 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' For the Delta strain, the susceptibility is ordered from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='15  for a complete vaccination scheme in 3 doses to 1, the unvaccinated reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' For Omicron, on the other  hand, the susceptibility is mostly 1 within noise except for people vaccinated with two doses but delaying  or refusing the third one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' This may point to biases introduced by different social behaviors correlated with  the vaccination status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  During three weeks, the growth rate of Omicron was σ+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='23 day−1 vs σ− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='0 day−1 for Delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The transmissibility of the strain Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1, as measured by ¯h, is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='8 times larger than the Delta strain,  6 times larger than the wild strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Figure S7 (a) shows the number of cases of Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1 and BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2 in  France during the replacement period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' An exponential phase is observed during two short periods of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  The transmissibility of the strain Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='7 larger for BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2 than BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1, consistently between the  two estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  9  Dec 1, 2019  Apr 1, 2020  Aug 1, 2020  Dec 1,2020  Apr 1, 2021  Aug 1, 2021  Dec 1, 2021  date  Wuhan-1  Alpha  (21J)  (21I)  Delta  Beta  Gamma  Omicron  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1 (21K)  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2 (21L)  (21A)  Alpha  Delta (21I)  Delta (21A)  Delta (21J)  Beta  Gamma  Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1 (21K)  Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2 (21L)  Figure S8: Phylogenetic tree up for the strains included in the article: Alpha, Beta, Gamma, Delta, Omicron  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1 and Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Source: GISAID/Nextstrain [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  10  Dec 1, 2019  May 1, 2020  Oct 1, 2020  Mar 1,2021  Aug 1, 2021  Jan 1, 2022  Nov 1, 2022  date  Wuhan-1  Alpha  Delta  Beta  Gamma  Omicron  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2 (21L)  BQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1  (22E)  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='75 (22D)  XBB (22F)  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='4 (22A)  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='5  (22B)  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1 (21K)  Alpha  Delta (21I)  Delta (21A)  Delta (21J)  Beta  Gamma  Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1 (21K)  Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2 (21L)  Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='4 (22A)  Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='5 (22B)  Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='75 (22D)  Omicron BQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1 (22E)  Omicron XBB (22F)  Figure S9: Phylogenetic tree including the Omicron variants: BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1, BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2, BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='4, BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='5, BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='75 and XBB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  Source: GISAID/Nextstrain [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  11  6  Supplementary Tables  Table 1: References used for viral kinetics  Strain  Maximum  viral  load  (log10 copies/mL)  Growth  time  (days)  Decay time (days)  Reference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  com-  ments  Wuhan  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='62  [2]  Wuhan  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='22  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='7  [9]  Wuhan  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='53  [10]  Wuhan  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='89  [18]  Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='4  [19]  Table 2: Parameters of the viral kinetics model  Strain  log ψm  (log10 copies/mL)  a  (log10 copies/mL)  b (days−1)  Tm (days)  Wuhan  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='0  Alpha  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='33  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='8  Delta  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='40  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='9  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='51  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='30  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='51  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='30  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1  Table 3: References used for the PFU/GU ratio  Strain  PFU/GU  Cells  Replication  Reference  Wuhan  85000  Vero  In vivo  [2]  Wuhan throat  94000  Vero  In vivo  [2]  Wuhan  13000000  Vero  In vivo  [12]  Alpha  430000  Vero  In vivo  [20]  Delta  43000  Vero  In vivo  [20]  Delta  590000  Vero  In vivo  [12]  Omicron BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1500000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Vero  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='In vivo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='[12]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Delta  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='unvacci-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='nated  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1300000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Vero  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='In vivo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='[12]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Delta vaccinated  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2500000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Vero  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='In vivo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='[12]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Delta  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='unvacci-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='nated  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1400000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Vero  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='In vivo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='2100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='600  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Calu  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='In vitro  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='Vero  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='In vitro  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='22000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='hNEC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='17000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='hNEC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='In vitro  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='[22]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Alpha  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='360  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Vero-TMPRSS2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='In vitro  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='[20]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Delta  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='90  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Vero-TMPRSS2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='In vitro  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='[20]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Epsilon  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Vero-TMPRSS2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='In vitro  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='[20]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Alpha  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='210  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Calu-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='In vitro  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='[20]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Delta  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='In vitro  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='[20]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Beta  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1700 − 2800  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Vero  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='In vitro  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='[23]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='SARS-CoV-1 Ur-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='bani  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Vero  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='In vitro  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='[24]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Alpha  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='700  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Vero  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='In vitro  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='[25]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Wuhan  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='6400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Vero  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='In vivo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='[26]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='Strain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Susceptibility  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Transmissibility  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Odds ratio type  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Reference  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='Alpha  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='32  adjusted  [27]  Alpha  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='29  adjusted  [28]  Wuhan and Alpha  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='53  adjusted  [29]  Delta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='15  raw  This study  Delta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='16  adjusted  [30]  Delta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='4  raw  [30]  Delta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='5  adjusted  [27]  Delta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='45  raw  [31]  Delta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='48  adjusted  [32]  Delta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='61  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1  raw  [33]  Delta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='91  raw  [34]  Delta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='29  adjusted  [35]  BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='5  [50]  Omicron  10  [52]  15  References  [1] Florian Poydenot, Ismael Abdourahamane, Elsa Caplain, Samuel Der, Jacques Haiech, Antoine Jal-  lon, In´es Khoutami, Amir Loucif, Emil Marinov, and Bruno Andreotti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content=' ISSN 2752-6542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='  The Lancet Infectious Dis-  eases, 21(10):1351, October 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='1016/S1473-3099(21)00580-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+page_content='com/retrieve/pii/S1473309921005806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  [49] Elizabeth Bast, Fei Tang, Jason Dahn, and Ana Palacio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Increased risk of hospitalisation and death  with the delta variant in the USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' The Lancet Infectious Diseases, 21(12):1629–1630, December  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' ISSN 14733099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1016/S1473-3099(21)00685-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' URL https://linkinghub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='com/retrieve/pii/S147330992100685X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  [50] Tommy Nyberg, Neil M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Ferguson, Sophie G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Nash, Harriet H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Webster, Seth Flaxman, Nick An-  drews, Wes Hinsley, Jamie Lopez Bernal, Meaghan Kall, Samir Bhatt, Paula Bianca Blomquist, Asad  Zaidi, Erik Volz, Nurin Abdul Aziz, Katie Harman, Russell Hope, Andre Charlett, Meera A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Chand,  Azra Ghani, Shaun Seaman, Gavin Dabrera, Daniela DeAngelis, Anne M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Presanis, and Simon Thel-  wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Comparative Analysis of the Risks of Hospitalisation and Death Associated with SARS-CoV-2  Omicron (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='529) and Delta (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2) Variants in England.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' SSRN Electronic Journal, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  ISSN 1556-5068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='2139/ssrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='4025932.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='ssrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='com/abstract=  4025932.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  [51] UK Health Security Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' SARS-CoV-2 variants of concern and variants under investigation in  England - Technical briefing 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Technical report, UK Health Security Agency, UK, February 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  [52] UK Health Security Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' COVID-19 vaccine surveillance report - week 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content=' Technical report, UK  Health Security Agency, UK, February 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
+page_content='  24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFOT4oBgHgl3EQf-zSY/content/2301.12975v1.pdf'}
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+Self-testing composite measurements and bound entangled state in a unified framework
+Shubhayan Sarkar,1, 2 Chandan Datta,3 Saronath Halder,3 and Remigiusz Augusiak1
+1Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland
+2Laboratoire d’Information Quantique, Université libre de Bruxelles (ULB), Av. F. D. Roosevelt 50, 1050 Bruxelles, Belgium
+3Centre for Quantum Optical Technologies, Centre of New Technologies,
+University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland
+Within the quantum networks scenario we introduce a single scheme allowing to certify three dif-
+ferent types of composite projective measurements acting on a three-qubit Hilbert space: one con-
+structed from genuinely entangled GHZ-like states, one constructed from fully product vectors that
+exhibit the phenomenon of nonlocality without entanglement (NLWE), and a hybrid measurement
+obtained from an unextendible product basis (UPB). Noticeably, we certify a basis exhibiting NLWE
+in the smallest dimension capable of supporting this phenomenon. On the other hand, the possibility
+of certification of a measurement obtained from a UPB has an interesting implication that one can
+also self-test a bound entangled state in the considered quantum network. Such a possibility does not
+seem to exist in the standard Bell scenario.
+Introduction.—With the advancement of new tech-
+nologies such as quantum cryptography [1], device-
+independent (DI) certification of quantum devices is be-
+coming increasingly important, allowing one to certify
+certain features of an underlying device in a “black-box”
+scenario [2–4], which requires basically no assumptions
+about the device except that it is governed by quantum
+theory.
+The key ingredient for DI certification is Bell
+nonlocality, i.e., the existence of quantum correlations
+that cannot be explained by local hidden variable mod-
+els [5, 6]. The most comprehensive form of DI certifi-
+cation is self-testing [7], which allows for almost com-
+plete characterisation of the underlying quantum state
+and the measurements performed on it. From an appli-
+cation standpoint, this type of certification is crucial as
+it allows to verify whether a quantum device works as
+expected without knowing its internal mechanism.
+Since its introduction in [7], self-testing has been in-
+vestigated in various scenarios and shown to have nu-
+merous advantages [8]. However, while much attention
+has been paid to the self-testing of bipartite and multi-
+partite states [9–22], the problem of certifying quantum
+measurements, in particular those acting on composite
+Hilbert spaces, has been largely unexplored. Apart from
+a few classes of local measurements [18, 20, 23] and a
+few composite ones [24–26], no general scheme exists
+allowing to certify composite measurements even in the
+simplest case of multi-qubit Hilbert spaces.
+Our aim here is to fill this gap and introduce a uni-
+fied scheme in the quantum networks scenario that
+allows for self-testing, in a single experiment, three
+types of projective measurements:
+one composed of
+genuinely entangled states, one constructed from fully
+product states exhibiting nonlocality without entangle-
+ment (NLWE) [27], and a hybrid one which is con-
+structed from an unextendible product basis (UPB) [28]
+and a projector onto the completely entangled subspace
+orthogonal to the UPB. Notice that UPBs are interesting
+mathematical objects that have found numerous appli-
+cations, for instance, in constructing bound entangled
+states [28] or Bell inequalities with no quantum viola-
+tion [29].
+While a network-based self-testing scheme of a two-
+qutrit product basis exhibiting NLWE was previously
+introduced in [26], our approach enables one to self-test
+such a basis in the smallest possible dimension capable
+of supporting the notion of NLWE, which is eight. Fur-
+thermore, our scheme utilises fewer measurements and
+is thus more efficient as compared to [26]. Finally, it also
+allows to simultaneously self-test a measurement of a
+hybrid type, which is constructed from a UPB. An inter-
+esting implication of this fact is that one can also (indi-
+rectly) certify in the network, a mixed bound entangled
+state constructed from the UPB. While a network-based
+certification scheme for pure states has already been in-
+troduced in Ref. [30], our work seems to be the first to
+address the question of certification of mixed entangled
+states (see nevertheless Refs. [31, 32]).
+Preliminaries.—Consider a Hilbert space H = H1 ⊗
+. . . ⊗ HN and a set of mutually orthogonal fully prod-
+uct states from H, S = {|ψ1
+i ⟩ ⊗ . . . ⊗ |ψN
+i ⟩}k
+i=1, where
+|ψm
+i ⟩ ∈ Hm and k ≤ D = dim H. Following Ref. [27] we
+say that this set exhibits NLWE if the vectors |ψi⟩ can-
+not be perfectly distinguished by local operations and
+classical communications (LOCC). An exemplary such
+set is the following basis of the three-qubit Hilbert space
+H = (C2)⊗3 [27]:
+|δ0⟩ = |0⟩|1⟩|+⟩, |δ1⟩ = |0⟩|1⟩|−⟩, |δ2⟩ = |+⟩|0⟩|1⟩,
+|δ3⟩ = |−⟩|0⟩|1⟩, |δ4⟩ = |1⟩|+⟩|0⟩, |δ5⟩ = |1⟩|−⟩|0⟩,
+|δ6⟩ = |0⟩|0⟩|0⟩,
+|δ7⟩ = |1⟩|1⟩|1⟩,
+(1)
+where |i⟩, |i⟩ (i = 0, 1) and |±⟩, |±⟩ are the eigenvec-
+tors of Z, (X + Z)/
+√
+2 and X, (X − Z)/
+√
+2, respectively,
+where Z and X are the Pauli matrices.
+While the above set forms a complete basis in the
+corresponding Hilbert space, there also exist sets S ex-
+hibiting NLWE that do not span the underlying Hilbert
+space.
+These are called UPB and were introduced to
+provide one of the first constructions of bound entan-
+gled states, which are entangled states from which no
+arXiv:2301.11409v1  [quant-ph]  26 Jan 2023
+
+2
+pure entanglement can be distilled [27]. To be more pre-
+cise, a collection S is a UPB if k < D, i.e., S spans a
+proper subspace V in H, and the subspace complemen-
+tary to V is completely entangled [33, 34], i.e., contains
+no fully product vectors. An excellent example of a UPB
+in H = (C2)⊗3 are the following four vectors
+|τ0⟩ = |0⟩|1⟩|+⟩,
+|τ1⟩ = |+⟩|0⟩|1⟩,
+|τ2⟩ = |1⟩|+⟩|0⟩,
+|τ3⟩ = |−⟩|−⟩|−⟩
+(2)
+which are equivalent under local unitary transforma-
+tions to the Shifts UPB introduced in [28]. The mixed
+state ρ = Γ/4, where
+Γ = 1 −
+3
+∑
+i=0
+|τi⟩⟨τi|
+(3)
+stands for the projector onto a subspace complementary
+to the UPB, is bound entangled; in fact, by the very con-
+struction it is entangled and all its partial transpositions
+are nonnegative [35].
+The last set of vectors that we consider here are the
+following GHZ-like pure states
+|φl⟩ =
+1
+√
+2
+(|l1l2l3⟩ + (−1)l1|l1l2l3⟩),
+(4)
+where l ≡ l1l2l3 with l1, l2, l3 = 0, 1 and li is the negation
+of the bit li, i.e., li = 1 − li. It is worth noting that, unlike
+the previous vectors |δi⟩ or |τi⟩, the GHZ states are all
+genuinely multipartite entangled.
+Composite measurements.— From each of the consid-
+ered sets of vectors one can construct a projective mea-
+surement acting on H3 = (C2)⊗3. First, the product ba-
+sis |δi⟩ gives rise to a separable eight-outcome measure-
+ment MNLWE = {|δi⟩⟨δi|}7
+i=0. This measurement cannot
+be implemented in terms of the LOCC, and, simultane-
+ously, cannot produce entanglement if applied to a state.
+On the other extreme, we have the eight-outcome mea-
+surement MGHZ = {|φl⟩⟨φl|}7
+l=0 constructed from the
+GHZ-like states which are all entangled. Thus, unlike
+MNLWE, this measurement leads to entangled states for
+all outcomes l.
+Let us finally move to the UPB in Eq.
+(3).
+It al-
+lows constructing a five-outcome hybrid measurement
+MUPB = {|τi⟩⟨τi|}3
+i=0 ∪ {Γ} that lies in between the GHZ
+measurement and the separable measurement. In fact,
+four of its outcomes correspond to projections onto fully
+product states |τi⟩, whereas the last outcome is repre-
+sented by Γ that projects onto a completely (but not gen-
+uinely) entangled four-dimensional subspace.
+Setting the scenario.—We consider a quantum network
+scenario consisting of three external parties Alice, Bob
+and Charlie and a central party Eve (see Fig. 1). The
+scenario also comprises of three independent sources Pi
+that distribute bipartite quantum states among the par-
+ties. We denote these states by ρss with s = A, B, C,
+where the subsystems A, B and C belong to the exter-
+nal parties, whereas the other three systems A, B and
+C go to Eve; in what follows we simplify the notation
+by using E := ABC. On their shares of the joint state
+ρABCE = ρAA ⊗ ρBB ⊗ ρCC, each party can choose to per-
+form one of the measurements Ax, By, Cz and Ee, where
+the measurement choices are labelled x, y, z, e = 0, 1, 2.
+We assume that each of the external party’s measure-
+ments has two outcomes, denoted a, b, c = 0, 1. The first
+two Eve’s measurements yield eight outputs, whereas
+the third one results in five outcomes. During the exper-
+iment, the parties cannot communicate classically.
+The correlations obtained by repeatedly performing
+these measurements are captured by a set of proba-
+bility distributions ⃗p = {p(abcl|xyze)}, where each
+p(abcl|xyze) is the probability of observing outcomes a,
+b, c and l by Alice, Bob, Charlie and Eve after perform-
+ing measurements labelled by x, y, z, and e, respectively;
+it is given by the well-known formula
+p(abcl|xyze) = Tr
+�
+ρABCENA
+a|x ⊗ NB
+b|y ⊗ NC
+c|z ⊗ NE
+l|e
+�
+,
+(5)
+where NA
+a|x, NB
+b|y etc. are the measurement elements rep-
+resenting the measurements of the observers; these are
+positive semi-definite and satisfy ∑a Ns
+a|x = 1 for every
+measurement choice x and every party s.
+It will be beneficial to use another representation of
+the observed correlations, that is, in terms of the ex-
+pectation values of observables of the external parties,
+which are defined as
+⟨AxByCzNE
+l|e⟩ =
+∑
+a,b,c=0,1
+(−1)a+b+cp(abcl|xyze).
+(6)
+Notice that by employing Eq. (5) one can express them
+as ⟨AxByCzNE
+l|e⟩
+=
+Tr[(Ax ⊗ By ⊗ Cz ⊗ NE
+l|e)ρABCE],
+where Ax, By and Cz are quantum operators that are
+defined through the measurement elements as sk =
+Ns
+0|k − Ns
+1|k, where s = A, B, C and k = x, y, z. In the
+particular case of projective measurements these opera-
+tors sk become unitary and thus represent the standard
+quantum observables.
+Self-testing.— The quantum networks scenario has re-
+cently been harnessed to propose self-testing schemes
+for few quantum measurements defined in composite
+Hilbert spaces such as the measurement correspond-
+ing to the two-qubit Bell basis composed of four maxi-
+mally entangled vectors [24], or the nine-outcome pro-
+jective measurement corresponding to a complete ba-
+sis in C3 ⊗ C3 that exhibits NLWE [26]. Our aim here
+is to employ these ideas to design a general frame-
+work for quantum networks-based device-independent
+(NDI) certification of various interesting types of quan-
+tum measurements, concentrating on the particular case
+of three-qubit Hilbert spaces.
+To define the task of self-testing in more precise terms,
+let us consider again the scenario depicted on Fig. 1, but
+
+3
+FIG. 1. Schematic of the considered quantum network sce-
+nario. It consists of four parties A, B, C and E and three in-
+dependent sources distributing bipartite quantum states ρSS
+(S = A, B, C) among the parties as shown on the figure. The
+central party E shares quantum states with each of the other
+parties. Each party performs one of the available measure-
+ments on their share of the state obtaining an outcome. The ob-
+tained correlations {p(abcl|xyze)} are used to certify that each
+source distributes the maximally entangled state of two qubits
+and that E’s measurements are MGHZ, MUPB and MNLWE.
+now we assume that both the states ρss ∈ L(Hs ⊗ Hs)
+and the measurements performed by the parties are un-
+known. Due to the fact that the dimensions of the under-
+lying Hilbert spaces Hs ⊗ Hs are unspecified we can em-
+ploy the standard dilation arguments and assume that
+the shared states are pure, that is, ρss = |ψss⟩⟨ψss| and
+that the measurements are projective. These states and
+measurements generate correlations that we denote by
+⃗p.
+Consider then a reference experiment involving some
+known pure states |ψ′
+ss⟩ ∈ Hs′ ⊗ Hs′ and known projec-
+tive measurements represented by the observables A′
+x,
+B′
+y, C′
+z and E′
+e that generate the same correlations ⃗p. We
+say that both experiments are equivalent, or, alterna-
+tively, that |ψ′
+ss⟩ and A′
+x, B′
+y, C′
+z, E′
+e are self-tested from
+⃗p if one can prove that the local Hilbert spaces admit
+the product form Hs = Hs′ ⊗ Hs′′ and Hs = Hs′ ⊗ Hs′′
+(s = A, B, C) for some auxiliary Hilbert spaces Hs′′
+and Hs′′, and that there are local unitary operations
+Us : Hs → Hs′ ⊗ Hs′′ and Us : Hs → Hs′ ⊗ Hs′′
+(Us ⊗ Us)|ψss⟩ = |ψ′
+s′s′⟩ ⊗ |junks′′s′′⟩,
+(7)
+where |junks′′s′′⟩ belongs to Hs′′ ⊗ Hs′′ and
+Us si U†
+s = s′
+i ⊗ 1s′′,
+UE Ee U†
+E = E′
+e ⊗ 1E′′,
+(8)
+where 1E′′ is the identity acting on the auxiliary systems
+Hs′′ and UE = UA ⊗ UB ⊗ UC; recall that E′′ = A′′B′′C′′.
+It is important to note here that, since the measure-
+ments can only be characterised on the local states, a
+natural assumption that we make throughout this work
+is that the latter are full-rank.
+Results.—We propose a scheme that allows to device-
+independently certify in a single experiment the three
+different measurements introduced above, i.e., MGHZ,
+MNLWE, and the hybrid one MUPB. Consider again the
+network scenario in which the unknown pure states
+|ψss⟩ with s
+=
+A, B, C are distributed by indepen-
+dent sources among four parties who perform unknown
+measurements Ax, By, Cz and Ee on their shares of those
+states. Now, their aim is to exploit the observed cor-
+relations ⃗p to certify in the network the ideal reference
+experiment in which each source distributes the max-
+imally entangled state of two qubits |ψ′
+ss⟩ = |φ+⟩ =
+(|00⟩ + |11⟩)/
+√
+2, the external observers measure the
+following observables on their shares of the joint state
+A′
+0/1/2 = X ± Z
+√
+2
+/Y,
+s′
+0/1/2 = Z/X/Y (s = B, C).
+(9)
+and Eve performs the three measurements mentioned
+above, i.e., E′
+0 = MGHZ, E′
+1 = MNLWE and E′
+2 = MUPB.
+To make our considerations easier to follow we divide
+them into three parts, each devoted to one of Eve’s mea-
+surements. We begin with E0 = {Rl|0}7
+l=0. To certify
+that it is equivalent to the GHZ measurement MGHZ, the
+observed correlations ⃗p must be such that for each out-
+come l of E0, {p(abcl|xyz0)} (x, y, z = 0, 1) maximally
+violate the Bell inequality
+√
+2(−1)l1
+�
+2 �A1B1C1 + (−1)l2 �A0B0 + (−1)l3 �A0C0
+�
+≤ 4,
+(10)
+where �A0/1 = (A0 ± A1)/
+√
+2 and l ≡ l1l2l3 with
+l1, l2, l3 = 0, 1 is the binary representation of l, and the
+probability of observing the outcome l by Eve must obey
+P(l|e = 0) = 1/8.
+The Bell inequality corresponding to l = 0 was intro-
+duced in [36] and is maximally violated by |φ0⟩, whereas
+those corresponding to l ̸= 0 are its modifications that
+are adjusted to be maximally violated by the remaining
+GHZ states |φl⟩ and the quantum observables given in
+Eq. (9). In fact, these Bell violations can be achieved in
+the reference quantum network described above.
+Let us now state our first result on self-testing the
+GHZ measurement (see Appendix B for proof).
+Theorem 1. Assume that the observed correlations ⃗p ob-
+tained in the network are such that the Bell inequalities in
+Eq. (10) are maximally violated for each outcome l of Eve’s
+measurement E0 and that each outcome occurs with proba-
+bility P(l|e = 0) = 1/8. Then, (i) the Hilbert spaces de-
+compose as Hs = Hs′ ⊗ Hs′′ and Hs = Hs′ ⊗ Hs′′; (ii)
+There exist local unitary transformations Us : Hs → Hs and
+Us : Hs → Hs such that
+(Us ⊗ Us)|ψss⟩ = |φ+
+s′s′⟩ ⊗ |ξs′′s′′⟩
+(11)
+for some |ξs′′s′′⟩ ∈ Hs′′ ⊗ Hs′′, and the measurements of all
+parties are certified as
+U Rl|0 U† = |φl⟩⟨φl|E′ ⊗ 1E′′,
+Us si U†
+s = s′
+i ⊗ 1s′′ (12)
+
+4
+for all l and i = 0, 1 where U = ⊗sUs such that s = A, B, C
+and E = ABC. The states |φl⟩ and the observables s′
+i are
+given in Eqs. (4) and (9) respectively.
+In what follows we build on this result to show how to
+certify the other Eve’s measurements E1 and E2 and also
+the third measurements of the external parties A2, B2
+and C2. Let us then consider Eve’s second measurement
+E1 = {Rl|1}7
+l=0. In order to certify that it is equivalent to
+the separable measurement MNLWE, the observed corre-
+lations ⃗p, apart from the conditions stated in Theorem 1,
+must additionally satisfy
+p(0100|0011) = p(0111|0011) = p(0012|1001)
+= p(1013|1001) = p(1004|0101) = p(1105|0101)
+= p(0006|0001) = p(1117|0001) = 1
+8.
+(13)
+Notice that these conditions are met in the ideal experi-
+ment outlined above.
+Let us state formally our second result that together
+with Theorem 1 provides a scheme for DI certification
+of the separable measurement exhibiting NLWE in the
+least possible dimension (cf. Appendix C for a proof).
+Theorem 2. Suppose that ⃗p generated in the network satis-
+fies the assumptions of Theorem 1 as well as the conditions
+in Eq.
+(13).
+Then, for any l it holds that U Rl|1 U† =
+|δl⟩⟨δl|E′ ⊗ 1E′′ where U is the same unitary as in Theorem 1
+and E = ABC.
+Before proceeding to the final result which is self-
+testing of MUPB in E2, we need to introduce another
+condition that is necessary to prove a self-testing state-
+ment for A2, B2 and C2.
+Precisely, the correlations
+{p(abc0|xyz0)} with x, y, z = 1, 2 corresponding to the
+situation in which Eve observes the first outcome l = 0
+of E0, the following condition is satisfied
+⟨ �A1B1C1 − �A1B2C2 − A2B1C2 − A2B2C1⟩ = 4,
+(14)
+where �A1 = (A0 − A1)/
+√
+2 and the above functional is
+inspired by the Mermin inequality [37]. This along with
+Theorem 1 implies that (see Appendix B 3)
+Us s2 U†
+s = ±Ys′ ⊗ 1s′′
+(s = A, B, C).
+(15)
+With the above characterisation at hand, we can fi-
+nally move onto showing how to certify MUPB in E2 =
+{Rl|2}4
+l=0. To this end, the observed correlations must
+satisfy
+p(0100|0012) = p(0011|1002) = p(1002|0102)
+= p(1113|1112) = 1
+8
+(16)
+along with four other conditions stated in Appendix D
+as Eqs. (D4a)-(D4d); we refer to them as Pr2. Notice
+again that correlations obtained within the reference ex-
+periments fulfil the above conditions. Let us now state
+the following theorem.
+Theorem 3. Assume that the assumptions of Theorem 2
+and the conditions in Eq. (16) and Pr2 are satisfied. Then,
+the measurement E2 = {Rl|2} is certified as U Rl|2 U† =
+|τl⟩⟨τl|E′ ⊗ 1E′′ for l = 0, 1, 2, 3, and, U R4|2 U† = ΓE′ ⊗
+1E′′, where |τl⟩ and Γ are defined in Eqs. (2) and (3), respec-
+tively, and U is the same unitary operation as in Theorem 1.
+The proof can be found in Appendix D. This final
+result shows that the hybrid separable-entangled mea-
+surement constructed from a UPB can also be self-tested
+using our scheme. Most importantly, this is the minimal
+scenario possible to self-test such a measurement.
+Bound entangled state.—An interesting consequence of
+Theorem 3 is that the considered network allows one to
+self-test a bound entangled state shared between the ex-
+ternal parties. Assume that the states and Eve’s mea-
+surement E2 are certified as in Eq. (9) and as in Theorem
+5, respectively. The post-measurement state shared by
+the external parties that correspond to the last outcome
+of E2 is then given by [see Appendix D]
+U ρABC U† = 1
+4ΓA′B′C′ ⊗ ˜ρA′′B′′C′′,
+(17)
+where U = �
+s Us and the unitaries Us are the same as in
+Theorem 1. As mentioned above, the state Γ/4 is bound
+entangled [27] and can be prepared by Eve in the exter-
+nal parties labs with a simple post-processing strategy.
+She first broadcasts her outcome of the measurement E2
+and then the external parties discard those runs of the
+experiment for which Eve observes any other outcome
+than the last one.
+Outlook.—Inspired by the notion of NLWE, we can
+identify a larger set of orthogonal projectors which can
+be referred to as nonlocality without distillable entan-
+glement (NLWDE). NLWDE is defined as a set of pro-
+jectors that cannot be perfectly distinguished using local
+operations and classical communication, such that their
+normalised versions are positive under partial transpose
+with respect to every bipartition. In this work, the mea-
+surement composed of UPB and bound entanglement
+falls under this category. It will be interesting to further
+analyse the properties of such sets.
+Several other follow-up problems arise from our
+work. Designing certification schemes for GHZ bases
+and product bases exhibiting NLWE for N qubits will
+be straightforward from our work. A more interesting
+question would be to generalize our scheme to certify
+any hybrid or even any composite projective measure-
+ment in the case of any number of qubits. An even more
+challenging problem will be to propose a quantum-
+networks-based scheme for non-projective composite
+measurements. Our work also establishes a way to self-
+test mixed entangled states.
+A natural question here
+would be to construct schemes to self-test any mixed en-
+tangled state; a possibility that does not seem to exist in
+the standard Bell scenario.
+
+5
+ACKNOWLEDGMENTS
+This project was funded within the QuantERA II Pro-
+gramme (VERIqTAS project) that has received fund-
+ing from the European Union’s Horizon 2020 research
+and innovation programme under Grant Agreement No
+101017733 and from the Polish National Science Center
+(project No 2021/03/Y/ST2/00175). C.D. and S.H. ac-
+knowledge the support by the “Quantum Optical Tech-
+nologies” project, carried out within the International
+Research Agendas program of the Foundation for Pol-
+ish Science co-financed by the European Union under
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+Appendix A: General result
+Before proceeding to the proofs of the main results, we introduce an important lemma that is required to derive
+our results.
+Lemma 1. Consider a positive semi-definite matrix M such that M ≤ 1 that for a given density matrix ρ which is full-rank
+satisfies Tr(Mρ) = 1. Then, M is an identity matrix acting on the support of ρ.
+Proof. Let us consider the eigendecomposition of ρ, ρ = ∑k pk|ψk⟩⟨ψk| such that pk > 0. After putting it into the
+condition Tr(Mρ) = 1, one obtains
+∑
+k
+pk⟨ψk|M|ψk⟩ = 1,
+(A1)
+which by employing the fact that ∑k pk = 1 can further be rewritten as
+∑
+k
+pk(1 − ⟨ψk|M|ψk⟩) = 0.
+(A2)
+Due to the facts that 0 ≤ M ≤ 1 and pk > 0, the above equation can hold true only if ⟨ψk|M|ψk⟩ = 1. Thus, every
+|ψk⟩ is an eigenstate of M with eigenvalue 1. As a result, M = ∑k |ψk⟩⟨ψk| = 1ρ, where 1ρ is an identity acting on the
+support of ρ.
+Appendix B: Self-testing of GHZ bases, measurements of external parties and the states prepared by the preparation devices
+In this section, we explain the certification of the GHZ basis, measurements of the external parties and the states
+distributed by the sources.
+1.
+The GHZ basis and measurements Ax, By and Cz with x, y, z = 0, 1
+Let us first consider the following eight Bell inequalities
+Il = 2(−1)l1 ⟨(A0 + A1)B1C1⟩ + (−1)l2+l1 ⟨(A0 − A1)B0⟩ + (−1)l3+l1 ⟨(A0 − A1)C0⟩ ≤ 4,
+(B1)
+where l = l1l2l3 with li ∈ {0, 1} for each i = 1, 2, 3. The inequality for l1 = l2 = l3 = 0 was constructed in
+[36], whereas the remaining seven are its variants obtained by making the signs in front of each expectation value
+depend on the parameters li. We can now state the following fact which is concerned with Tsirelson’s bound of the
+inequalities in Eq. (B1).
+Fact 1. The maximal quantum value of the Bell expression Il is 4
+√
+2 and it is achieved by the following observables
+A0 = X + Z
+√
+2
+,
+A1 = X − Z
+√
+2
+,
+B0 = Z,
+B1 = X,
+C0 = Z,
+C1 = X
+(B2)
+as well as the GHZ-like state
+|φl⟩ =
+1
+√
+2
+(|l1l2l3⟩ + (−1)l1|l1l2l3⟩),
+(B3)
+where l ≡ l1l2l3 with l1, l2, l3 = 0, 1 and li is the negation of the bit li, i.e., li = 1 − li.
+
+7
+Proof. We follow the results of Ref. [36]. First, to each of the Bell expressions Il we associate a Bell operator of the
+following form
+ˆIl1l2l3 = 2(−1)l1(A0 + A1) ⊗ B1 ⊗ C1 + (−1)l2+l1(A0 − A1) ⊗ B0 + (−1)l3+l1(A0 − A1) ⊗ C0,
+(B4)
+where Ax, By and Cz are arbitrary ±1-valued quantum observables of arbitrary dimensions. Second, for each ˆIl we
+construct the following sum-of-squares (SOS) decomposition,
+4
+√
+2 1 − ˆIl1l2l3 =
+√
+2
+�
+1 − P1,l1
+�2 + 1
+√
+2
+��
+1 − P2,l1,l2
+�2 +
+�
+1 − P3,l1,l3
+�2�
+,
+(B5)
+where
+P1,l1 = (−1)l1 A0 + A1
+√
+2
+⊗ B1 ⊗ C1,
+(B6a)
+P2,l1,l2 = (−1)l1+l2 A0 − A1
+√
+2
+⊗ B0,
+(B6b)
+P3,l1,l3 = (−1)l1+l2 A0 − A1
+√
+2
+⊗ C0.
+(B6c)
+It directly follows from Eq. (B5) that 4
+√
+2 1 − ˆIl ≥ 0 and thus 4
+√
+2 is an upper bound on the maximal quantum value
+of Il, which means that ⟨ψ| ˆIl|ψ⟩ ≤ 4
+√
+2 for arbitrary state |ψ⟩. To finally show that the latter inequality is tight and
+that 4
+√
+2 is in fact Tsirelson’s bound of the inequalities in Eq. (B1) it suffices to observe that Il achieves the value
+4
+√
+2 for the GHZ-like state |φl⟩ and the observables given in Eq. (B2).
+Crucially, the SOS decomposition in Eq. (B5) implies that any state |ψ⟩ and any observables Ax, By and Cz that
+achieve the quantum bound βQ = 4
+√
+2 of Il must satisfy the following relations
+P1,l1|ψ⟩ = |ψ⟩
+and
+Pi,l1,li|ψ⟩ = |ψ⟩
+(i = 2, 3).
+(B7)
+which, by virtue of the relations in Eqs. (B6a), (B6b) and (B6c), can be rewritten as
+(−1)l1 A0 + A1
+√
+2
+⊗ B1 ⊗ C1|ψ⟩ = |ψ⟩,
+(B8a)
+(−1)l1+l2 A0 − A1
+√
+2
+⊗ B0|ψ⟩ = |ψ⟩,
+(B8b)
+(−1)l1+l3 A0 − A1
+√
+2
+⊗ C0|ψ⟩ = |ψ⟩.
+(B8c)
+The above relations are used in the proof of Theorem 1 stated in the main text as well as in the preceding subsection.
+2.
+Proof of self-testing
+Theorem 1. Consider the network scenario outlined in the main text and assume that the observed correlations ⃗p achieve the
+maximal quantum value of Il in Eq. (B1) for each outcome l of Eve’s first measurement E0 and that each outcome l occurs with
+probability P(l|e = 0) = 1/8. Then,
+(i) All six Hilbert spaces decompose as Hs = Hs′ ⊗ Hs′′, and Hs = Hs′ ⊗ Hs′′ with s = A, B, C, where Hs′ and Hs′ are
+one-qubit Hilbert spaces.
+(ii) There exist local unitary transformations Us : Hs → Hs and Us : Hs → Hs such that
+Us ⊗ Us|ψss⟩ = |φ+
+s′s′⟩ ⊗ |ξs′′s′′⟩
+(B9)
+for each s = A, B, C.
+
+8
+(iii) Then, Eve’s first measurement E0 = {Rl|0} satisfies
+(UA ⊗ UB ⊗ UC) Rl|0 (UA ⊗ UB ⊗ UC)† = |φl⟩⟨φl|E′ ⊗ 1E′′,
+(B10)
+where E = ABC, |φl⟩ are the GHZ-like states given in Eq. (B3) and the measurements of all other parties are given by
+UA A0 U†
+A =
+� X + Z
+√
+2
+�
+A′ ⊗ 1A′′,
+UA A1 U†
+A =
+� X − Z
+√
+2
+�
+A′ ⊗ 1A′′,
+UB B0 U†
+B = ZB′ ⊗ 1B′′,
+UB B1 U†
+B = XB′ ⊗ 1B′′,
+UC C0 U†
+C = ZC′ ⊗ 1C′′,
+UC C1 U†
+C = XC′ ⊗ 1C′′.
+(B11)
+Proof. Before we proceed with the proof we first notice that the post-measurement state that A, B and C share after
+Eve performs her first measurement E0 and obtains the outcome l is given by
+ρl
+ABC =
+1
+P(l)TrABC
+��
+1ABC ⊗ Rl|0
+�
+�
+s=A,B,C
+|ψss⟩⟨ψss|
+�
+.
+(B12)
+We divide the proof into a few steps and the first one is concerned with determining the form of the states ρl
+ABC
+for any l from the observed maximal violations of the inequalities in Eq. (B1). Building on this result we then find
+the form of the states generated by the sources |ψss⟩ for any s = A, B, C. Finally, using both of these results we obtain
+the form of the entangled measurement {Rl|0}. For simplicity, in the rest of the proof, we represent Rl|0 as Rl.
+(a) Post-measurement states ρl
+ABC. To determine the form of the post-measurement states ρl
+ABC we exploit the
+relations given in Eq. (B8). First, let us consider a purification of ρl
+ABC, denoted |ψl⟩ABCG, which is a pure state
+satisfying
+ρl
+ABC = TrG (|ψl⟩⟨ψl|ABCG) .
+(B13)
+For simplicity, in what follows we drop the subscript from the above state. From the assumption that ρl
+ABC maximally
+violates the Bell inequality in Eq. (B1) it follows that the relations in Eq. (B8) are satisfied by the purification |ψl⟩,
+which after taking into account that B2
+y = C2
+z = 1 can be stated as
+(−1)l1 A0 + A1
+√
+2
+|ψl⟩ = B1 ⊗ C1|ψl⟩
+(B14a)
+(−1)l1+l2 A0 − A1
+√
+2
+|ψl⟩ = B0|ψl⟩,
+(B14b)
+(−1)l1+l3 A0 − A1
+√
+2
+|ψl⟩ = C0|ψl⟩.
+(B14c)
+In the above equations as well as in the following considerations we omit the identities acting on the remaining
+subsystems, including the G one.
+Now, by applying B1 ⊗ C1 to both sides of Eq. (B14a), we obtain
+(A0 + A1)2|ψl⟩ = 2|ψl⟩.
+(B15)
+As the measurements can be characterised only on the support of the local reduced density matrices of every party
+we can always assume them to be full rank. Consequently, from the above Eq. (B15), we find that
+(A0 + A1)2 = 2 1A.
+(B16)
+Expanding the above equation and using the fact that A2
+0 = A2
+1 = 1A, we get that {A0, A1} = 0. As proven in
+Ref. [18] or Ref. [36], for a pair of unitary observables with eigenvalues ±1 that anti-commute there exist a unitary
+operation UA : HA → HA such that
+UA A0 U†
+A = X + Z
+√
+2
+⊗ 1A′′,
+UA A1 U†
+A = X − Z
+√
+2
+⊗ 1A′′.
+(B17)
+
+9
+Let us now move on to characterizing the other parties’ observables and use Eq. (B17) to rewrite the relations in
+Eqs. (B14) as
+(−1)l1 XA′|ψ′
+l⟩ = B1 ⊗ C1|ψ′
+l⟩,
+(B18a)
+(−1)l1+l2 ZA′|ψ′
+l⟩ = B0|ψ′
+l⟩,
+(B18b)
+(−1)l1+l3ZA′|ψ′
+l⟩ = C0|ψ′
+l⟩.
+(B18c)
+where |ψ′
+l⟩ = UA|ψl⟩ and we omitted the identity acting on the A′′ subsystem. Let us then consider the relation in
+Eq. (B18a) and multiply it with ZA. Then using the relation in Eq. (B18b) on the right-hand side of the obtained
+expression, we get
+(−1)l2(ZX)A′|ψ′
+l⟩ = B1B0 ⊗ C1|ψ′
+l⟩
+(B19)
+Then, after multiplying Eq. (B18b) with XA′ and using Eq. (B18a) on the right-hand side of the obtained expression,
+we get
+(−1)l2(XZ)A′|ψ′
+l⟩ = B0B1 ⊗ C1|ψ′
+l⟩.
+(B20)
+Adding Eqs. (B19) and (B20), and using the fact that ZX + XZ = 0, we finally arrive at
+{B1, B0} ⊗ C1|ψ′
+l⟩ = 0.
+(B21)
+Exploiting the facts that C1 is invertible and that the local density matrices are full-rank, we get that {B1, B0} = 0.
+Employing again the result of [18], we conclude that HB = (C2)B′ ⊗ HB′′ and that there exists a unitary operation
+UB : HB → HB for which
+UB B0 U†
+B = ZB′ ⊗ 1B′′,
+UB B1 U†
+B = XB′ ⊗ 1B′′.
+(B22)
+Proceeding exactly in the same manner, we can conclude that C0 and C1 that appear in Eqs. (B18a) and (B18c) also
+anticommute. Thus, as before, HC = (C2)C′ ⊗ HC′′ and there exists a unitary UC : HC → HC such that
+UC C0 U†
+C = ZC′ ⊗ 1C′′,
+UC C1 U†
+C = XC′ ⊗ 1C′′.
+(B23)
+Let us now characterise the state ρl
+ABC. For this purpose, we plug the forms of the observables from Eqs. (B17),
+(B22) and (B23) into Eqs. (B7) to obtain
+(−1)l1XA′ ⊗ XB′ ⊗ XC′| ˜ψl⟩ = | ˜ψl⟩,
+(B24a)
+(−1)l1+l2ZA′ ⊗ ZB′| ˜ψl⟩ = | ˜ψl⟩,
+(B24b)
+(−1)l1+l3ZA′ ⊗ ZC′| ˜ψl⟩ = | ˜ψl⟩,
+(B24c)
+where | ˜ψl⟩ = UA ⊗ UB ⊗ UC|ψl⟩. As already concluded above each local Hilbert space Hs (s = A, B, C) decomposes
+as Hs = (C2)s′ ⊗ Hs′′. Thus, | ˜ψl⟩ can be decomposed as
+| ˜ψl⟩ =
+∑
+i1,i2,i3=0,1
+|i1i2i3⟩A′B′C′|φl
+i1i2i3⟩A′′B′′C′′G,
+(B25)
+where the normalisation factors are included in |φl
+i1i2i3⟩. Putting the above form of | ˜ψl⟩ in Eqs. (B24b) and (B24c), we
+obtain
+(−1)l1+l2
+∑
+i1,i2,i3=0,1
+(−1)i1+i2|i1i2i3⟩|φl
+i1i2i3⟩ =
+∑
+i1,i2,i3=0,1
+|i1i2i3⟩|φl
+i1i2i3⟩,
+(B26)
+and
+(−1)l1+l3
+∑
+i1,i2,i3=0,1
+(−1)i1+i3|i1i2i3⟩|φl
+i1i2i3⟩ =
+∑
+i1,i2,i3=0,1
+|i1i2i3⟩|φl
+i1i2i3⟩.
+(B27)
+For brevity, we dropped subscripts denoting the subsystems. Projecting both the above formulas on ⟨i1i2i3|, we
+obtain the following relations
+(−1)l1+l2(−1)i1+i2|φl
+i1i2i3⟩ = |φl
+i1i2i3⟩
+and
+(−1)l1+l3(−1)i1+i3|φl
+i1i2i3⟩ = |φl
+i1i2i3⟩,
+(B28)
+
+10
+which allow us to conclude that |φl
+i1i2i3⟩ = 0 whenever (l1 + l2 + i1 + i2) mod 2 = 1 and (l1 + l3 + i1 + i3) mod 2 = 1.
+Thus, the state in Eq. (B25) that satisfies the conditions in Eq. (B28) must be of the form
+| ˜ψl⟩ = |l1l2l3⟩|φl1l2l3⟩ + |l1l2l3⟩|φl1l2l3⟩
+(B29)
+where li = 0, 1 for any i = 1, 2, 3 and li = 1 − li. Now, putting this state in Eq. (B24a), we obtain the following
+relation
+(−1)l1|l1l2l3⟩|φl1l2l3⟩ + (−1)l1|l1l2l3⟩|φl1l2l3⟩ = |l1l2l3⟩|φl1l2l3⟩ + |l1l2l3⟩|φl1l2l3⟩,
+(B30)
+which implies that
+(−1)l1|φl1l2l3⟩ = |φl1l2l3⟩.
+(B31)
+Thus, from Eq. (B29) we conclude that the state | ˜ψl⟩, by putting the appropriate normalisation constant, is given by
+| ˜ψl⟩ =
+1
+√
+2
+�
+|l1l2l3⟩ + (−1)l1|l1l2l3⟩
+�
+A′B′C′ ⊗ |φl1l2l3⟩A′′B′′C′′G.
+(B32)
+Tracing out the ancillary subsystem G, we finally obtain
+UA ⊗ UB ⊗ UC ρl
+ABC (UA ⊗ UB ⊗ UC)† = |φl⟩⟨φl|A′B′C′ ⊗ ˜ρl
+A′′B′′C′′,
+(B33)
+where ˜ρl
+A′′B′′C′′ denotes the auxiliary state acting on HA′′ ⊗ HB′′ ⊗ HC′′. Putting the above relation back into Eq. (B12)
+and taking the unitaries to the right-hand side, we see that
+|φl⟩⟨φl|A′B′C′ ⊗ ˜ρl
+A′′B′′C′′ = 8 TrABC
+�
+(1ABC ⊗ Rl)
+�
+s=A,B,C
+| ˜ψss⟩⟨ ˜ψss|
+�
+,
+(B34)
+where we denoted | ˜ψss⟩ = Us|ψss⟩ and used the fact that P(l) = 1/8.
+(b) States |ψss⟩ generated by the sources. Let us now exploit the relations in Eq. (B34) to determine the form of the
+the states | ˜ψss⟩ (s = A, B, C). To this end, we first express them using their Schmidt decompositions,
+| ˜ψss⟩ =
+ds−1
+∑
+i=0
+αs
+i |ei⟩s| fi⟩s,
+(B35)
+where ds denotes the dimension of the Hilbert space Hs and {|ei⟩s} and {| fi⟩s} are some local bases corresponding
+to the subsystems s and s, respectively; recall that, as proven above, ds is even for any s as the Hilbert spaces of the
+external parties decompose as Hs = C2 ⊗ Hs′′. Moreover, the Schmidt coefficients satisfy αs
+i > 0 and ∑i(αs
+i )2 = 1.
+Let us now consider a unitary operation Us : Hs → Hs such that Us| fi⟩s = |e∗
+i ⟩s for any i, where the asterisk stands
+for the complex conjugation. By using this unitary we can bring Eq. (B35) to the following form
+|ψss⟩ = 1s ⊗ Us| ˜ψss⟩ =
+ds−1
+∑
+i=0
+αs
+i |ei⟩s|e∗
+i ⟩s.
+(B36)
+Introducing then the following matrix
+Ps =
+ds−1
+∑
+i=0
+αs
+i |e∗
+i ⟩⟨e∗
+i |s,
+(B37)
+we can rewrite the state in Eq. (B36) as
+|ψss⟩ =
+�
+ds (1s ⊗ Ps) |φ+
+ds⟩ss
+(s = A, B, C),
+(B38)
+where |φ+
+ds⟩ss denotes the maximally entangled state of local dimension ds, that is,
+|φ+
+ds⟩ss =
+1
+√
+ds
+ds−1
+∑
+i=0
+|ei⟩s|e∗
+i ⟩s =
+1
+√
+ds
+ds−1
+∑
+i=0
+|i⟩s|i⟩s,
+(B39)
+where |e∗
+i ⟩ is the complex conjugate of |ei⟩ and the last equality follows from the fact that the maximally entangled
+state is invariant under the action of U ⊗ U∗ for any unitary operation U.
+
+11
+Putting now Eq. (B39) back into Eq. (B34), we conclude that
+|φl⟩⟨φl|A′B′C′ ⊗ ˜ρl
+A′′B′′C′′ = 8TrE
+�
+�
+1ABC ⊗ Rl
+� �
+s
+|φ+
+ds⟩⟨φ+
+ds|ss
+�
+,
+(B40)
+where we denoted
+Rl = dAdBdC
+�
+�
+s
+Ps Us
+�
+Rl
+�
+�
+s
+U†
+s Ps
+�
+.
+(B41)
+Now, notice that the tensor product of the three maximally entangled states appearing in (B41) can also be under-
+stood as a single maximally entangled state across the bipartition ABC|ABC whose local dimension is dAdBdC, that
+is,
+|φ+
+ds⟩AA ⊗ |φ+
+ds⟩BB ⊗ |φ+
+ds⟩CC = |φ+
+dAdBdC⟩ABC|ABC,
+(B42)
+|φ+
+dAdBdC⟩ABC|ABC =
+1
+√dAdBdC
+dAdBdC−1
+∑
+i=0
+|i⟩ABC|i⟩ABC.
+(B43)
+Substituting Eq. (B42) in Eq. (B40) and using the well known fact that 1A ⊗ QB|φ+
+D⟩A|B = QT
+A ⊗ 1B|φ+
+D⟩A|B for any
+matrix Q, we can rewrite (B40) as
+|φl⟩⟨φl|A′B′C′ ⊗ ˜ρl
+A′′B′′C′′ = 8TrABC
+��
+RT
+l
+�
+ABC ⊗ 1ABC|φ+
+dAdBdC⟩⟨φ+
+dAdBdC|ABC|ABC
+�
+,
+(B44)
+which implies that
+|φl⟩⟨φl|A′B′C′ ⊗ ˜ρl
+A′′B′′C′′ =
+8
+dAdBdC
+RT
+l .
+(B45)
+Now, applying the transposition map to both sides and then using Eq. (B41), we arrive at
+|φl⟩⟨φl|A′B′C′ ⊗
+�
+˜ρl
+A′′B′′C′′
+�T
+= 8
+�
+�
+s
+PsUs
+�
+Rl
+�
+�
+s
+U†
+s Ps
+�
+,
+(B46)
+which after summing over l and using the fact that ∑7
+l=0 Rl = 1 gives
+7
+∑
+l=0
+|φl⟩⟨φl|A′B′C′ ⊗
+�
+˜ρl
+A′′B′′C′′
+�T
+= 8 P2
+A ⊗ P2
+B ⊗ P2
+C.
+(B47)
+We can now employ the fact that Tr(P2
+B) = Tr(P2
+C) = 1 to take a partial trace of the above expression over the
+subsystems BC and obtain
+1A′
+2 ⊗ σA′′ = P2
+A,
+(B48)
+which, by virtue of the fact that, by definition, PA ≥ 0, directly implies that
+PA = 1A′ ⊗
+�
+σA′′
+2 ,
+(B49)
+where
+σA′′ = 1
+8
+7
+∑
+l=0
+TrB′′C′′
+��
+˜ρl
+A′′B′′C′′
+�T�
+.
+(B50)
+In an analogous manner we can determine the other matrices Ps with s = B, C. Precisely, by taking partial traces
+of Eq. (B47) over all subsystems except the sth one, one obtains
+Ps = 1s′ ⊗
+�
+σs′′
+2
+(s = A, B, C)
+(B51)
+
+12
+with
+σs′′ = 1
+8
+7
+∑
+l=0
+TrA′′B′′C′′\{s′′}
+��
+˜ρl
+A′′B′′C′′
+�T�
+,
+(B52)
+where TrA′′B′′C′′\{s′′} represents the partial trace over the systems A′′B′′C′′ except the one labelled by s′′; For instance
+TrA′′B′′C′′\{A′′} ≡ TrB′′C′′.
+Now, we can substitute Ps given by Eq. (B51) into Eq. (B38) and also use the fact that ds = 2d′′
+s for some positive
+integer d′′
+s to obtain
+|ψss⟩ =
+�
+d′′s 1s′s′ ⊗ 1s′′ ⊗ √σs′′ |φ+
+ds⟩ss
+(B53)
+for any s. Using again the fact that |φ+
+ds⟩ss = |φ+⟩s′s′|φ+
+d′′s ⟩s′′s′′, we finally conclude that the states generated by the
+sources admit the following form
+Us ⊗ Us|ψss⟩ = |ψss⟩ = |φ+⟩s′s′|ξs′′s′′⟩
+(s = A, B, C),
+(B54)
+where the auxiliary state |ξs′′s′′⟩ is given by
+|ξs′′s′′⟩ =
+�
+d′′s 1s′′ ⊗ √σs′′ |φ+
+d′′s ⟩s′′s′′.
+(B55)
+(c) Entangled measurement E0. Let us now characterise the measurement E0 = {Rl}. For this purpose, we notice
+that the states σs′′ are invertible because we assumed all the reduced density matrices of |ψl⟩ to be full rank. This
+implies that the matrices Ps [cf. Eq. (B51)] are invertible too and therefore we can act with P−1
+s
+on both sides of Eq.
+(B46) to bring to the following form
+�
+�
+s
+Us
+�
+Rl
+�
+�
+s
+U†
+s
+�
+= |φl⟩⟨φl|A′B′C′ ⊗
+�
+�
+s
+σ−1/2
+s′′
+�
+˜ρl,T
+A′′B′′C′′
+�
+�
+s
+σ−1/2
+s′′
+�
+.
+(B56)
+We thus conclude that
+�
+�
+s
+Us
+�
+Rl
+�
+�
+s
+U†
+s
+�
+= |φl⟩⟨φl|A′B′C′ ⊗
+� ˜Rl
+�
+A′′B′′C′′ ,
+(B57)
+for all l = 0, . . . , 7, where ˜Rl is defined as
+˜Rl =
+�
+�
+s
+σ−1/2
+s′′
+� �
+˜ρl
+A′′B′′C′′
+�T
+�
+�
+s
+σ−1/2
+s′′
+�
+.
+(B58)
+Notice that ˜Rl ≥ 0. Moreover, the fact that Rl ≤ 1 implies via Eq. (B57) that ˜Rl ≤ 1. Taking then the sum over l on
+both sides of Eq. (B57) and employing the fact that ∑l Rl = 1 allows us to conclude that
+1ABC =
+7
+∑
+l=0
+|φl⟩⟨φl|A′B′C′ ⊗
+� ˜Rl
+�
+A′′B′′C′′ .
+(B59)
+Now, we can take advantage of the fact that the GHZ-like states are mutually orthogonal and therefore by projecting
+the A′B′C′ subsystems onto |φk⟩ we deduce that for every k,
+˜Rk = 1A′′B′′C′′.
+(B60)
+After plugging the above relation into Eq. (B57) we finally conclude that there exist local unitary transformations
+such that the measurement operators Rl admit the following form
+(UA ⊗ UB ⊗ UB) Rl (UA ⊗ UB ⊗ UB)† = |φl⟩⟨φl|A′B′C′ ⊗ 1A′′B′′C′′
+(l = 0, . . . , 7),
+(B61)
+which is exactly what we promised in Eq. (B10). This completes the proof.
+An interesting consequence of the above theorem is that the states ˜ρl
+A′′B′′C′′ in Eq. (B33) are separable for all l.
+
+13
+Corollary 2. Assume that the sources P1, P2, P3 generate states that are certified as in Eq. (B9) and the measurements of all the
+parties are certified as in Eqs. (B10) and (B11). Then, the post-measurement state ρl
+ABC when Eve gets an outcome l is given by
+�
+�
+s
+Us
+�
+ρl
+ABC
+�
+�
+s
+U†
+s
+�
+= ρl
+ABC = |φl⟩⟨φl|A′B′C′ ⊗ ˜ρA′′ ⊗ ˜ρB′′ ⊗ ˜ρC′′
+∀l.
+(B62)
+Proof. Let us first observe that by combining Eqs. (B58) and (B60) we can write
+�
+�
+s
+σ−1/2
+s′′
+� �
+˜ρl
+A′′B′′C′′
+�T
+�
+�
+s
+σ−1/2
+s′′
+�
+= ˜Rl = 1A′′B′′C′′,
+(B63)
+which after rearranging the terms and taking transposition gives us
+˜ρl
+A′′B′′C′′ =
+�
+s
+σT
+s′′
+(l = 0, . . . , 7).
+(B64)
+Recall that σs′′ from Eq. (B52) are valid quantum states as they positive semi-definite and satisfy Tr(σs′′) = 1 for any
+s = A, B, C. This completes the proof.
+3.
+Self-testing of the extra measurements A2, B2 and C2
+Now using the above theorem, let us self-test the additional measurements A2, B2 and C2. For this purpose, we
+consider a functional inspired by the well-known Mermin Bell inequality [37]:
+I2 =
+� 1
+√
+2
+(A0 + A1)B1C1 − 1
+√
+2
+(A0 + A1)B2C2 − A2B1C2 − A2B2C1
+�
+.
+(B65)
+Theorem 3. Assume that the sources P1, P2, P3 generate states that are certified as in Eq. (B9) and the measurements of all the
+external parties are certified as in Eqs. (B10) and (B11). If I2 achieves the value four for the state shared by A, B and C that
+corresponds to the outcome l = 000 of Eve’s first measurement E0, then the observables A2, B2 and C2 can have two possible
+forms
+UA A2 U†
+A = YA′ ⊗ 1A′′,
+UB B2 U†
+B = YB′ ⊗ 1B′′,
+UC C2 U†
+C = YC′ ⊗ 1C′′,
+(B66)
+or
+UA A2 U†
+A = −YA′ ⊗ 1A′′,
+UB B2 U†
+B = −YB′ ⊗ 1B′′,
+UC C2 U†
+C = −YC′ ⊗ 1C′′.
+(B67)
+Proof. Let us first consider the functional I2 for the particular observables A0, A1, B1 and C1 that are given by Eq.
+(B11) as well as a particular state corresponding to the l = 000 outcome of Eve’s first measurement, i.e., ρ000
+ABC,
+I′
+2 = ⟨XA′ ⊗ XB′ ⊗ XC′ ⊗ 1A′′B′′C′′⟩ − ⟨XA′ ⊗ B′
+2 ⊗ C′
+2 ⊗ 1A′′⟩ − ⟨A′
+2 ⊗ XB′ ⊗ C′
+2 ⊗ 1B′′⟩
+−⟨A′
+2 ⊗ B′
+2 ⊗ XC′ ⊗ 1C′′⟩ρ000
+ABC,
+(B68)
+where A′
+2 = UA A2 U†
+A etc., and ρ000
+ABC is a ’rotated version’ of ρ000
+ABC and is given in Eq. (B62); to simplify the notation
+from now on we denote ϱ0 ≡ ρ000
+ABC.
+Now, taking into account the fact that the observables A′
+2, B′
+2 and C′
+2 are unitary it is not difficult to realise that I′
+2
+attains the value four if and only if the first expectation value in Eq. (B68) equals 1 whereas the remaining three are
+−1. Again, given that A′
+2, B′
+2 and C′
+2 are unitary this implies that the following conditions are satisfied,
+(XA′ ⊗ XB′ ⊗ XC′ ⊗ 1A′′B′′C′′) ϱ0 = ϱ0,
+(B69)
+(XA′ ⊗ B′
+2 ⊗ C′
+2 ⊗ 1A′′) ϱ0 = −ϱ0,
+(B70)
+(A′
+2 ⊗ XB′ ⊗ C′
+2 ⊗ 1B′′) ϱ0 = − ϱ0,
+(B71)
+(A′
+2 ⊗ B′
+2 ⊗ XC′ ⊗ 1C′′) ϱ0 = − ϱ0.
+(B72)
+Let us now consider the relation in Eq. (B70). After acting on it with XA′ ⊗ XB′ ⊗ XC′ and then using Eq. (B69), we
+obtain
+�
+1A ⊗ (XB′ ⊗ 1B′′) B′
+2 ⊗ (XC′ ⊗ 1C′′) C′
+2
+�
+ϱ0 = − ϱ0.
+(B73)
+
+14
+Next, after multiplication by C′
+2 (XC′ ⊗ 1C′′), the above relation can be brought to
+(XB′ ⊗ 1B′′) B′
+2 ϱ0 = −C′
+2 (XC′ ⊗ 1C′′) ϱ0.
+(B74)
+Let us then consider the relation in Eq. (B69). After multiplying it with XA′ ⊗ B′
+2 ⊗ C′
+2, using Eq. (B70), and then
+multiplying the resulting relation with (XC′ ⊗ 1C′′) C′
+2, we arrive at
+B′
+2 (XB′ ⊗ 1B′′) ϱ0 = − (XC′ ⊗ 1C′′) C′
+2 ϱ0.
+(B75)
+By adding Eqs. (B74) and (B75), one obtains
+{B′
+2, (XB′ ⊗ 1B′′)} ϱ0 = −{C′
+2, (XC′ ⊗ 1C′′)} ϱ0.
+(B76)
+Now, we consider Eq. (B71). By acting on it with A′
+2 ⊗ B′
+2 ⊗ XC′, using Eq. (B72), and then multiplying the resulting
+formula with C′
+2 (XC′ ⊗ 1C′′), we obtain
+B′
+2 (XB′ ⊗ 1B′′) ϱ0 = C′
+2 (XC′ ⊗ 1C′′) ϱ0.
+(B77)
+Similarly, we then consider Eq. (B72), multiply it with A′
+2 ⊗ (XB′ ⊗ 1B′′) ⊗ C′
+2, use Eq. (B71), and finally multiply
+resulting formula with (XC′ ⊗ 1C′′)C′
+2 to obtain
+(XB′ ⊗ 1B′′) B′
+2 ϱ0 = (XC′ ⊗ 1C′′) C′
+2 ϱ0.
+(B78)
+Adding Eqs. (B77) and (B78), we get
+{B′
+2, (XB′ ⊗ 1B′′)} ϱ0 = {C′
+2, (XC′ ⊗ 1C′′)} ϱ0
+(B79)
+We have thus obtained two similar relations, (B76) and (B79), but with the opposite signs. By adding them we thus
+conclude that
+{B′
+2, (XB′ ⊗ 1B′′)} ϱ0 = 0,
+(B80)
+which, by taking into account that all reduced density matrices of ϱ0 are full rank eventually implies that
+{B′
+2, (XB′ ⊗ 1B′′)} = 0.
+(B81)
+In the exactly same manner, one can use Eqs. (B69)–(B72) to obtain similar relations for the observables A′
+2, and
+C′
+2:
+{A′
+2, (XA′ ⊗ 1A′′)} = 0,
+{C′
+2, (XC′ ⊗ 1C′′)} = 0.
+(B82)
+Let us exploit the above anticommutation relations to determine the forms of A′
+2, B′
+2 and C′
+2. We begin with A′
+2.
+As characterised before, the Hilbert space of Alice is given by HA = C2 ⊗ HA′′. Thus, any observable acting on such
+a Hilbert space can be decomposed as
+A′
+2 = 12 ⊗ Q0 + Z ⊗ Q1 + X ⊗ Q2 + Y ⊗ Q3,
+(B83)
+where for simplicity we have omitted the subscripts A′ and A′′. Putting it back into Eq. (B82), we get that
+X ⊗ Q0 + 1 ⊗ Q2 = 0,
+(B84)
+which implies that Q0 = Q2 = 0, and consequently A2 expresses as
+A′
+2 = Z ⊗ Q1 + Y ⊗ Q3,
+(B85)
+where, due to the fact that A2
+2 = 1, the matrices Q1 and Q3 obey the following relations
+Q2
+1 + Q2
+3 = 1,
+[Q1, Q3] = 0.
+(B86)
+One can similarly find that
+B′
+2 = Z ⊗ R1 + Y ⊗ R3,
+C′
+2 = Z ⊗ S1 + Y ⊗ S3
+(B87)
+for some matrices R1, R3, S1 and S3 such that R2
+1 + R2
+3 = 1 and [R1, R3] = 0, and S2
+1 + S2
+3 = 1 and [S1, S3] = 0.
+Now, after putting these forms of the observables into Eq. (B70), we get
+[XA′ ⊗
+�
+ZB′ ⊗ R1,B′′ + YB′ ⊗ R3,B′′
+� ⊗
+�
+ZC′ ⊗ S1,C′′ + YC′ ⊗ S3,C′′
+� ⊗ 1A′′] ϱ0 = − ϱ0,
+(B88)
+
+15
+which on expansion and substituting the state ˜ρ000
+ABC from Eq. (B62) gives
+�
+XA′ZB′ZC′ ⊗ 1A′′R1,B′′S1,C′′ + XA′ZB′YC′ ⊗ 1A′′R1,B′′S3,C′′ + XA′YB′ZC′ ⊗ 1A′′R3,B′′S1,C′′
++XA′YB′YC′ ⊗ 1A′′R3,B′′S3,C′′
+� |φ0⟩⟨φ0|A′B′C′ ⊗ ˜ρA′′ ˜ρB′′ ˜ρC′′ = −|φ0⟩⟨φ0|A′B′C′ ⊗ ˜ρA′′ ˜ρB′′ ˜ρC′′,
+(B89)
+where for simplicity, we are representing the index 000 as 0 and the symbol of the tensor products are removed.
+Notice that the following relations hold true
+XA′ZB′ZC′|φ000⟩A′B′C′ = |φ011⟩A′B′C′,
+(B90a)
+XA′ZB′YC′|φ000⟩A′B′C′ = i|φ010⟩A′B′C′,
+(B90b)
+XA′YB′ZC′|φ000⟩A′B′C′ = i|φ001⟩A′B′C′,
+(B90c)
+XA′YB′YC′|φ000⟩A′B′C′ = −|φ000⟩A′B′C′
+(B90d)
+where |φl⟩ for any l can be found in Eqs. (B3). Using these relations and the condition in Eq. (B89), we get four
+different relations
+R1,B′′ ˜ρB′′ ⊗ S1,C′′ ˜ρC′′ = 0,
+R1,B′′ ˜ρB′′ ⊗ S3,C′′ ˜ρC′′ = 0,
+R3,B′′ ˜ρB′′ ⊗ S1,C′′ ˜ρC′′ = 0,
+(B91)
+and
+˜ρA′′ ⊗ R3,B′′ ˜ρB′′ ⊗ S3,C′′ ˜ρC′′ = ˜ρA′′ ⊗ ˜ρB′′ ⊗ ˜ρC′′.
+(B92)
+All three reduced density matrices ˜ρs′′ (s = A, B, C) are full rank, it follows from Eq. (B92) that R3 and S3 are non-
+zero. Consequently, the last two relations in Eq. (B91) imply that S1 = 0 and R1 = 0. Analogously, after plugging A2
+as given in Eq. (B85) and C2 = Y ⊗ S3 into Eq. (B71) we infer that Q1 = 0 and
+Q3,A′′ ˜ρA′′ ⊗ S3,C′′ ˜ρC′′ = ˜ρA′′ ⊗ ˜ρC′′.
+(B93)
+Thus, we obtain that
+A′
+2 = Y ⊗ Q,
+B′
+2 = Y ⊗ R,
+C′
+2 = Y ⊗ S,
+(B94)
+where Q, R and S are some hermitian matrices such that Q2 = 1, R2 = 1 and S2 = 1, which makes them also
+unitary; for simplicity we dropped the subscripts from them.
+Let us now exploit the fact that Q, R and S are hermitian and square to the identity to decompose them as
+Q = Q+ − Q−,
+R = R+ − R−,
+S = S+ − S−,
+where Q±, R± and S± are projectors onto the eigenspaces of A′
+2, B′
+2 and C′
+2 corresponding to eigenvalues ±1. Now
+using the fact that Q2 = Q+ + Q− = 1 and S2 = S+ + S− = 1 and tracing the A′′ out, we obtain from Eqs. (B92) that
+2(R+ ˜ρB′′) ⊗ (S+ ˜ρC′′) = (R+ ˜ρB′′) ⊗ ˜ρC′′ + ˜ρB′′ ⊗ (S+ ˜ρC′′).
+(B95)
+One concludes from the above relation that (R+ ˜ρB′′) ⊗ (S− ˜ρC′′) = 0 and (R− ˜ρB′′) ⊗ (S+ ˜ρC′′) = 0, which by taking
+into account the fact that both ˜ρB′′ and ˜ρC′′ are full rank, implies that either R+ = 0 and S+ = 0 or R− = 0 and
+S− = 0. In the same spirit, one can exploit the second relation in Eq. (B93) to conclude that either Q+ = 0 and
+S+ = 0 or Q− = 0 and S− = 0. Taking into account all the possibilities listed above, one deduces that either
+Q− = R− = S− = 0 (in which case Q+ = 1A′′, R+ = 1B′′ and S+ = 1C′′) or Q+ = R+ = S+ = 0 (in which case
+Q− = 1A′′, R− = 1B′′ and S− = 1C′′), which directly leads us to two possible forms that the observables A′
+2, B′
+2 and
+C′
+2 can take:
+A2 = YA′ ⊗ 1A′′,
+B2 = YB′ ⊗ 1B′′,
+C2 = YC′ ⊗ 1C′′
+(B96)
+or
+A2 = −YA′ ⊗ 1A′′,
+B2 = −YB′ ⊗ 1B′′,
+C2 = −YC′ ⊗ 1C′′,
+(B97)
+from which one recovers Eqs. (B66) and (B67). This completes the proof.
+
+16
+Appendix C: Self-testing the three-qubit NLWE basis
+In this section, we show that using the certified states in Eq. (B9) generated by the sources Pi (i = 1, 2, 3) and
+measurements in Eqs. (B10) and (B11) along with some additional statistics, one can self-test the measurement
+corresponding to the input e = 1 with the central party to be NLWE basis given below in Eq. (C3) upto some
+additional degrees of freedom.
+Before proceeding, let us denote the eigenvectors of (X + Z)/
+√
+2 as
+|0⟩ = cos(π/8)|0⟩ + sin(π/8)|1⟩,
+|1⟩ = − sin(π/8)|0⟩ + cos(π/8)|1⟩,
+(C1)
+and the eigenvectors of (X − Z)/
+√
+2 as
+|+⟩ = sin(π/8)|0⟩ + cos(π/8)|1⟩,
+|−⟩ = cos(π/8)|0⟩ − sin(π/8)|1⟩.
+(C2)
+Recall also that the NLWE measurement MNLWE = {|δl⟩⟨δl|} is defined via the following fully product vectors
+|δ0⟩ = |0⟩|1⟩|+⟩,
+|δ1⟩ = |0⟩|1⟩|−⟩,
+|δ2⟩ = |+⟩|0⟩|1⟩,
+|δ3⟩ = |−⟩|0⟩|1⟩,
+|δ4⟩ = |1⟩|+⟩|0⟩,
+|δ5⟩ = |1⟩|−⟩|0⟩,
+|δ6⟩ = |0⟩|0⟩|0⟩,
+|δ7⟩ = |1⟩|1⟩|1⟩.
+(C3)
+Notice that the above vectors are equivalent to the standard NLWE vectors introduced in Ref. [27] up to a local
+unitary transformation applied to the first qubit.
+To certify the second Eve’s measurement E1, the correlations observed by the parties {p(a, b, c, e|x, y, z, 1)} must
+satisfy the following conditions,
+p(0, 1, 0, 0|0, 0, 1, 1) = 1
+8,
+p(0, 1, 1, 1|0, 0, 1, 1) = 1
+8,
+p(0, 0, 1, 2|1, 0, 0, 1) = 1
+8,
+p(1, 0, 1, 3|1, 0, 0, 1) = 1
+8,
+p(1, 0, 0, 4|0, 1, 0, 1) = 1
+8,
+p(1, 1, 0, 5|0, 1, 0, 1) = 1
+8,
+p(0, 0, 0, 6|0, 0, 0, 1) = 1
+8,
+p(1, 1, 1, 7|0, 0, 0, 1) = 1
+8.
+(C4)
+Notice that the above distribution can be realised if the sources Pi (i = 1, 2, 3) generate the maximally entangled
+state of two qubits |φ+⟩ = (1/
+√
+2)(|00⟩ + |11⟩) and the measurement E1 is exactly MNLWE = {|δl⟩⟨δl|}, whereas the
+external parties perform the following measurements
+A0 = X + Z
+√
+2
+,
+A1 = X − Z
+√
+2
+,
+B0 = Z,
+B1 = X,
+C0 = Z,
+C1 = X.
+(C5)
+Next, we show that the above probabilities along with certification of the states and measurements presented in
+Theorem 1 are sufficient to fully characterise the unknown measurement E1 = {Rl|1} where Rl denotes the measure-
+ment element corresponding to outcome l. The theorem stated below is labelled as Theorem 2 in the manuscript.
+Theorem 4. Assume that the correlations ⃗p generated in the network satisfy the assumptions of Theorem 1 as well as the
+conditions in Eq. (C4). Then, for any l it holds that
+URl|1U† = |δl⟩⟨δl|E′ ⊗ 1E′′,
+(C6)
+where U is the same unitary as in Theorem 1 and E = ABC.
+Proof. For simplicity, we represent Rl|1 as Rl throughout the proof. Let us first consider the first relation in Eq. (C4),
+that is,
+p(0, 1, 0, 0|0, 0, 1, 1) = 1
+8
+(C7)
+
+17
+and then expand it by using the fact that the observables Ai, Bi, Ci for i = 0, 1 are certified as in Eq. (B11). This gives
+us
+��
+�
+s=A,B,C
+U†
+s
+�
+�|0⟩⟨0|A′ ⊗ |1⟩⟨1|B′ ⊗ |+⟩⟨+|C′ ⊗ 1A′′B′′C′′
+�
+�
+�
+s=A,B,C
+Us
+�
+⊗ R0
+�
+ψABCE
+= 1
+8.
+(C8)
+Let us then use Theorem 1 to represent the global state |ψABCE⟩ as [cf. Eq. (B9)]
+|ψABCE⟩ =
+�
+s
+|ψss⟩ =
+�
+s
+U†
+s ⊗ U†
+s |φ+
+s′s′⟩ ⊗ |ξs′′s′′⟩.
+(C9)
+Notice also that by virtue of Eq. (B55) the junk states |ξs′′s′′⟩ can be represented as
+|ξs′′s′′⟩ = (1s′′ ⊗ Ps′′)|φ+
+d′′s ⟩s′′s′′,
+(C10)
+where Ps′′ =
+�
+d′′s σs′′. The joint state in Eq. (C9) can be further written as [c.f. Eq. (B42)],
+�
+�
+s=A,B,C
+Us ⊗ Us
+�
+|ψABC|E⟩ =
+�
+PA′′ ⊗ PB′′ ⊗ PC′′
+�
+|φ+
+8d′′
+Ad′′
+Bd′′
+C⟩ABC|E
+(C11)
+where the bipartition is between the subsystem ABC and E ≡ ABC and the local dimension of the state is 8d′′
+Ad′′
+Bd′′
+C.
+After plugging this state into the Eq. (C8), and then using the fact that (1 ⊗ Q)|φ+
+d ⟩ = (QT ⊗ 1)|φ+
+d ⟩ for any matrix
+Q, we get that
+��
+|0⟩⟨0| ⊗ |1⟩⟨1| ⊗ |+⟩⟨+| ⊗
+�
+PT
+A′′
+�2
+⊗
+�
+PT
+B′′
+�2
+⊗
+�
+PT
+C′′
+�2�
+RT
+0 ⊗ 1E
+�
+|φ+
+8dAdBdC ⟩ABC|E
+= 1
+8,
+(C12)
+where
+R0 =
+�
+�
+s=A,B,C
+Us
+�
+R0
+�
+�
+s=A,B,C
+U†
+s
+�
+.
+(C13)
+Expanding the above term, we arrive at
+1
+d′′
+Ad′′
+Bd′′
+C
+Tr
+��
+|0⟩⟨0| ⊗ |1⟩⟨1| ⊗ |+⟩⟨+| ⊗
+�
+PT
+A′′
+�2
+⊗
+�
+PT
+B′′
+�2
+⊗
+�
+PT
+C′′
+�2�
+RT
+0
+�
+= 1.
+(C14)
+Then using the fact that Ps′′ =
+�
+d′′s σs′′ for any s, we obtain
+Tr
+��
+|0⟩⟨0| ⊗ |1⟩⟨1| ⊗ |+⟩⟨+| ⊗ σT
+A′′ ⊗ σT
+B′′ ⊗ σT
+C′′
+�
+RT
+0
+�
+= 1.
+(C15)
+Recall that one can characterise measurements only on the support of the state or equivalently the states σs′′ are
+full-rank. Since RT
+0 acts on the Hilbert space C8 ⊗ HA′′B′′C′′, we can express it using the basis given in Eq. (C3) as
+RT
+0 =
+7
+∑
+l,l′=0
+|δl⟩⟨δl′| ⊗ ˜Rl,l′
+(C16)
+where ˜Rl,l′ act on HA′′B′′C′′. Then, using the cyclic property of trace and Lemma 1 we conclude from Eq. (C15) that
+˜R0,0 = 1 and thus we finally get that
+RT
+0 = |0⟩⟨0| ⊗ |1⟩⟨1| ⊗ |+⟩⟨+| ⊗ 1A′′B′′C′′ + L0
+= |δ0⟩⟨δ0|A′B′C′ ⊗ 1A′′B′′C′′ + L0,
+(C17)
+where L0 stands for an operator given by
+L0 =
+7
+∑
+l,l′=0
+l̸=0, l′̸=0
+|δl⟩⟨δl′| ⊗ ˜Rl,l′.
+(C18)
+
+18
+Let us now show that L0 is positive semi-definite. For this purpose, we consider the state |δ0⟩ ⊗ |ξ⟩, where |ξ⟩ is an
+arbitrary state from HA′′B′′C′′, and act on this state with RT
+0 . Taking into account Eqs. (C17) and (C18), we obtain
+RT
+0 |δ0⟩|ξ⟩ = |δ0⟩|ξ⟩ +
+7
+∑
+l=1
+|δl⟩ ˜Rl,0|ξ⟩.
+(C19)
+Now, multiplying the above formula in Eq. (C19) with its complex conjugate we obtain that
+⟨δ0|⟨ξ|
+�
+RT
+0
+�2
+|δ0⟩|ξ⟩ = 1 +
+7
+∑
+l=1
+⟨ξ| ˜R†
+l,0 ˜Rl,0|ξ⟩
+(C20)
+which after using the fact that
+�
+RT
+0
+�2
+≤ RT
+0 ≤ 1, gives us
+7
+∑
+l=1
+⟨ξ| ˜R†
+l,0 ˜Rl,0|ξ⟩ ≤ 0.
+(C21)
+As ˜R†
+l,0 ˜Rl,0 ≥ 0, it follows that ⟨ξ| ˜R†
+l,0 ˜Rl,0|ξ⟩ = 0 for any |ξ⟩ ∈ HA′′B′′C′′, and consequently ˜Rl,0 = 0 for any l =
+1, . . . , 7. Since L0 is hermitian, the above implies also that ˜R0,l = 0 for l = 1, . . . , 7, and hence
+L0 =
+7
+∑
+l,l′=1
+|δl⟩⟨δl′| ⊗ ˜Rl,l′.
+(C22)
+This means that RT
+0 can now be expressed in the block form as
+RT
+0 = |δ0⟩⟨δ0| ⊗ 1A′′B′′C′′ + L0 =
+�
+1
+0
+0 L0
+�
+,
+(C23)
+where both components act on orthogonal subspaces of the three-qubit Hilbert space. Owing to the fact that RT
+0 ≥ 0,
+we thus obtain L0 ≥ 0.
+Similar analysis using all the other probabilities in Eq. (C4) can be done for the other operators RT
+l , and thus we
+arrive at
+RT
+l = |δl⟩⟨δl| ⊗ 1A′′B′′C′′ + Ll
+(l = 0, . . . , 7),
+(C24)
+where Ll is a positive semi-definite operator that with respect to A′B′C′ subsystem is defined a subspace of C8 which
+is orthogonal to |δl⟩.
+Now, the fact that ∑l RT
+l = 1 implies that
+∑
+l
+Ll = 0.
+(C25)
+As each Ll is positive semi-definite, the only way the above condition is satisfied is that Ll = 0 for any l. Thus, we
+finally obtain from Eq. (C24) that
+Rl = |δl⟩⟨δl| ⊗ 1,
+(C26)
+which by taking into account Eq. (C13) gives the desired result form Eq. (C6), completing the proof.
+Appendix D: Self-testing the measurement constructed from the UPB
+Let us finally provide the proof of Theorem 4 stated in the main text. To this end, we recall that the measurement
+constructed from the UPB is defined as MUPB = {|τ0⟩⟨τ0|, |τ1⟩⟨τ1|, |τ2⟩⟨τ2|, |τ3⟩⟨τ3|, Γ}, where
+|τ0⟩ = |0⟩|1⟩|+⟩,
+|τ1⟩ = |+⟩|0⟩|1⟩
+|τ2⟩ = |1⟩|+⟩|0⟩,
+|τ3⟩ = |−⟩|−⟩|−⟩,
+(D1)
+
+19
+and
+Γ = 1 −
+3
+∑
+i=0
+|τi⟩⟨τi|.
+(D2)
+Notice that |τi⟩ form a four-element UPB which is obtained from the UPB introduced in [28] by applying a local
+unitary to Alice’s qubit.
+Now, to self-test the five-outcome measurement E2, the correlations observed by the parties {p(abce|xyz2)} must
+satisfy
+p(0100|0012) = 1
+8,
+p(0011|1002) = 1
+8,
+p(1002|0102) = 1
+8,
+p(1113|1112) = 1
+8,
+(D3)
+as well as the following conditions that are expressed in the correlation picture:
+⟨(1 + A0B0 + B0C0 + A0C0 + A1B1C1 − A1B2C2 − A2B1C2 − A2B2C1) R4⟩ = 1
+(D4a)
+�
+(1 + A0)
+�
+−B1C1 + B2C2 + B1(1 + C0) − 1
+2(1 − B0)C1 + (1 + B0)C0 + 1
+2 B0 + 3
+21
+�
+R4
+�
+= 3
+2,
+(D4b)
+�
+(1 + B0)
+�
+−A1C1 + A2C2 + (1 + A0)C1 − 1
+2 A1(1 − C0) + A0(1 + C0) + 1
+2C0 + 3
+21
+�
+R4
+�
+= 3
+2,
+(D4c)
+�
+(1 + C0)
+�
+−A1B1 + A2B2 + A1(1 + B0) − 1
+2(1 − A0)B1 + (1 + A0)B0 + 1
+2 A0 + 3
+21
+�
+R4
+�
+= 3
+2,
+(D4d)
+where E2 = {Rl}4
+l=0.
+Notice that the above conditions are met in a situation in which the sources Pi (i = 1, 2, 3) distribute the state |φ+
+2 ⟩
+and Eve’s measurements E2 is the ideal measurement MUPB given in Eq. (D1), whereas the external parties measure
+the following observables
+A0 = X + Z
+√
+2
+,
+A1 = X − Z
+√
+2
+,
+A2 = Y,
+B0 = Z,
+B1 = X,
+B2 = Y,
+C0 = Z,
+C1 = X
+C2 = Y.
+(D5)
+In what follows we show that Theorem 1 and Theorem 3 together with the conditions in Eqs. (D3) and (D4) enable
+self-testing MUPB is E2 = {Rl} where Rl denotes the measurement element corresponding to outcome l. The theorem
+stated below is labelled as Theorem 3 in the manuscript.
+Theorem 5. Assume that the correlations ⃗p observed in the network satisfy the assumptions of Theorems 1 and 3 as well as
+conditions in Eqs. (D3) and (D4). Then,
+(UA ⊗ UB ⊗ UC) Rl|2 (UA ⊗ UB ⊗ UC)† = |τl⟩⟨τl|E′ ⊗ 1E′′
+(l = 0, 1, 2, 3),
+(D6)
+and,
+(UA ⊗ UB ⊗ UC) R4|2 (UA ⊗ UB ⊗ UC) = ΓE′ ⊗ 1E′′,
+(D7)
+where the unitary operations Us (s = A, B, C) are the same as in Theorem 1.
+Proof. For simplicity, we represent Rl|2 as Rl throughout the proof. Let us first consider the conditions in Eq. (D3).
+As was done in the previous section in the proof of Theorem 4, we can conclude from these conditions that for an
+unknown measurement {Rl} we have that
+RT
+l = |τl⟩⟨τl| ⊗ 1A′′B′′C′′ + Ll
+l = 0, 1, 2, 3
+(D8)
+where
+Rl =
+�
+�
+s=A,B,C
+Us
+�
+Rl
+�
+�
+s=A,B,C
+U†
+s
+�
+l = 0, 1, 2, 3, 4
+(D9)
+and Ll ≥ 0. Let us now consider Eq. (D4a)
+⟨(1 + A0B0 + B0C0 + A0C0 + A1B1C1 − A1B2C2 − A2B1C2 − C1A2B2) R4⟩ = 1
+(D10)
+
+20
+and expand it by using the fact that the observables Ai, Bi, Ci for i = 0, 1, 2 are certified as in Eqs. (B11) and (B66).
+This gives us
+��
+�
+s=A,B,C
+U†
+s
+�
+S1,A′B′C′ ⊗ 1A′′B′′C′′
+�
+�
+s=A,B,C
+Us
+�
+⊗ R4
+�
+= 1
+(D11)
+where
+S1,A′B′C′ = 1A′B′C′ + (X + Z)A′
+√
+2
+ZB′ + ZB′ZC′ + (X + Z)A′
+√
+2
+ZC′ + (X − Z)A′
+√
+2
+XB′XC′ − (X − Z)A′
+√
+2
+YB′YC′ − YA′XB′YC′
+− YA′YB′XC′.
+(D12)
+The states generated by the sources P1, P2, P3 have already been certified to be of the form in Eq. (B9). Thus, the joint
+state of all the parties can be represented as
+|ψABCE⟩ =
+�
+s
+|ψss⟩ =
+�
+s
+(U†
+s ⊗ U†
+s )|φ+⟩s′s′|ξs′′s′′⟩
+(D13)
+which can be simplified to
+�
+�
+s
+Us ⊗ Us
+�
+|ψABC|E⟩ =
+�
+PA′′ ⊗ PB′′ ⊗ PC′′
+�
+|φ+
+8d′′
+Ad′′
+Bd′′
+C⟩ABC|E
+(D14)
+where Ps′′ =
+�
+d′′s σs′′ [c. f. Eqs. (C9)-(C11)]. Now, following exactly the same steps from Eqs. (C12)-(C15), we obtain
+from Eq. (D11) that
+1
+8Tr
+��
+S1,A′B′C′ ⊗ σT
+A′′ ⊗ σT
+B′′ ⊗ σT
+C′′
+�
+RT
+4
+�
+= 1.
+(D15)
+The state |ψ1,4⟩ given by
+|ψ1,4⟩ =
+1
+√
+2
+(|000⟩ + |111⟩)
+(D16)
+and then one can check that
+S1,A′B′C′ = 8|ψ1,4⟩⟨ψ1,4|.
+(D17)
+The above fact also explains why we imposed the condition in Eq. (D10). Taking into account Eq. (D17) we can
+rewrite Eq. (D15) as
+Tr
+�(|ψ1,4⟩⟨ψ1,4|A′B′C′ ⊗ σA′′ ⊗ σB′′ ⊗ σC′′) R4
+� = 1,
+(D18)
+where we have also used the fact that Tr[XTYT] = Tr[XY] for any pair of matrices X and Y and that the state |ψ1,4⟩ is
+real.
+Consider now an orthonormal basis {|ϕl⟩} in C8 in which |ϕl⟩ = |τl⟩ for l = 0, 1, 2, 3, where |τl⟩ form the UPB
+given in Eq. (D1), and |φl⟩ = |ψ1,l⟩ (l = 4, 5, 6, 7), where |ψ1,4⟩ is given in (D16) whereas the remaining vectors are
+defined as
+|ψ1,5⟩ =
+1
+√
+6(−2|000⟩ − |010⟩ + |011⟩),
+|ψ1,6⟩ =
+1
+√
+6
+�−2|000⟩ − |001⟩ + |101⟩
+�
+,
+|ψ1,7⟩ =
+1
+√
+6
+�−2|000⟩ − |100⟩ + |110⟩
+�
+.
+(D19)
+Notice that ∑7
+l=4 |ψ1,l⟩⟨ψ1,l| = Γ. Since R4 acts on the Hilbert space C8 ⊗ HA′′B′′C′′, we can express it using the above
+mentioned basis as
+R0 =
+7
+∑
+l,l′=0
+|ϕl⟩⟨ϕl′| ⊗ ˜Rl,l′,
+(D20)
+
+21
+where ˜Rl,l′ are some matrices acting on HA′′B′′C′′. Recalling that the local states are full-rank and then using the cyclic
+property of trace and Lemma 1 we conclude from Eq. (C15) that ˜R0,0 = 1 and thus we get that
+R4 = |ψ1,4⟩⟨ψ1,4| ⊗ 1A′′B′′C′′ + L4,
+(D21)
+where
+L4 =
+7
+∑
+l,l′=0
+l=l′̸=4
+|ϕl⟩⟨ϕl′| ⊗ ˜Rl,l′.
+(D22)
+Let us now consider Eq. (D4b)
+�
+(1 + A0)
+�
+−B1C1 + B2C2 + B1(1 + C0) − 1
+2(1 − B0)C1 + (1 + B0)C0 + 1
+2 B0 + 3
+21
+�
+R4
+�
+= 3
+2
+(D23)
+and then expand it by using the fact that the observables Ai, Bi, Ci for i = 0, 1, 2 are certified as in Eqs. (B11) and
+(B66). This gives us
+��
+�
+s=A,B,C
+U†
+s
+�
+S2,A′B′C′ ⊗ 1A′′B′′C′′
+�
+�
+s=A,B,C
+Us
+�
+⊗ R4
+�
+= 3
+4
+(D24)
+where
+S2,A′B′C′ = |0⟩⟨0|A′
+�
+−XB′XC′ + YB′YC′ + 2XB′|0⟩⟨0|C′ − |1⟩⟨1|B′XC′ + 2|0⟩⟨0|B′ZC′ + 1
+2ZB′ + 3
+21A′B′C′
+�
+.
+(D25)
+It is direct to verify that S2,A′B′C′ is proportional to the projection onto |ψ1,5⟩; precisely,
+S2,A′B′C′ = 6|ψ1,5⟩⟨ψ1,5|.
+(D26)
+Using then the above form of S2,A′B′C′ as well as the fact that |ψ1,5⟩ is real, we can simplify Eq. (D24) to
+Tr
+�(|ψ1,5⟩⟨ψ1,5| ⊗ σA′′ ⊗ σB′′ ⊗ σC′′) R4
+� = 1.
+(D27)
+Now, expanding R4 using Eq. (D21), we can conclude that
+R4 = |ψ1,4⟩⟨ψ1,4| ⊗ 1A′′B′′C′′ + |ψ1,5⟩⟨ψ1,5| ⊗ 1A′′B′′C′′ + L5,
+(D28)
+where
+L5 =
+7
+∑
+l,l′=0
+l=l′̸=4,5
+|ϕl⟩⟨ϕl′| ⊗ ˜Rl,l′.
+(D29)
+Similarly, we can conclude from the other two conditions in Eqs. (D4c) and (D4d), and using Eq. (D28) that
+R4 =
+7
+∑
+i=4
+|ψ1,i⟩⟨ψ1,i| ⊗ 1A′′B′′C′′ + L′
+= Γ ⊗ 1A′′B′′C′′ + L′,
+(D30)
+where
+L′ =
+7
+∑
+l,l′=0
+l=l′̸=4,5,6,7
+|ϕl⟩⟨ϕl′| ⊗ ˜Rl,l′.
+(D31)
+Now, following the same steps as between Eqs. (C19)-(C23) using the states |ϕl⟩|ξ⟩ for l = 4, 5, 6, 7 and any |ξ⟩ ∈
+HA′′B′′C′′ we can conclude that ˜Rl,l′ = 0 for any l ̸= l′ such that l′, l = 4, 5, 6, 7, and L′ simplifies to
+L′ =
+3
+∑
+l,l′=0
+|ϕl⟩⟨ϕl′| ⊗ ˜Rl,l′.
+(D32)
+
+22
+The fact that R4 ≥ 0 implies that L′ ≥ 0. Now, adding Eqs. (D8) and (D30) and using the fact that ∑l Rl = 1, we get
+that
+3
+∑
+l=0
+Ll + L′ = 0,
+(D33)
+which, after taking into account the fact that Ll as well as L′ are positive semi-definite, implies that Ll = L′ = 0.
+Thus, from Eqs. (D8) and (D30), we obtain
+Rl = |τl⟩⟨τl|E′ ⊗ 1E′′
+(l = 0, 1, 2, 3)
+(D34)
+and,
+R4 =
+�
+1 −
+3
+∑
+i=0
+|τi⟩⟨τi|
+�
+E′
+⊗ 1E′′ = ΓE′ ⊗ 1E′′.
+(D35)
+By virtue of Eq. (D9) we finally arrive at the desired forms of Eqs. (D6) and (D7), completing the proof.
+An interesting consequence of the above theorem is the certification of bound entangled state when Eve observes
+the final outcome of her measurement E2.
+Corollary 6. Assume that the states are certified as in Eq. (B9) and Eve’s measurement E2 is certified as in Eqs. (D6) and (D7).
+Consequently, when Eve observes the last outcome of her measurement, the post-measurement state with the external parties is
+given by
+U ρABC U† = 1
+4ΓA′B′C′ ⊗ ˜ρA′′B′′C′′,
+(D36)
+where U = �
+s Us and the unitaries Us are the same as in Eq. (B9).
+Proof. The post-measurement state when Eve observes the last outcome of her measurement E2 is given by
+ρABC =
+1
+P(4|e = 2)TrE
+��
+1ABC ⊗ R4|2
+�
+�
+s=A,B,C
+|ψss⟩⟨ψss|
+�
+.
+(D37)
+Now, substituting the states |ψss⟩ from Eq. (B9) and the measurement element R4|2 from Eq. (D7) and then using the
+fact that P(4|e = 2) = 1/2, we get that
+U ρABC U† = 2 TrE
+�
+(1ABC ⊗ ΓE′ ⊗ 1E′′)
+�
+s=A,B,C
+|φ+⟩⟨φ+|s′s′ ⊗ |ξs′′s′′⟩⟨ξs′′s′′|
+�
+.
+(D38)
+Again using the identity (1 ⊗ Q)|φ+⟩ = (QT ⊗ 1)|φ+⟩, we get
+U ρABC U† = 2 TrE
+�
+(ΓA′B′C′ ⊗ 1A′′B′′C′′ ⊗ 1E)
+�
+s=A,B,C
+|φ+⟩⟨φ+|s′s′ ⊗ |ξs′′s′′⟩⟨ξs′′s′′|
+�
+,
+(D39)
+where we also used the fact that ΓT = Γ. After tracing the E subsystem we arrive at
+U ρABC U† = 1
+4ΓA′B′C′ ⊗ ˜ρA′′B′′C′′,
+(D40)
+where ˜ρA′′B′′C′′ = TrE′′
+��
+s=A,B,C |ξs′′s′′⟩⟨ξs′′s′′|
+�
+.
+
diff --git a/_tFIT4oBgHgl3EQf-Cuq/content/tmp_files/load_file.txt b/_tFIT4oBgHgl3EQf-Cuq/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9b01ce97f26eed58be57d232200809d39f1310ba
--- /dev/null
+++ b/_tFIT4oBgHgl3EQf-Cuq/content/tmp_files/load_file.txt
@@ -0,0 +1,885 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf,len=884
+page_content='Self-testing composite measurements and bound entangled state in a unified framework  Shubhayan Sarkar,1, 2 Chandan Datta,3 Saronath Halder,3 and Remigiusz Augusiak1  1Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland  2Laboratoire d’Information Quantique, Université libre de Bruxelles (ULB), Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Roosevelt 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' 1050 Bruxelles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Belgium  3Centre for Quantum Optical Technologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Centre of New Technologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  University of Warsaw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Banacha 2c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' 02-097 Warsaw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Poland  Within the quantum networks scenario we introduce a single scheme allowing to certify three dif-  ferent types of composite projective measurements acting on a three-qubit Hilbert space: one con-  structed from genuinely entangled GHZ-like states,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' one constructed from fully product vectors that  exhibit the phenomenon of nonlocality without entanglement (NLWE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' and a hybrid measurement  obtained from an unextendible product basis (UPB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Noticeably, we certify a basis exhibiting NLWE  in the smallest dimension capable of supporting this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' On the other hand, the possibility  of certification of a measurement obtained from a UPB has an interesting implication that one can  also self-test a bound entangled state in the considered quantum network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Such a possibility does not  seem to exist in the standard Bell scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='—With the advancement of new tech-  nologies such as quantum cryptography [1], device-  independent (DI) certification of quantum devices is be-  coming increasingly important, allowing one to certify  certain features of an underlying device in a “black-box”  scenario [2–4], which requires basically no assumptions  about the device except that it is governed by quantum  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  The key ingredient for DI certification is Bell  nonlocality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=', the existence of quantum correlations  that cannot be explained by local hidden variable mod-  els [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' The most comprehensive form of DI certifi-  cation is self-testing [7], which allows for almost com-  plete characterisation of the underlying quantum state  and the measurements performed on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' From an appli-  cation standpoint, this type of certification is crucial as  it allows to verify whether a quantum device works as  expected without knowing its internal mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Since its introduction in [7], self-testing has been in-  vestigated in various scenarios and shown to have nu-  merous advantages [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' However, while much attention  has been paid to the self-testing of bipartite and multi-  partite states [9–22], the problem of certifying quantum  measurements, in particular those acting on composite  Hilbert spaces, has been largely unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Apart from  a few classes of local measurements [18, 20, 23] and a  few composite ones [24–26], no general scheme exists  allowing to certify composite measurements even in the  simplest case of multi-qubit Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Our aim here is to fill this gap and introduce a uni-  fied scheme in the quantum networks scenario that  allows for self-testing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' in a single experiment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' three  types of projective measurements:  one composed of  genuinely entangled states,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' one constructed from fully  product states exhibiting nonlocality without entangle-  ment (NLWE) [27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' and a hybrid one which is con-  structed from an unextendible product basis (UPB) [28]  and a projector onto the completely entangled subspace  orthogonal to the UPB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Notice that UPBs are interesting  mathematical objects that have found numerous appli-  cations, for instance, in constructing bound entangled  states [28] or Bell inequalities with no quantum viola-  tion [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  While a network-based self-testing scheme of a two-  qutrit product basis exhibiting NLWE was previously  introduced in [26], our approach enables one to self-test  such a basis in the smallest possible dimension capable  of supporting the notion of NLWE, which is eight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Fur-  thermore, our scheme utilises fewer measurements and  is thus more efficient as compared to [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Finally, it also  allows to simultaneously self-test a measurement of a  hybrid type, which is constructed from a UPB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' An inter-  esting implication of this fact is that one can also (indi-  rectly) certify in the network, a mixed bound entangled  state constructed from the UPB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' While a network-based  certification scheme for pure states has already been in-  troduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' [30], our work seems to be the first to  address the question of certification of mixed entangled  states (see nevertheless Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' [31, 32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='—Consider a Hilbert space H = H1 ⊗  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' ⊗ HN and a set of mutually orthogonal fully prod-  uct states from H, S = {|ψ1  i ⟩ ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' ⊗ |ψN  i ⟩}k  i=1, where  |ψm  i ⟩ ∈ Hm and k ≤ D = dim H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' [27] we  say that this set exhibits NLWE if the vectors |ψi⟩ can-  not be perfectly distinguished by local operations and  classical communications (LOCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' An exemplary such  set is the following basis of the three-qubit Hilbert space  H = (C2)⊗3 [27]:  |δ0⟩ = |0⟩|1⟩|+⟩, |δ1⟩ = |0⟩|1⟩|−⟩, |δ2⟩ = |+⟩|0⟩|1⟩,  |δ3⟩ = |−⟩|0⟩|1⟩, |δ4⟩ = |1⟩|+⟩|0⟩, |δ5⟩ = |1⟩|−⟩|0⟩,  |δ6⟩ = |0⟩|0⟩|0⟩,  |δ7⟩ = |1⟩|1⟩|1⟩,  (1)  where |i⟩, |i⟩ (i = 0, 1) and |±⟩, |±⟩ are the eigenvec-  tors of Z, (X + Z)/  √  2 and X, (X − Z)/  √  2, respectively,  where Z and X are the Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  While the above set forms a complete basis in the  corresponding Hilbert space, there also exist sets S ex-  hibiting NLWE that do not span the underlying Hilbert  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  These are called UPB and were introduced to  provide one of the first constructions of bound entan-  gled states, which are entangled states from which no  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='11409v1  [quant-ph]  26 Jan 2023  2  pure entanglement can be distilled [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' To be more pre-  cise, a collection S is a UPB if k < D, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=', S spans a  proper subspace V in H, and the subspace complemen-  tary to V is completely entangled [33, 34], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=', contains  no fully product vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' An excellent example of a UPB  in H = (C2)⊗3 are the following four vectors  |τ0⟩ = |0⟩|1⟩|+⟩,  |τ1⟩ = |+⟩|0⟩|1⟩,  |τ2⟩ = |1⟩|+⟩|0⟩,  |τ3⟩ = |−⟩|−⟩|−⟩  (2)  which are equivalent under local unitary transforma-  tions to the Shifts UPB introduced in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' The mixed  state ρ = Γ/4, where  Γ = 1 −  3  ∑  i=0  |τi⟩⟨τi|  (3)  stands for the projector onto a subspace complementary  to the UPB, is bound entangled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' in fact, by the very con-  struction it is entangled and all its partial transpositions  are nonnegative [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  The last set of vectors that we consider here are the  following GHZ-like pure states  |φl⟩ =  1  √  2  (|l1l2l3⟩ + (−1)l1|l1l2l3⟩),  (4)  where l ≡ l1l2l3 with l1, l2, l3 = 0, 1 and li is the negation  of the bit li, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=', li = 1 − li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' It is worth noting that, unlike  the previous vectors |δi⟩ or |τi⟩, the GHZ states are all  genuinely multipartite entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Composite measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='— From each of the consid-  ered sets of vectors one can construct a projective mea-  surement acting on H3 = (C2)⊗3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' First, the product ba-  sis |δi⟩ gives rise to a separable eight-outcome measure-  ment MNLWE = {|δi⟩⟨δi|}7  i=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' This measurement cannot  be implemented in terms of the LOCC, and, simultane-  ously, cannot produce entanglement if applied to a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  On the other extreme, we have the eight-outcome mea-  surement MGHZ = {|φl⟩⟨φl|}7  l=0 constructed from the  GHZ-like states which are all entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Thus, unlike  MNLWE, this measurement leads to entangled states for  all outcomes l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Let us finally move to the UPB in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  It al-  lows constructing a five-outcome hybrid measurement  MUPB = {|τi⟩⟨τi|}3  i=0 ∪ {Γ} that lies in between the GHZ  measurement and the separable measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' In fact,  four of its outcomes correspond to projections onto fully  product states |τi⟩, whereas the last outcome is repre-  sented by Γ that projects onto a completely (but not gen-  uinely) entangled four-dimensional subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Setting the scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='—We consider a quantum network  scenario consisting of three external parties Alice, Bob  and Charlie and a central party Eve (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' The  scenario also comprises of three independent sources Pi  that distribute bipartite quantum states among the par-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' We denote these states by ρss with s = A, B, C,  where the subsystems A, B and C belong to the exter-  nal parties, whereas the other three systems A, B and  C go to Eve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' in what follows we simplify the notation  by using E := ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' On their shares of the joint state  ρABCE = ρAA ⊗ ρBB ⊗ ρCC, each party can choose to per-  form one of the measurements Ax, By, Cz and Ee, where  the measurement choices are labelled x, y, z, e = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  We assume that each of the external party’s measure-  ments has two outcomes, denoted a, b, c = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' The first  two Eve’s measurements yield eight outputs, whereas  the third one results in five outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' During the exper-  iment, the parties cannot communicate classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  The correlations obtained by repeatedly performing  these measurements are captured by a set of proba-  bility distributions ⃗p = {p(abcl|xyze)}, where each  p(abcl|xyze) is the probability of observing outcomes a,  b, c and l by Alice, Bob, Charlie and Eve after perform-  ing measurements labelled by x, y, z, and e, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  it is given by the well-known formula  p(abcl|xyze) = Tr  �  ρABCENA  a|x ⊗ NB  b|y ⊗ NC  c|z ⊗ NE  l|e  �  ,  (5)  where NA  a|x, NB  b|y etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' are the measurement elements rep-  resenting the measurements of the observers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' these are  positive semi-definite and satisfy ∑a Ns  a|x = 1 for every  measurement choice x and every party s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  It will be beneficial to use another representation of  the observed correlations, that is, in terms of the ex-  pectation values of observables of the external parties,  which are defined as  ⟨AxByCzNE  l|e⟩ =  ∑  a,b,c=0,1  (−1)a+b+cp(abcl|xyze).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (6)  Notice that by employing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (5) one can express them  as ⟨AxByCzNE  l|e⟩  =  Tr[(Ax ⊗ By ⊗ Cz ⊗ NE  l|e)ρABCE],  where Ax, By and Cz are quantum operators that are  defined through the measurement elements as sk =  Ns  0|k − Ns  1|k, where s = A, B, C and k = x, y, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' In the  particular case of projective measurements these opera-  tors sk become unitary and thus represent the standard  quantum observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Self-testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='— The quantum networks scenario has re-  cently been harnessed to propose self-testing schemes  for few quantum measurements defined in composite  Hilbert spaces such as the measurement correspond-  ing to the two-qubit Bell basis composed of four maxi-  mally entangled vectors [24], or the nine-outcome pro-  jective measurement corresponding to a complete ba-  sis in C3 ⊗ C3 that exhibits NLWE [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Our aim here  is to employ these ideas to design a general frame-  work for quantum networks-based device-independent  (NDI) certification of various interesting types of quan-  tum measurements, concentrating on the particular case  of three-qubit Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  To define the task of self-testing in more precise terms,  let us consider again the scenario depicted on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' 1, but  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Schematic of the considered quantum network sce-  nario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' It consists of four parties A, B, C and E and three in-  dependent sources distributing bipartite quantum states ρSS  (S = A, B, C) among the parties as shown on the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' The  central party E shares quantum states with each of the other  parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Each party performs one of the available measure-  ments on their share of the state obtaining an outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' The ob-  tained correlations {p(abcl|xyze)} are used to certify that each  source distributes the maximally entangled state of two qubits  and that E’s measurements are MGHZ, MUPB and MNLWE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  now we assume that both the states ρss ∈ L(Hs ⊗ Hs)  and the measurements performed by the parties are un-  known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Due to the fact that the dimensions of the under-  lying Hilbert spaces Hs ⊗ Hs are unspecified we can em-  ploy the standard dilation arguments and assume that  the shared states are pure, that is, ρss = |ψss⟩⟨ψss| and  that the measurements are projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' These states and  measurements generate correlations that we denote by  ⃗p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Consider then a reference experiment involving some  known pure states |ψ′  ss⟩ ∈ Hs′ ⊗ Hs′ and known projec-  tive measurements represented by the observables A′  x,  B′  y, C′  z and E′  e that generate the same correlations ⃗p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' We  say that both experiments are equivalent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' alterna-  tively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' that |ψ′  ss⟩ and A′  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' B′  y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' C′  z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' E′  e are self-tested from  ⃗p if one can prove that the local Hilbert spaces admit  the product form Hs = Hs′ ⊗ Hs′′ and Hs = Hs′ ⊗ Hs′′  (s = A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' C) for some auxiliary Hilbert spaces Hs′′  and Hs′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' and that there are local unitary operations  Us : Hs → Hs′ ⊗ Hs′′ and Us : Hs → Hs′ ⊗ Hs′′  (Us ⊗ Us)|ψss⟩ = |ψ′  s′s′⟩ ⊗ |junks′′s′′⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (7)  where |junks′′s′′⟩ belongs to Hs′′ ⊗ Hs′′ and  Us si U†  s = s′  i ⊗ 1s′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  UE Ee U†  E = E′  e ⊗ 1E′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (8)  where 1E′′ is the identity acting on the auxiliary systems  Hs′′ and UE = UA ⊗ UB ⊗ UC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' recall that E′′ = A′′B′′C′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  It is important to note here that, since the measure-  ments can only be characterised on the local states, a  natural assumption that we make throughout this work  is that the latter are full-rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='—We propose a scheme that allows to device-  independently certify in a single experiment the three  different measurements introduced above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=', MGHZ,  MNLWE, and the hybrid one MUPB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Consider again the  network scenario in which the unknown pure states  |ψss⟩ with s  =  A, B, C are distributed by indepen-  dent sources among four parties who perform unknown  measurements Ax, By, Cz and Ee on their shares of those  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Now, their aim is to exploit the observed cor-  relations ⃗p to certify in the network the ideal reference  experiment in which each source distributes the max-  imally entangled state of two qubits |ψ′  ss⟩ = |φ+⟩ =  (|00⟩ + |11⟩)/  √  2, the external observers measure the  following observables on their shares of the joint state  A′  0/1/2 = X ± Z  √  2  /Y,  s′  0/1/2 = Z/X/Y (s = B, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (9)  and Eve performs the three measurements mentioned  above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=', E′  0 = MGHZ, E′  1 = MNLWE and E′  2 = MUPB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  To make our considerations easier to follow we divide  them into three parts, each devoted to one of Eve’s mea-  surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' We begin with E0 = {Rl|0}7  l=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' To certify  that it is equivalent to the GHZ measurement MGHZ, the  observed correlations ⃗p must be such that for each out-  come l of E0, {p(abcl|xyz0)} (x, y, z = 0, 1) maximally  violate the Bell inequality  √  2(−1)l1  �  2 �A1B1C1 + (−1)l2 �A0B0 + (−1)l3 �A0C0  �  ≤ 4,  (10)  where �A0/1 = (A0 ± A1)/  √  2 and l ≡ l1l2l3 with  l1, l2, l3 = 0, 1 is the binary representation of l, and the  probability of observing the outcome l by Eve must obey  P(l|e = 0) = 1/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  The Bell inequality corresponding to l = 0 was intro-  duced in [36] and is maximally violated by |φ0⟩, whereas  those corresponding to l ̸= 0 are its modifications that  are adjusted to be maximally violated by the remaining  GHZ states |φl⟩ and the quantum observables given in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' In fact, these Bell violations can be achieved in  the reference quantum network described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Let us now state our first result on self-testing the  GHZ measurement (see Appendix B for proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Assume that the observed correlations ⃗p ob-  tained in the network are such that the Bell inequalities in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (10) are maximally violated for each outcome l of Eve’s  measurement E0 and that each outcome occurs with proba-  bility P(l|e = 0) = 1/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Then, (i) the Hilbert spaces de-  compose as Hs = Hs′ ⊗ Hs′′ and Hs = Hs′ ⊗ Hs′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (ii)  There exist local unitary transformations Us : Hs → Hs and  Us : Hs → Hs such that  (Us ⊗ Us)|ψss⟩ = |φ+  s′s′⟩ ⊗ |ξs′′s′′⟩  (11)  for some |ξs′′s′′⟩ ∈ Hs′′ ⊗ Hs′′, and the measurements of all  parties are certified as  U Rl|0 U† = |φl⟩⟨φl|E′ ⊗ 1E′′,  Us si U†  s = s′  i ⊗ 1s′′ (12)  4  for all l and i = 0, 1 where U = ⊗sUs such that s = A, B, C  and E = ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' The states |φl⟩ and the observables s′  i are  given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (4) and (9) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  In what follows we build on this result to show how to  certify the other Eve’s measurements E1 and E2 and also  the third measurements of the external parties A2, B2  and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Let us then consider Eve’s second measurement  E1 = {Rl|1}7  l=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' In order to certify that it is equivalent to  the separable measurement MNLWE, the observed corre-  lations ⃗p, apart from the conditions stated in Theorem 1,  must additionally satisfy  p(0100|0011) = p(0111|0011) = p(0012|1001)  = p(1013|1001) = p(1004|0101) = p(1105|0101)  = p(0006|0001) = p(1117|0001) = 1  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (13)  Notice that these conditions are met in the ideal experi-  ment outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Let us state formally our second result that together  with Theorem 1 provides a scheme for DI certification  of the separable measurement exhibiting NLWE in the  least possible dimension (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Appendix C for a proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Suppose that ⃗p generated in the network satis-  fies the assumptions of Theorem 1 as well as the conditions  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Then, for any l it holds that U Rl|1 U† =  |δl⟩⟨δl|E′ ⊗ 1E′′ where U is the same unitary as in Theorem 1  and E = ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Before proceeding to the final result which is self-  testing of MUPB in E2, we need to introduce another  condition that is necessary to prove a self-testing state-  ment for A2, B2 and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Precisely, the correlations  {p(abc0|xyz0)} with x, y, z = 1, 2 corresponding to the  situation in which Eve observes the first outcome l = 0  of E0, the following condition is satisfied  ⟨ �A1B1C1 − �A1B2C2 − A2B1C2 − A2B2C1⟩ = 4,  (14)  where �A1 = (A0 − A1)/  √  2 and the above functional is  inspired by the Mermin inequality [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' This along with  Theorem 1 implies that (see Appendix B 3)  Us s2 U†  s = ±Ys′ ⊗ 1s′′  (s = A, B, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (15)  With the above characterisation at hand, we can fi-  nally move onto showing how to certify MUPB in E2 =  {Rl|2}4  l=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' To this end, the observed correlations must  satisfy  p(0100|0012) = p(0011|1002) = p(1002|0102)  = p(1113|1112) = 1  8  (16)  along with four other conditions stated in Appendix D  as Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D4a)-(D4d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' we refer to them as Pr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Notice  again that correlations obtained within the reference ex-  periments fulfil the above conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Let us now state  the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Assume that the assumptions of Theorem 2  and the conditions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (16) and Pr2 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Then,  the measurement E2 = {Rl|2} is certified as U Rl|2 U† =  |τl⟩⟨τl|E′ ⊗ 1E′′ for l = 0, 1, 2, 3, and, U R4|2 U† = ΓE′ ⊗  1E′′, where |τl⟩ and Γ are defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (2) and (3), respec-  tively, and U is the same unitary operation as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  The proof can be found in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' This final  result shows that the hybrid separable-entangled mea-  surement constructed from a UPB can also be self-tested  using our scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Most importantly, this is the minimal  scenario possible to self-test such a measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Bound entangled state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='—An interesting consequence of  Theorem 3 is that the considered network allows one to  self-test a bound entangled state shared between the ex-  ternal parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Assume that the states and Eve’s mea-  surement E2 are certified as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (9) and as in Theorem  5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' The post-measurement state shared by  the external parties that correspond to the last outcome  of E2 is then given by [see Appendix D]  U ρABC U† = 1  4ΓA′B′C′ ⊗ ˜ρA′′B′′C′′,  (17)  where U = �  s Us and the unitaries Us are the same as in  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' As mentioned above, the state Γ/4 is bound  entangled [27] and can be prepared by Eve in the exter-  nal parties labs with a simple post-processing strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  She first broadcasts her outcome of the measurement E2  and then the external parties discard those runs of the  experiment for which Eve observes any other outcome  than the last one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='—Inspired by the notion of NLWE, we can  identify a larger set of orthogonal projectors which can  be referred to as nonlocality without distillable entan-  glement (NLWDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' NLWDE is defined as a set of pro-  jectors that cannot be perfectly distinguished using local  operations and classical communication, such that their  normalised versions are positive under partial transpose  with respect to every bipartition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' In this work, the mea-  surement composed of UPB and bound entanglement  falls under this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' It will be interesting to further  analyse the properties of such sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Several other follow-up problems arise from our  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Designing certification schemes for GHZ bases  and product bases exhibiting NLWE for N qubits will  be straightforward from our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' A more interesting  question would be to generalize our scheme to certify  any hybrid or even any composite projective measure-  ment in the case of any number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' An even more  challenging problem will be to propose a quantum-  networks-based scheme for non-projective composite  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Our work also establishes a way to self-  test mixed entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  A natural question here  would be to construct schemes to self-test any mixed en-  tangled state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' a possibility that does not seem to exist in  the standard Bell scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  5  ACKNOWLEDGMENTS  This project was funded within the QuantERA II Pro-  gramme (VERIqTAS project) that has received fund-  ing from the European Union’s Horizon 2020 research  and innovation programme under Grant Agreement No  101017733 and from the Polish National Science Center  (project No 2021/03/Y/ST2/00175).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' ac-  knowledge the support by the “Quantum Optical Tech-  nologies” project, carried out within the International  Research Agendas program of the Foundation for Pol-  ish Science co-financed by the European Union under  the European Regional Development Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Gisin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Ribordy, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Tittel, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Zbinden, Quantum  cryptography, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
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+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Acín, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Brunner, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Gisin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Massar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Pironio,  and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Scarani, Device-independent security of quantum  cryptography against collective attacks, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  98, 230501 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
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+page_content=' Horodecki, Mixed-  state entanglement and distillation: Is there a “bound”  entanglement in nature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' 80, 5239 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  [36] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Baccari, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Augusiak, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Šupi´c, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Tura, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Acín, Scal-  able Bell inequalities for qubit graph states and robust  self-testing, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' 124, 020402 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  [37] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Mermin, Extreme quantum entanglement in a su-  perposition of macroscopically distinct states, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' 65, 1838 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Appendix A: General result  Before proceeding to the proofs of the main results, we introduce an important lemma that is required to derive  our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Consider a positive semi-definite matrix M such that M ≤ 1 that for a given density matrix ρ which is full-rank  satisfies Tr(Mρ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Then, M is an identity matrix acting on the support of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Let us consider the eigendecomposition of ρ, ρ = ∑k pk|ψk⟩⟨ψk| such that pk > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' After putting it into the  condition Tr(Mρ) = 1, one obtains  ∑  k  pk⟨ψk|M|ψk⟩ = 1,  (A1)  which by employing the fact that ∑k pk = 1 can further be rewritten as  ∑  k  pk(1 − ⟨ψk|M|ψk⟩) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (A2)  Due to the facts that 0 ≤ M ≤ 1 and pk > 0, the above equation can hold true only if ⟨ψk|M|ψk⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Thus, every  |ψk⟩ is an eigenstate of M with eigenvalue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' As a result, M = ∑k |ψk⟩⟨ψk| = 1ρ, where 1ρ is an identity acting on the  support of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Appendix B: Self-testing of GHZ bases, measurements of external parties and the states prepared by the preparation devices  In this section, we explain the certification of the GHZ basis, measurements of the external parties and the states  distributed by the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  The GHZ basis and measurements Ax, By and Cz with x, y, z = 0, 1  Let us first consider the following eight Bell inequalities  Il = 2(−1)l1 ⟨(A0 + A1)B1C1⟩ + (−1)l2+l1 ⟨(A0 − A1)B0⟩ + (−1)l3+l1 ⟨(A0 − A1)C0⟩ ≤ 4,  (B1)  where l = l1l2l3 with li ∈ {0, 1} for each i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' The inequality for l1 = l2 = l3 = 0 was constructed in  [36], whereas the remaining seven are its variants obtained by making the signs in front of each expectation value  depend on the parameters li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' We can now state the following fact which is concerned with Tsirelson’s bound of the  inequalities in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' The maximal quantum value of the Bell expression Il is 4  √  2 and it is achieved by the following observables  A0 = X + Z  √  2  ,  A1 = X − Z  √  2  ,  B0 = Z,  B1 = X,  C0 = Z,  C1 = X  (B2)  as well as the GHZ-like state  |φl⟩ =  1  √  2  (|l1l2l3⟩ + (−1)l1|l1l2l3⟩),  (B3)  where l ≡ l1l2l3 with l1, l2, l3 = 0, 1 and li is the negation of the bit li, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=', li = 1 − li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  7  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' We follow the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' First, to each of the Bell expressions Il we associate a Bell operator of the  following form  ˆIl1l2l3 = 2(−1)l1(A0 + A1) ⊗ B1 ⊗ C1 + (−1)l2+l1(A0 − A1) ⊗ B0 + (−1)l3+l1(A0 − A1) ⊗ C0,  (B4)  where Ax, By and Cz are arbitrary ±1-valued quantum observables of arbitrary dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Second, for each ˆIl we  construct the following sum-of-squares (SOS) decomposition,  4  √  2 1 − ˆIl1l2l3 =  √  2  �  1 − P1,l1  �2 + 1  √  2  ��  1 − P2,l1,l2  �2 +  �  1 − P3,l1,l3  �2�  ,  (B5)  where  P1,l1 = (−1)l1 A0 + A1  √  2  ⊗ B1 ⊗ C1,  (B6a)  P2,l1,l2 = (−1)l1+l2 A0 − A1  √  2  ⊗ B0,  (B6b)  P3,l1,l3 = (−1)l1+l2 A0 − A1  √  2  ⊗ C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B6c)  It directly follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B5) that 4  √  2 1 − ˆIl ≥ 0 and thus 4  √  2 is an upper bound on the maximal quantum value  of Il, which means that ⟨ψ| ˆIl|ψ⟩ ≤ 4  √  2 for arbitrary state |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' To finally show that the latter inequality is tight and  that 4  √  2 is in fact Tsirelson’s bound of the inequalities in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B1) it suffices to observe that Il achieves the value  4  √  2 for the GHZ-like state |φl⟩ and the observables given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Crucially, the SOS decomposition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B5) implies that any state |ψ⟩ and any observables Ax, By and Cz that  achieve the quantum bound βQ = 4  √  2 of Il must satisfy the following relations  P1,l1|ψ⟩ = |ψ⟩  and  Pi,l1,li|ψ⟩ = |ψ⟩  (i = 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B7)  which, by virtue of the relations in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B6a), (B6b) and (B6c), can be rewritten as  (−1)l1 A0 + A1  √  2  ⊗ B1 ⊗ C1|ψ⟩ = |ψ⟩,  (B8a)  (−1)l1+l2 A0 − A1  √  2  ⊗ B0|ψ⟩ = |ψ⟩,  (B8b)  (−1)l1+l3 A0 − A1  √  2  ⊗ C0|ψ⟩ = |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B8c)  The above relations are used in the proof of Theorem 1 stated in the main text as well as in the preceding subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Proof of self-testing  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Consider the network scenario outlined in the main text and assume that the observed correlations ⃗p achieve the  maximal quantum value of Il in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B1) for each outcome l of Eve’s first measurement E0 and that each outcome l occurs with  probability P(l|e = 0) = 1/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Then,  (i) All six Hilbert spaces decompose as Hs = Hs′ ⊗ Hs′′, and Hs = Hs′ ⊗ Hs′′ with s = A, B, C, where Hs′ and Hs′ are  one-qubit Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (ii) There exist local unitary transformations Us : Hs → Hs and Us : Hs → Hs such that  Us ⊗ Us|ψss⟩ = |φ+  s′s′⟩ ⊗ |ξs′′s′′⟩  (B9)  for each s = A, B, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  8  (iii) Then, Eve’s first measurement E0 = {Rl|0} satisfies  (UA ⊗ UB ⊗ UC) Rl|0 (UA ⊗ UB ⊗ UC)† = |φl⟩⟨φl|E′ ⊗ 1E′′,  (B10)  where E = ABC, |φl⟩ are the GHZ-like states given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B3) and the measurements of all other parties are given by  UA A0 U†  A =  � X + Z  √  2  �  A′ ⊗ 1A′′,  UA A1 U†  A =  � X − Z  √  2  �  A′ ⊗ 1A′′,  UB B0 U†  B = ZB′ ⊗ 1B′′,  UB B1 U†  B = XB′ ⊗ 1B′′,  UC C0 U†  C = ZC′ ⊗ 1C′′,  UC C1 U†  C = XC′ ⊗ 1C′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B11)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Before we proceed with the proof we first notice that the post-measurement state that A, B and C share after  Eve performs her first measurement E0 and obtains the outcome l is given by  ρl  ABC =  1  P(l)TrABC  ��  1ABC ⊗ Rl|0  �  �  s=A,B,C  |ψss⟩⟨ψss|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B12)  We divide the proof into a few steps and the first one is concerned with determining the form of the states ρl  ABC  for any l from the observed maximal violations of the inequalities in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Building on this result we then find  the form of the states generated by the sources |ψss⟩ for any s = A, B, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Finally, using both of these results we obtain  the form of the entangled measurement {Rl|0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' For simplicity, in the rest of the proof, we represent Rl|0 as Rl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (a) Post-measurement states ρl  ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' To determine the form of the post-measurement states ρl  ABC we exploit the  relations given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' First, let us consider a purification of ρl  ABC, denoted |ψl⟩ABCG, which is a pure state  satisfying  ρl  ABC = TrG (|ψl⟩⟨ψl|ABCG) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B13)  For simplicity, in what follows we drop the subscript from the above state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' From the assumption that ρl  ABC maximally  violates the Bell inequality in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B1) it follows that the relations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B8) are satisfied by the purification |ψl⟩,  which after taking into account that B2  y = C2  z = 1 can be stated as  (−1)l1 A0 + A1  √  2  |ψl⟩ = B1 ⊗ C1|ψl⟩  (B14a)  (−1)l1+l2 A0 − A1  √  2  |ψl⟩ = B0|ψl⟩,  (B14b)  (−1)l1+l3 A0 − A1  √  2  |ψl⟩ = C0|ψl⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B14c)  In the above equations as well as in the following considerations we omit the identities acting on the remaining  subsystems, including the G one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Now, by applying B1 ⊗ C1 to both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B14a), we obtain  (A0 + A1)2|ψl⟩ = 2|ψl⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B15)  As the measurements can be characterised only on the support of the local reduced density matrices of every party  we can always assume them to be full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Consequently, from the above Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B15), we find that  (A0 + A1)2 = 2 1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B16)  Expanding the above equation and using the fact that A2  0 = A2  1 = 1A, we get that {A0, A1} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' As proven in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' [18] or Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' [36], for a pair of unitary observables with eigenvalues ±1 that anti-commute there exist a unitary  operation UA : HA → HA such that  UA A0 U†  A = X + Z  √  2  ⊗ 1A′′,  UA A1 U†  A = X − Z  √  2  ⊗ 1A′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B17)  9  Let us now move on to characterizing the other parties’ observables and use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B17) to rewrite the relations in  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B14) as  (−1)l1 XA′|ψ′  l⟩ = B1 ⊗ C1|ψ′  l⟩,  (B18a)  (−1)l1+l2 ZA′|ψ′  l⟩ = B0|ψ′  l⟩,  (B18b)  (−1)l1+l3ZA′|ψ′  l⟩ = C0|ψ′  l⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B18c)  where |ψ′  l⟩ = UA|ψl⟩ and we omitted the identity acting on the A′′ subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Let us then consider the relation in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B18a) and multiply it with ZA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Then using the relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B18b) on the right-hand side of the obtained  expression, we get  (−1)l2(ZX)A′|ψ′  l⟩ = B1B0 ⊗ C1|ψ′  l⟩  (B19)  Then, after multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B18b) with XA′ and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B18a) on the right-hand side of the obtained expression,  we get  (−1)l2(XZ)A′|ψ′  l⟩ = B0B1 ⊗ C1|ψ′  l⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B20)  Adding Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B19) and (B20), and using the fact that ZX + XZ = 0, we finally arrive at  {B1, B0} ⊗ C1|ψ′  l⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B21)  Exploiting the facts that C1 is invertible and that the local density matrices are full-rank, we get that {B1, B0} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Employing again the result of [18], we conclude that HB = (C2)B′ ⊗ HB′′ and that there exists a unitary operation  UB : HB → HB for which  UB B0 U†  B = ZB′ ⊗ 1B′′,  UB B1 U†  B = XB′ ⊗ 1B′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B22)  Proceeding exactly in the same manner, we can conclude that C0 and C1 that appear in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B18a) and (B18c) also  anticommute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Thus, as before, HC = (C2)C′ ⊗ HC′′ and there exists a unitary UC : HC → HC such that  UC C0 U†  C = ZC′ ⊗ 1C′′,  UC C1 U†  C = XC′ ⊗ 1C′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B23)  Let us now characterise the state ρl  ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' For this purpose, we plug the forms of the observables from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B17),  (B22) and (B23) into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B7) to obtain  (−1)l1XA′ ⊗ XB′ ⊗ XC′| ˜ψl⟩ = | ˜ψl⟩,  (B24a)  (−1)l1+l2ZA′ ⊗ ZB′| ˜ψl⟩ = | ˜ψl⟩,  (B24b)  (−1)l1+l3ZA′ ⊗ ZC′| ˜ψl⟩ = | ˜ψl⟩,  (B24c)  where | ˜ψl⟩ = UA ⊗ UB ⊗ UC|ψl⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' As already concluded above each local Hilbert space Hs (s = A, B, C) decomposes  as Hs = (C2)s′ ⊗ Hs′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Thus, | ˜ψl⟩ can be decomposed as  | ˜ψl⟩ =  ∑  i1,i2,i3=0,1  |i1i2i3⟩A′B′C′|φl  i1i2i3⟩A′′B′′C′′G,  (B25)  where the normalisation factors are included in |φl  i1i2i3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Putting the above form of | ˜ψl⟩ in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B24b) and (B24c), we  obtain  (−1)l1+l2  ∑  i1,i2,i3=0,1  (−1)i1+i2|i1i2i3⟩|φl  i1i2i3⟩ =  ∑  i1,i2,i3=0,1  |i1i2i3⟩|φl  i1i2i3⟩,  (B26)  and  (−1)l1+l3  ∑  i1,i2,i3=0,1  (−1)i1+i3|i1i2i3⟩|φl  i1i2i3⟩ =  ∑  i1,i2,i3=0,1  |i1i2i3⟩|φl  i1i2i3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B27)  For brevity, we dropped subscripts denoting the subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Projecting both the above formulas on ⟨i1i2i3|, we  obtain the following relations  (−1)l1+l2(−1)i1+i2|φl  i1i2i3⟩ = |φl  i1i2i3⟩  and  (−1)l1+l3(−1)i1+i3|φl  i1i2i3⟩ = |φl  i1i2i3⟩,  (B28)  10  which allow us to conclude that |φl  i1i2i3⟩ = 0 whenever (l1 + l2 + i1 + i2) mod 2 = 1 and (l1 + l3 + i1 + i3) mod 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Thus, the state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B25) that satisfies the conditions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B28) must be of the form  | ˜ψl⟩ = |l1l2l3⟩|φl1l2l3⟩ + |l1l2l3⟩|φl1l2l3⟩  (B29)  where li = 0, 1 for any i = 1, 2, 3 and li = 1 − li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Now, putting this state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B24a), we obtain the following  relation  (−1)l1|l1l2l3⟩|φl1l2l3⟩ + (−1)l1|l1l2l3⟩|φl1l2l3⟩ = |l1l2l3⟩|φl1l2l3⟩ + |l1l2l3⟩|φl1l2l3⟩,  (B30)  which implies that  (−1)l1|φl1l2l3⟩ = |φl1l2l3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B31)  Thus, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B29) we conclude that the state | ˜ψl⟩, by putting the appropriate normalisation constant, is given by  | ˜ψl⟩ =  1  √  2  �  |l1l2l3⟩ + (−1)l1|l1l2l3⟩  �  A′B′C′ ⊗ |φl1l2l3⟩A′′B′′C′′G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B32)  Tracing out the ancillary subsystem G, we finally obtain  UA ⊗ UB ⊗ UC ρl  ABC (UA ⊗ UB ⊗ UC)† = |φl⟩⟨φl|A′B′C′ ⊗ ˜ρl  A′′B′′C′′,  (B33)  where ˜ρl  A′′B′′C′′ denotes the auxiliary state acting on HA′′ ⊗ HB′′ ⊗ HC′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Putting the above relation back into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B12)  and taking the unitaries to the right-hand side, we see that  |φl⟩⟨φl|A′B′C′ ⊗ ˜ρl  A′′B′′C′′ = 8 TrABC  �  (1ABC ⊗ Rl)  �  s=A,B,C  | ˜ψss⟩⟨ ˜ψss|  �  ,  (B34)  where we denoted | ˜ψss⟩ = Us|ψss⟩ and used the fact that P(l) = 1/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (b) States |ψss⟩ generated by the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Let us now exploit the relations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B34) to determine the form of the  the states | ˜ψss⟩ (s = A, B, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' To this end, we first express them using their Schmidt decompositions,  | ˜ψss⟩ =  ds−1  ∑  i=0  αs  i |ei⟩s| fi⟩s,  (B35)  where ds denotes the dimension of the Hilbert space Hs and {|ei⟩s} and {| fi⟩s} are some local bases corresponding  to the subsystems s and s, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' recall that, as proven above, ds is even for any s as the Hilbert spaces of the  external parties decompose as Hs = C2 ⊗ Hs′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Moreover, the Schmidt coefficients satisfy αs  i > 0 and ∑i(αs  i )2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Let us now consider a unitary operation Us : Hs → Hs such that Us| fi⟩s = |e∗  i ⟩s for any i, where the asterisk stands  for the complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' By using this unitary we can bring Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B35) to the following form  |ψss⟩ = 1s ⊗ Us| ˜ψss⟩ =  ds−1  ∑  i=0  αs  i |ei⟩s|e∗  i ⟩s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B36)  Introducing then the following matrix  Ps =  ds−1  ∑  i=0  αs  i |e∗  i ⟩⟨e∗  i |s,  (B37)  we can rewrite the state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B36) as  |ψss⟩ =  �  ds (1s ⊗ Ps) |φ+  ds⟩ss  (s = A, B, C),  (B38)  where |φ+  ds⟩ss denotes the maximally entangled state of local dimension ds, that is,  |φ+  ds⟩ss =  1  √  ds  ds−1  ∑  i=0  |ei⟩s|e∗  i ⟩s =  1  √  ds  ds−1  ∑  i=0  |i⟩s|i⟩s,  (B39)  where |e∗  i ⟩ is the complex conjugate of |ei⟩ and the last equality follows from the fact that the maximally entangled  state is invariant under the action of U ⊗ U∗ for any unitary operation U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  11  Putting now Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B39) back into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B34), we conclude that  |φl⟩⟨φl|A′B′C′ ⊗ ˜ρl  A′′B′′C′′ = 8TrE  �  �  1ABC ⊗ Rl  � �  s  |φ+  ds⟩⟨φ+  ds|ss  �  ,  (B40)  where we denoted  Rl = dAdBdC  �  �  s  Ps Us  �  Rl  �  �  s  U†  s Ps  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B41)  Now, notice that the tensor product of the three maximally entangled states appearing in (B41) can also be under-  stood as a single maximally entangled state across the bipartition ABC|ABC whose local dimension is dAdBdC, that  is,  |φ+  ds⟩AA ⊗ |φ+  ds⟩BB ⊗ |φ+  ds⟩CC = |φ+  dAdBdC⟩ABC|ABC,  (B42)  |φ+  dAdBdC⟩ABC|ABC =  1  √dAdBdC  dAdBdC−1  ∑  i=0  |i⟩ABC|i⟩ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B43)  Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B42) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B40) and using the well known fact that 1A ⊗ QB|φ+  D⟩A|B = QT  A ⊗ 1B|φ+  D⟩A|B for any  matrix Q, we can rewrite (B40) as  |φl⟩⟨φl|A′B′C′ ⊗ ˜ρl  A′′B′′C′′ = 8TrABC  ��  RT  l  �  ABC ⊗ 1ABC|φ+  dAdBdC⟩⟨φ+  dAdBdC|ABC|ABC  �  ,  (B44)  which implies that  |φl⟩⟨φl|A′B′C′ ⊗ ˜ρl  A′′B′′C′′ =  8  dAdBdC  RT  l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B45)  Now, applying the transposition map to both sides and then using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B41), we arrive at  |φl⟩⟨φl|A′B′C′ ⊗  �  ˜ρl  A′′B′′C′′  �T  = 8  �  �  s  PsUs  �  Rl  �  �  s  U†  s Ps  �  ,  (B46)  which after summing over l and using the fact that ∑7  l=0 Rl = 1 gives  7  ∑  l=0  |φl⟩⟨φl|A′B′C′ ⊗  �  ˜ρl  A′′B′′C′′  �T  = 8 P2  A ⊗ P2  B ⊗ P2  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B47)  We can now employ the fact that Tr(P2  B) = Tr(P2  C) = 1 to take a partial trace of the above expression over the  subsystems BC and obtain  1A′  2 ⊗ σA′′ = P2  A,  (B48)  which, by virtue of the fact that, by definition, PA ≥ 0, directly implies that  PA = 1A′ ⊗  �  σA′′  2 ,  (B49)  where  σA′′ = 1  8  7  ∑  l=0  TrB′′C′′  ��  ˜ρl  A′′B′′C′′  �T�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B50)  In an analogous manner we can determine the other matrices Ps with s = B, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Precisely, by taking partial traces  of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B47) over all subsystems except the sth one, one obtains  Ps = 1s′ ⊗  �  σs′′  2  (s = A, B, C)  (B51)  12  with  σs′′ = 1  8  7  ∑  l=0  TrA′′B′′C′′\\{s′′}  ��  ˜ρl  A′′B′′C′′  �T�  ,  (B52)  where TrA′′B′′C′′\\{s′′} represents the partial trace over the systems A′′B′′C′′ except the one labelled by s′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' For instance  TrA′′B′′C′′\\{A′′} ≡ TrB′′C′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Now, we can substitute Ps given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B51) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B38) and also use the fact that ds = 2d′′  s for some positive  integer d′′  s to obtain  |ψss⟩ =  �  d′′s 1s′s′ ⊗ 1s′′ ⊗ √σs′′ |φ+  ds⟩ss  (B53)  for any s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Using again the fact that |φ+  ds⟩ss = |φ+⟩s′s′|φ+  d′′s ⟩s′′s′′, we finally conclude that the states generated by the  sources admit the following form  Us ⊗ Us|ψss⟩ = |ψss⟩ = |φ+⟩s′s′|ξs′′s′′⟩  (s = A, B, C),  (B54)  where the auxiliary state |ξs′′s′′⟩ is given by  |ξs′′s′′⟩ =  �  d′′s 1s′′ ⊗ √σs′′ |φ+  d′′s ⟩s′′s′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B55)  (c) Entangled measurement E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Let us now characterise the measurement E0 = {Rl}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' For this purpose, we notice  that the states σs′′ are invertible because we assumed all the reduced density matrices of |ψl⟩ to be full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' This  implies that the matrices Ps [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B51)] are invertible too and therefore we can act with P−1  s  on both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B46) to bring to the following form  �  �  s  Us  �  Rl  �  �  s  U†  s  �  = |φl⟩⟨φl|A′B′C′ ⊗  �  �  s  σ−1/2  s′′  �  ˜ρl,T  A′′B′′C′′  �  �  s  σ−1/2  s′′  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B56)  We thus conclude that  �  �  s  Us  �  Rl  �  �  s  U†  s  �  = |φl⟩⟨φl|A′B′C′ ⊗  � ˜Rl  �  A′′B′′C′′ ,  (B57)  for all l = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' , 7, where ˜Rl is defined as  ˜Rl =  �  �  s  σ−1/2  s′′  � �  ˜ρl  A′′B′′C′′  �T  �  �  s  σ−1/2  s′′  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B58)  Notice that ˜Rl ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Moreover, the fact that Rl ≤ 1 implies via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B57) that ˜Rl ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Taking then the sum over l on  both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B57) and employing the fact that ∑l Rl = 1 allows us to conclude that  1ABC =  7  ∑  l=0  |φl⟩⟨φl|A′B′C′ ⊗  � ˜Rl  �  A′′B′′C′′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B59)  Now, we can take advantage of the fact that the GHZ-like states are mutually orthogonal and therefore by projecting  the A′B′C′ subsystems onto |φk⟩ we deduce that for every k,  ˜Rk = 1A′′B′′C′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B60)  After plugging the above relation into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B57) we finally conclude that there exist local unitary transformations  such that the measurement operators Rl admit the following form  (UA ⊗ UB ⊗ UB) Rl (UA ⊗ UB ⊗ UB)† = |φl⟩⟨φl|A′B′C′ ⊗ 1A′′B′′C′′  (l = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' , 7),  (B61)  which is exactly what we promised in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  An interesting consequence of the above theorem is that the states ˜ρl  A′′B′′C′′ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B33) are separable for all l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  13  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Assume that the sources P1, P2, P3 generate states that are certified as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B9) and the measurements of all the  parties are certified as in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B10) and (B11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Then, the post-measurement state ρl  ABC when Eve gets an outcome l is given by  �  �  s  Us  �  ρl  ABC  �  �  s  U†  s  �  = ρl  ABC = |φl⟩⟨φl|A′B′C′ ⊗ ˜ρA′′ ⊗ ˜ρB′′ ⊗ ˜ρC′′  ∀l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B62)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Let us first observe that by combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B58) and (B60) we can write  �  �  s  σ−1/2  s′′  � �  ˜ρl  A′′B′′C′′  �T  �  �  s  σ−1/2  s′′  �  = ˜Rl = 1A′′B′′C′′,  (B63)  which after rearranging the terms and taking transposition gives us  ˜ρl  A′′B′′C′′ =  �  s  σT  s′′  (l = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' , 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B64)  Recall that σs′′ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B52) are valid quantum states as they positive semi-definite and satisfy Tr(σs′′) = 1 for any  s = A, B, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Self-testing of the extra measurements A2, B2 and C2  Now using the above theorem, let us self-test the additional measurements A2, B2 and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' For this purpose, we  consider a functional inspired by the well-known Mermin Bell inequality [37]:  I2 =  � 1  √  2  (A0 + A1)B1C1 − 1  √  2  (A0 + A1)B2C2 − A2B1C2 − A2B2C1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B65)  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Assume that the sources P1, P2, P3 generate states that are certified as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B9) and the measurements of all the  external parties are certified as in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B10) and (B11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' If I2 achieves the value four for the state shared by A, B and C that  corresponds to the outcome l = 000 of Eve’s first measurement E0, then the observables A2, B2 and C2 can have two possible  forms  UA A2 U†  A = YA′ ⊗ 1A′′,  UB B2 U†  B = YB′ ⊗ 1B′′,  UC C2 U†  C = YC′ ⊗ 1C′′,  (B66)  or  UA A2 U†  A = −YA′ ⊗ 1A′′,  UB B2 U†  B = −YB′ ⊗ 1B′′,  UC C2 U†  C = −YC′ ⊗ 1C′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B67)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Let us first consider the functional I2 for the particular observables A0, A1, B1 and C1 that are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B11) as well as a particular state corresponding to the l = 000 outcome of Eve’s first measurement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=', ρ000  ABC,  I′  2 = ⟨XA′ ⊗ XB′ ⊗ XC′ ⊗ 1A′′B′′C′′⟩ − ⟨XA′ ⊗ B′  2 ⊗ C′  2 ⊗ 1A′′⟩ − ⟨A′  2 ⊗ XB′ ⊗ C′  2 ⊗ 1B′′⟩  −⟨A′  2 ⊗ B′  2 ⊗ XC′ ⊗ 1C′′⟩ρ000  ABC,  (B68)  where A′  2 = UA A2 U†  A etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=', and ρ000  ABC is a ’rotated version’ of ρ000  ABC and is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B62);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' to simplify the notation  from now on we denote ϱ0 ≡ ρ000  ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Now, taking into account the fact that the observables A′  2, B′  2 and C′  2 are unitary it is not difficult to realise that I′  2  attains the value four if and only if the first expectation value in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B68) equals 1 whereas the remaining three are  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Again, given that A′  2, B′  2 and C′  2 are unitary this implies that the following conditions are satisfied,  (XA′ ⊗ XB′ ⊗ XC′ ⊗ 1A′′B′′C′′) ϱ0 = ϱ0,  (B69)  (XA′ ⊗ B′  2 ⊗ C′  2 ⊗ 1A′′) ϱ0 = −ϱ0,  (B70)  (A′  2 ⊗ XB′ ⊗ C′  2 ⊗ 1B′′) ϱ0 = − ϱ0,  (B71)  (A′  2 ⊗ B′  2 ⊗ XC′ ⊗ 1C′′) ϱ0 = − ϱ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B72)  Let us now consider the relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' After acting on it with XA′ ⊗ XB′ ⊗ XC′ and then using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B69), we  obtain  �  1A ⊗ (XB′ ⊗ 1B′′) B′  2 ⊗ (XC′ ⊗ 1C′′) C′  2  �  ϱ0 = − ϱ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B73)  14  Next, after multiplication by C′  2 (XC′ ⊗ 1C′′), the above relation can be brought to  (XB′ ⊗ 1B′′) B′  2 ϱ0 = −C′  2 (XC′ ⊗ 1C′′) ϱ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B74)  Let us then consider the relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' After multiplying it with XA′ ⊗ B′  2 ⊗ C′  2, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B70), and then  multiplying the resulting relation with (XC′ ⊗ 1C′′) C′  2, we arrive at  B′  2 (XB′ ⊗ 1B′′) ϱ0 = − (XC′ ⊗ 1C′′) C′  2 ϱ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B75)  By adding Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B74) and (B75), one obtains  {B′  2, (XB′ ⊗ 1B′′)} ϱ0 = −{C′  2, (XC′ ⊗ 1C′′)} ϱ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B76)  Now, we consider Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' By acting on it with A′  2 ⊗ B′  2 ⊗ XC′, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B72), and then multiplying the resulting  formula with C′  2 (XC′ ⊗ 1C′′), we obtain  B′  2 (XB′ ⊗ 1B′′) ϱ0 = C′  2 (XC′ ⊗ 1C′′) ϱ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B77)  Similarly, we then consider Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B72), multiply it with A′  2 ⊗ (XB′ ⊗ 1B′′) ⊗ C′  2, use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B71), and finally multiply  resulting formula with (XC′ ⊗ 1C′′)C′  2 to obtain  (XB′ ⊗ 1B′′) B′  2 ϱ0 = (XC′ ⊗ 1C′′) C′  2 ϱ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B78)  Adding Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B77) and (B78), we get  {B′  2, (XB′ ⊗ 1B′′)} ϱ0 = {C′  2, (XC′ ⊗ 1C′′)} ϱ0  (B79)  We have thus obtained two similar relations, (B76) and (B79), but with the opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' By adding them we thus  conclude that  {B′  2, (XB′ ⊗ 1B′′)} ϱ0 = 0,  (B80)  which, by taking into account that all reduced density matrices of ϱ0 are full rank eventually implies that  {B′  2, (XB′ ⊗ 1B′′)} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B81)  In the exactly same manner, one can use Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B69)–(B72) to obtain similar relations for the observables A′  2, and  C′  2:  {A′  2, (XA′ ⊗ 1A′′)} = 0,  {C′  2, (XC′ ⊗ 1C′′)} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B82)  Let us exploit the above anticommutation relations to determine the forms of A′  2, B′  2 and C′  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' We begin with A′  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  As characterised before, the Hilbert space of Alice is given by HA = C2 ⊗ HA′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Thus, any observable acting on such  a Hilbert space can be decomposed as  A′  2 = 12 ⊗ Q0 + Z ⊗ Q1 + X ⊗ Q2 + Y ⊗ Q3,  (B83)  where for simplicity we have omitted the subscripts A′ and A′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Putting it back into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B82), we get that  X ⊗ Q0 + 1 ⊗ Q2 = 0,  (B84)  which implies that Q0 = Q2 = 0, and consequently A2 expresses as  A′  2 = Z ⊗ Q1 + Y ⊗ Q3,  (B85)  where, due to the fact that A2  2 = 1, the matrices Q1 and Q3 obey the following relations  Q2  1 + Q2  3 = 1,  [Q1, Q3] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B86)  One can similarly find that  B′  2 = Z ⊗ R1 + Y ⊗ R3,  C′  2 = Z ⊗ S1 + Y ⊗ S3  (B87)  for some matrices R1, R3, S1 and S3 such that R2  1 + R2  3 = 1 and [R1, R3] = 0, and S2  1 + S2  3 = 1 and [S1, S3] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Now, after putting these forms of the observables into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B70), we get  [XA′ ⊗  �  ZB′ ⊗ R1,B′′ + YB′ ⊗ R3,B′′  � ⊗  �  ZC′ ⊗ S1,C′′ + YC′ ⊗ S3,C′′  � ⊗ 1A′′] ϱ0 = − ϱ0,  (B88)  15  which on expansion and substituting the state ˜ρ000  ABC from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B62) gives  �  XA′ZB′ZC′ ⊗ 1A′′R1,B′′S1,C′′ + XA′ZB′YC′ ⊗ 1A′′R1,B′′S3,C′′ + XA′YB′ZC′ ⊗ 1A′′R3,B′′S1,C′′  +XA′YB′YC′ ⊗ 1A′′R3,B′′S3,C′′  � |φ0⟩⟨φ0|A′B′C′ ⊗ ˜ρA′′ ˜ρB′′ ˜ρC′′ = −|φ0⟩⟨φ0|A′B′C′ ⊗ ˜ρA′′ ˜ρB′′ ˜ρC′′,  (B89)  where for simplicity, we are representing the index 000 as 0 and the symbol of the tensor products are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Notice that the following relations hold true  XA′ZB′ZC′|φ000⟩A′B′C′ = |φ011⟩A′B′C′,  (B90a)  XA′ZB′YC′|φ000⟩A′B′C′ = i|φ010⟩A′B′C′,  (B90b)  XA′YB′ZC′|φ000⟩A′B′C′ = i|φ001⟩A′B′C′,  (B90c)  XA′YB′YC′|φ000⟩A′B′C′ = −|φ000⟩A′B′C′  (B90d)  where |φl⟩ for any l can be found in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Using these relations and the condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B89), we get four  different relations  R1,B′′ ˜ρB′′ ⊗ S1,C′′ ˜ρC′′ = 0,  R1,B′′ ˜ρB′′ ⊗ S3,C′′ ˜ρC′′ = 0,  R3,B′′ ˜ρB′′ ⊗ S1,C′′ ˜ρC′′ = 0,  (B91)  and  ˜ρA′′ ⊗ R3,B′′ ˜ρB′′ ⊗ S3,C′′ ˜ρC′′ = ˜ρA′′ ⊗ ˜ρB′′ ⊗ ˜ρC′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B92)  All three reduced density matrices ˜ρs′′ (s = A, B, C) are full rank, it follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B92) that R3 and S3 are non-  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Consequently, the last two relations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B91) imply that S1 = 0 and R1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Analogously, after plugging A2  as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B85) and C2 = Y ⊗ S3 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B71) we infer that Q1 = 0 and  Q3,A′′ ˜ρA′′ ⊗ S3,C′′ ˜ρC′′ = ˜ρA′′ ⊗ ˜ρC′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B93)  Thus, we obtain that  A′  2 = Y ⊗ Q,  B′  2 = Y ⊗ R,  C′  2 = Y ⊗ S,  (B94)  where Q, R and S are some hermitian matrices such that Q2 = 1, R2 = 1 and S2 = 1, which makes them also  unitary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' for simplicity we dropped the subscripts from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Let us now exploit the fact that Q, R and S are hermitian and square to the identity to decompose them as  Q = Q+ − Q−,  R = R+ − R−,  S = S+ − S−,  where Q±, R± and S± are projectors onto the eigenspaces of A′  2, B′  2 and C′  2 corresponding to eigenvalues ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Now  using the fact that Q2 = Q+ + Q− = 1 and S2 = S+ + S− = 1 and tracing the A′′ out, we obtain from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B92) that  2(R+ ˜ρB′′) ⊗ (S+ ˜ρC′′) = (R+ ˜ρB′′) ⊗ ˜ρC′′ + ˜ρB′′ ⊗ (S+ ˜ρC′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (B95)  One concludes from the above relation that (R+ ˜ρB′′) ⊗ (S− ˜ρC′′) = 0 and (R− ˜ρB′′) ⊗ (S+ ˜ρC′′) = 0, which by taking  into account the fact that both ˜ρB′′ and ˜ρC′′ are full rank, implies that either R+ = 0 and S+ = 0 or R− = 0 and  S− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' In the same spirit, one can exploit the second relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B93) to conclude that either Q+ = 0 and  S+ = 0 or Q− = 0 and S− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Taking into account all the possibilities listed above, one deduces that either  Q− = R− = S− = 0 (in which case Q+ = 1A′′, R+ = 1B′′ and S+ = 1C′′) or Q+ = R+ = S+ = 0 (in which case  Q− = 1A′′, R− = 1B′′ and S− = 1C′′), which directly leads us to two possible forms that the observables A′  2, B′  2 and  C′  2 can take:  A2 = YA′ ⊗ 1A′′,  B2 = YB′ ⊗ 1B′′,  C2 = YC′ ⊗ 1C′′  (B96)  or  A2 = −YA′ ⊗ 1A′′,  B2 = −YB′ ⊗ 1B′′,  C2 = −YC′ ⊗ 1C′′,  (B97)  from which one recovers Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B66) and (B67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  16  Appendix C: Self-testing the three-qubit NLWE basis  In this section, we show that using the certified states in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B9) generated by the sources Pi (i = 1, 2, 3) and  measurements in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B10) and (B11) along with some additional statistics, one can self-test the measurement  corresponding to the input e = 1 with the central party to be NLWE basis given below in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (C3) upto some  additional degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Before proceeding, let us denote the eigenvectors of (X + Z)/  √  2 as  |0⟩ = cos(π/8)|0⟩ + sin(π/8)|1⟩,  |1⟩ = − sin(π/8)|0⟩ + cos(π/8)|1⟩,  (C1)  and the eigenvectors of (X − Z)/  √  2 as  |+⟩ = sin(π/8)|0⟩ + cos(π/8)|1⟩,  |−⟩ = cos(π/8)|0⟩ − sin(π/8)|1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (C2)  Recall also that the NLWE measurement MNLWE = {|δl⟩⟨δl|} is defined via the following fully product vectors  |δ0⟩ = |0⟩|1⟩|+⟩,  |δ1⟩ = |0⟩|1⟩|−⟩,  |δ2⟩ = |+⟩|0⟩|1⟩,  |δ3⟩ = |−⟩|0⟩|1⟩,  |δ4⟩ = |1⟩|+⟩|0⟩,  |δ5⟩ = |1⟩|−⟩|0⟩,  |δ6⟩ = |0⟩|0⟩|0⟩,  |δ7⟩ = |1⟩|1⟩|1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (C3)  Notice that the above vectors are equivalent to the standard NLWE vectors introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' [27] up to a local  unitary transformation applied to the first qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  To certify the second Eve’s measurement E1, the correlations observed by the parties {p(a, b, c, e|x, y, z, 1)} must  satisfy the following conditions,  p(0, 1, 0, 0|0, 0, 1, 1) = 1  8,  p(0, 1, 1, 1|0, 0, 1, 1) = 1  8,  p(0, 0, 1, 2|1, 0, 0, 1) = 1  8,  p(1, 0, 1, 3|1, 0, 0, 1) = 1  8,  p(1, 0, 0, 4|0, 1, 0, 1) = 1  8,  p(1, 1, 0, 5|0, 1, 0, 1) = 1  8,  p(0, 0, 0, 6|0, 0, 0, 1) = 1  8,  p(1, 1, 1, 7|0, 0, 0, 1) = 1  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (C4)  Notice that the above distribution can be realised if the sources Pi (i = 1, 2, 3) generate the maximally entangled  state of two qubits |φ+⟩ = (1/  √  2)(|00⟩ + |11⟩) and the measurement E1 is exactly MNLWE = {|δl⟩⟨δl|}, whereas the  external parties perform the following measurements  A0 = X + Z  √  2  ,  A1 = X − Z  √  2  ,  B0 = Z,  B1 = X,  C0 = Z,  C1 = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (C5)  Next, we show that the above probabilities along with certification of the states and measurements presented in  Theorem 1 are sufficient to fully characterise the unknown measurement E1 = {Rl|1} where Rl denotes the measure-  ment element corresponding to outcome l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' The theorem stated below is labelled as Theorem 2 in the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Assume that the correlations ⃗p generated in the network satisfy the assumptions of Theorem 1 as well as the  conditions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (C4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Then, for any l it holds that  URl|1U† = |δl⟩⟨δl|E′ ⊗ 1E′′,  (C6)  where U is the same unitary as in Theorem 1 and E = ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' For simplicity, we represent Rl|1 as Rl throughout the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Let us first consider the first relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (C4),  that is,  p(0, 1, 0, 0|0, 0, 1, 1) = 1  8  (C7)  17  and then expand it by using the fact that the observables Ai, Bi, Ci for i = 0, 1 are certified as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' This gives  us  ��  �  s=A,B,C  U†  s  �  �|0⟩⟨0|A′ ⊗ |1⟩⟨1|B′ ⊗ |+⟩⟨+|C′ ⊗ 1A′′B′′C′′  �  �  �  s=A,B,C  Us  �  ⊗ R0  �  ψABCE  = 1  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (C8)  Let us then use Theorem 1 to represent the global state |ψABCE⟩ as [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B9)]  |ψABCE⟩ =  �  s  |ψss⟩ =  �  s  U†  s ⊗ U†  s |φ+  s′s′⟩ ⊗ |ξs′′s′′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (C9)  Notice also that by virtue of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B55) the junk states |ξs′′s′′⟩ can be represented as  |ξs′′s′′⟩ = (1s′′ ⊗ Ps′′)|φ+  d′′s ⟩s′′s′′,  (C10)  where Ps′′ =  �  d′′s σs′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' The joint state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (C9) can be further written as [c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B42)],  �  �  s=A,B,C  Us ⊗ Us  �  |ψABC|E⟩ =  �  PA′′ ⊗ PB′′ ⊗ PC′′  �  |φ+  8d′′  Ad′′  Bd′′  C⟩ABC|E  (C11)  where the bipartition is between the subsystem ABC and E ≡ ABC and the local dimension of the state is 8d′′  Ad′′  Bd′′  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  After plugging this state into the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (C8), and then using the fact that (1 ⊗ Q)|φ+  d ⟩ = (QT ⊗ 1)|φ+  d ⟩ for any matrix  Q, we get that  ��  |0⟩⟨0| ⊗ |1⟩⟨1| ⊗ |+⟩⟨+| ⊗  �  PT  A′′  �2  ⊗  �  PT  B′′  �2  ⊗  �  PT  C′′  �2�  RT  0 ⊗ 1E  �  |φ+  8dAdBdC ⟩ABC|E  = 1  8,  (C12)  where  R0 =  �  �  s=A,B,C  Us  �  R0  �  �  s=A,B,C  U†  s  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (C13)  Expanding the above term, we arrive at  1  d′′  Ad′′  Bd′′  C  Tr  ��  |0⟩⟨0| ⊗ |1⟩⟨1| ⊗ |+⟩⟨+| ⊗  �  PT  A′′  �2  ⊗  �  PT  B′′  �2  ⊗  �  PT  C′′  �2�  RT  0  �  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (C14)  Then using the fact that Ps′′ =  �  d′′s σs′′ for any s, we obtain  Tr  ��  |0⟩⟨0| ⊗ |1⟩⟨1| ⊗ |+⟩⟨+| ⊗ σT  A′′ ⊗ σT  B′′ ⊗ σT  C′′  �  RT  0  �  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (C15)  Recall that one can characterise measurements only on the support of the state or equivalently the states σs′′ are  full-rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Since RT  0 acts on the Hilbert space C8 ⊗ HA′′B′′C′′, we can express it using the basis given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (C3) as  RT  0 =  7  ∑  l,l′=0  |δl⟩⟨δl′| ⊗ ˜Rl,l′  (C16)  where ˜Rl,l′ act on HA′′B′′C′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Then, using the cyclic property of trace and Lemma 1 we conclude from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (C15) that  ˜R0,0 = 1 and thus we finally get that  RT  0 = |0⟩⟨0| ⊗ |1⟩⟨1| ⊗ |+⟩⟨+| ⊗ 1A′′B′′C′′ + L0  = |δ0⟩⟨δ0|A′B′C′ ⊗ 1A′′B′′C′′ + L0,  (C17)  where L0 stands for an operator given by  L0 =  7  ∑  l,l′=0  l̸=0, l′̸=0  |δl⟩⟨δl′| ⊗ ˜Rl,l′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (C18)  18  Let us now show that L0 is positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' For this purpose, we consider the state |δ0⟩ ⊗ |ξ⟩, where |ξ⟩ is an  arbitrary state from HA′′B′′C′′, and act on this state with RT  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Taking into account Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (C17) and (C18), we obtain  RT  0 |δ0⟩|ξ⟩ = |δ0⟩|ξ⟩ +  7  ∑  l=1  |δl⟩ ˜Rl,0|ξ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (C19)  Now, multiplying the above formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (C19) with its complex conjugate we obtain that  ⟨δ0|⟨ξ|  �  RT  0  �2  |δ0⟩|ξ⟩ = 1 +  7  ∑  l=1  ⟨ξ| ˜R†  l,0 ˜Rl,0|ξ⟩  (C20)  which after using the fact that  �  RT  0  �2  ≤ RT  0 ≤ 1, gives us  7  ∑  l=1  ⟨ξ| ˜R†  l,0 ˜Rl,0|ξ⟩ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (C21)  As ˜R†  l,0 ˜Rl,0 ≥ 0, it follows that ⟨ξ| ˜R†  l,0 ˜Rl,0|ξ⟩ = 0 for any |ξ⟩ ∈ HA′′B′′C′′, and consequently ˜Rl,0 = 0 for any l =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' , 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Since L0 is hermitian, the above implies also that ˜R0,l = 0 for l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' , 7, and hence  L0 =  7  ∑  l,l′=1  |δl⟩⟨δl′| ⊗ ˜Rl,l′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (C22)  This means that RT  0 can now be expressed in the block form as  RT  0 = |δ0⟩⟨δ0| ⊗ 1A′′B′′C′′ + L0 =  �  1  0  0 L0  �  ,  (C23)  where both components act on orthogonal subspaces of the three-qubit Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Owing to the fact that RT  0 ≥ 0,  we thus obtain L0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Similar analysis using all the other probabilities in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (C4) can be done for the other operators RT  l , and thus we  arrive at  RT  l = |δl⟩⟨δl| ⊗ 1A′′B′′C′′ + Ll  (l = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' , 7),  (C24)  where Ll is a positive semi-definite operator that with respect to A′B′C′ subsystem is defined a subspace of C8 which  is orthogonal to |δl⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Now, the fact that ∑l RT  l = 1 implies that  ∑  l  Ll = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (C25)  As each Ll is positive semi-definite, the only way the above condition is satisfied is that Ll = 0 for any l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Thus, we  finally obtain from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (C24) that  Rl = |δl⟩⟨δl| ⊗ 1,  (C26)  which by taking into account Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (C13) gives the desired result form Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (C6), completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Appendix D: Self-testing the measurement constructed from the UPB  Let us finally provide the proof of Theorem 4 stated in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' To this end, we recall that the measurement  constructed from the UPB is defined as MUPB = {|τ0⟩⟨τ0|, |τ1⟩⟨τ1|, |τ2⟩⟨τ2|, |τ3⟩⟨τ3|, Γ}, where  |τ0⟩ = |0⟩|1⟩|+⟩,  |τ1⟩ = |+⟩|0⟩|1⟩  |τ2⟩ = |1⟩|+⟩|0⟩,  |τ3⟩ = |−⟩|−⟩|−⟩,  (D1)  19  and  Γ = 1 −  3  ∑  i=0  |τi⟩⟨τi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D2)  Notice that |τi⟩ form a four-element UPB which is obtained from the UPB introduced in [28] by applying a local  unitary to Alice’s qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' to self-test the five-outcome measurement E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' the correlations observed by the parties {p(abce|xyz2)} must  satisfy  p(0100|0012) = 1  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  p(0011|1002) = 1  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  p(1002|0102) = 1  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  p(1113|1112) = 1  8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D3)  as well as the following conditions that are expressed in the correlation picture:  ⟨(1 + A0B0 + B0C0 + A0C0 + A1B1C1 − A1B2C2 − A2B1C2 − A2B2C1) R4⟩ = 1  (D4a)  �  (1 + A0)  �  −B1C1 + B2C2 + B1(1 + C0) − 1  2(1 − B0)C1 + (1 + B0)C0 + 1  2 B0 + 3  21  �  R4  �  = 3  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D4b)  �  (1 + B0)  �  −A1C1 + A2C2 + (1 + A0)C1 − 1  2 A1(1 − C0) + A0(1 + C0) + 1  2C0 + 3  21  �  R4  �  = 3  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D4c)  �  (1 + C0)  �  −A1B1 + A2B2 + A1(1 + B0) − 1  2(1 − A0)B1 + (1 + A0)B0 + 1  2 A0 + 3  21  �  R4  �  = 3  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D4d)  where E2 = {Rl}4  l=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Notice that the above conditions are met in a situation in which the sources Pi (i = 1, 2, 3) distribute the state |φ+  2 ⟩  and Eve’s measurements E2 is the ideal measurement MUPB given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D1), whereas the external parties measure  the following observables  A0 = X + Z  √  2  ,  A1 = X − Z  √  2  ,  A2 = Y,  B0 = Z,  B1 = X,  B2 = Y,  C0 = Z,  C1 = X  C2 = Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D5)  In what follows we show that Theorem 1 and Theorem 3 together with the conditions in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D3) and (D4) enable  self-testing MUPB is E2 = {Rl} where Rl denotes the measurement element corresponding to outcome l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' The theorem  stated below is labelled as Theorem 3 in the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Assume that the correlations ⃗p observed in the network satisfy the assumptions of Theorems 1 and 3 as well as  conditions in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D3) and (D4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Then,  (UA ⊗ UB ⊗ UC) Rl|2 (UA ⊗ UB ⊗ UC)† = |τl⟩⟨τl|E′ ⊗ 1E′′  (l = 0, 1, 2, 3),  (D6)  and,  (UA ⊗ UB ⊗ UC) R4|2 (UA ⊗ UB ⊗ UC) = ΓE′ ⊗ 1E′′,  (D7)  where the unitary operations Us (s = A, B, C) are the same as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' For simplicity, we represent Rl|2 as Rl throughout the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Let us first consider the conditions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  As was done in the previous section in the proof of Theorem 4, we can conclude from these conditions that for an  unknown measurement {Rl} we have that  RT  l = |τl⟩⟨τl| ⊗ 1A′′B′′C′′ + Ll  l = 0, 1, 2, 3  (D8)  where  Rl =  �  �  s=A,B,C  Us  �  Rl  �  �  s=A,B,C  U†  s  �  l = 0, 1, 2, 3, 4  (D9)  and Ll ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Let us now consider Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D4a)  ⟨(1 + A0B0 + B0C0 + A0C0 + A1B1C1 − A1B2C2 − A2B1C2 − C1A2B2) R4⟩ = 1  (D10)  20  and expand it by using the fact that the observables Ai, Bi, Ci for i = 0, 1, 2 are certified as in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B11) and (B66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  This gives us  ��  �  s=A,B,C  U†  s  �  S1,A′B′C′ ⊗ 1A′′B′′C′′  �  �  s=A,B,C  Us  �  ⊗ R4  �  = 1  (D11)  where  S1,A′B′C′ = 1A′B′C′ + (X + Z)A′  √  2  ZB′ + ZB′ZC′ + (X + Z)A′  √  2  ZC′ + (X − Z)A′  √  2  XB′XC′ − (X − Z)A′  √  2  YB′YC′ − YA′XB′YC′  − YA′YB′XC′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D12)  The states generated by the sources P1, P2, P3 have already been certified to be of the form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Thus, the joint  state of all the parties can be represented as  |ψABCE⟩ =  �  s  |ψss⟩ =  �  s  (U†  s ⊗ U†  s )|φ+⟩s′s′|ξs′′s′′⟩  (D13)  which can be simplified to  �  �  s  Us ⊗ Us  �  |ψABC|E⟩ =  �  PA′′ ⊗ PB′′ ⊗ PC′′  �  |φ+  8d′′  Ad′′  Bd′′  C⟩ABC|E  (D14)  where Ps′′ =  �  d′′s σs′′ [c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (C9)-(C11)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Now, following exactly the same steps from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (C12)-(C15), we obtain  from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D11) that  1  8Tr  ��  S1,A′B′C′ ⊗ σT  A′′ ⊗ σT  B′′ ⊗ σT  C′′  �  RT  4  �  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D15)  The state |ψ1,4⟩ given by  |ψ1,4⟩ =  1  √  2  (|000⟩ + |111⟩)  (D16)  and then one can check that  S1,A′B′C′ = 8|ψ1,4⟩⟨ψ1,4|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D17)  The above fact also explains why we imposed the condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Taking into account Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D17) we can  rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D15) as  Tr  �(|ψ1,4⟩⟨ψ1,4|A′B′C′ ⊗ σA′′ ⊗ σB′′ ⊗ σC′′) R4  � = 1,  (D18)  where we have also used the fact that Tr[XTYT] = Tr[XY] for any pair of matrices X and Y and that the state |ψ1,4⟩ is  real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Consider now an orthonormal basis {|ϕl⟩} in C8 in which |ϕl⟩ = |τl⟩ for l = 0, 1, 2, 3, where |τl⟩ form the UPB  given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D1), and |φl⟩ = |ψ1,l⟩ (l = 4, 5, 6, 7), where |ψ1,4⟩ is given in (D16) whereas the remaining vectors are  defined as  |ψ1,5⟩ =  1  √  6(−2|000⟩ − |010⟩ + |011⟩),  |ψ1,6⟩ =  1  √  6  �−2|000⟩ − |001⟩ + |101⟩  �  ,  |ψ1,7⟩ =  1  √  6  �−2|000⟩ − |100⟩ + |110⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D19)  Notice that ∑7  l=4 |ψ1,l⟩⟨ψ1,l| = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Since R4 acts on the Hilbert space C8 ⊗ HA′′B′′C′′, we can express it using the above  mentioned basis as  R0 =  7  ∑  l,l′=0  |ϕl⟩⟨ϕl′| ⊗ ˜Rl,l′,  (D20)  21  where ˜Rl,l′ are some matrices acting on HA′′B′′C′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Recalling that the local states are full-rank and then using the cyclic  property of trace and Lemma 1 we conclude from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (C15) that ˜R0,0 = 1 and thus we get that  R4 = |ψ1,4⟩⟨ψ1,4| ⊗ 1A′′B′′C′′ + L4,  (D21)  where  L4 =  7  ∑  l,l′=0  l=l′̸=4  |ϕl⟩⟨ϕl′| ⊗ ˜Rl,l′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D22)  Let us now consider Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D4b)  �  (1 + A0)  �  −B1C1 + B2C2 + B1(1 + C0) − 1  2(1 − B0)C1 + (1 + B0)C0 + 1  2 B0 + 3  21  �  R4  �  = 3  2  (D23)  and then expand it by using the fact that the observables Ai, Bi, Ci for i = 0, 1, 2 are certified as in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B11) and  (B66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' This gives us  ��  �  s=A,B,C  U†  s  �  S2,A′B′C′ ⊗ 1A′′B′′C′′  �  �  s=A,B,C  Us  �  ⊗ R4  �  = 3  4  (D24)  where  S2,A′B′C′ = |0⟩⟨0|A′  �  −XB′XC′ + YB′YC′ + 2XB′|0⟩⟨0|C′ − |1⟩⟨1|B′XC′ + 2|0⟩⟨0|B′ZC′ + 1  2ZB′ + 3  21A′B′C′  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D25)  It is direct to verify that S2,A′B′C′ is proportional to the projection onto |ψ1,5⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' precisely,  S2,A′B′C′ = 6|ψ1,5⟩⟨ψ1,5|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D26)  Using then the above form of S2,A′B′C′ as well as the fact that |ψ1,5⟩ is real, we can simplify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D24) to  Tr  �(|ψ1,5⟩⟨ψ1,5| ⊗ σA′′ ⊗ σB′′ ⊗ σC′′) R4  � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D27)  Now, expanding R4 using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D21), we can conclude that  R4 = |ψ1,4⟩⟨ψ1,4| ⊗ 1A′′B′′C′′ + |ψ1,5⟩⟨ψ1,5| ⊗ 1A′′B′′C′′ + L5,  (D28)  where  L5 =  7  ∑  l,l′=0  l=l′̸=4,5  |ϕl⟩⟨ϕl′| ⊗ ˜Rl,l′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D29)  Similarly, we can conclude from the other two conditions in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D4c) and (D4d), and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D28) that  R4 =  7  ∑  i=4  |ψ1,i⟩⟨ψ1,i| ⊗ 1A′′B′′C′′ + L′  = Γ ⊗ 1A′′B′′C′′ + L′,  (D30)  where  L′ =  7  ∑  l,l′=0  l=l′̸=4,5,6,7  |ϕl⟩⟨ϕl′| ⊗ ˜Rl,l′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D31)  Now, following the same steps as between Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (C19)-(C23) using the states |ϕl⟩|ξ⟩ for l = 4, 5, 6, 7 and any |ξ⟩ ∈  HA′′B′′C′′ we can conclude that ˜Rl,l′ = 0 for any l ̸= l′ such that l′, l = 4, 5, 6, 7, and L′ simplifies to  L′ =  3  ∑  l,l′=0  |ϕl⟩⟨ϕl′| ⊗ ˜Rl,l′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D32)  22  The fact that R4 ≥ 0 implies that L′ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Now, adding Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D8) and (D30) and using the fact that ∑l Rl = 1, we get  that  3  ∑  l=0  Ll + L′ = 0,  (D33)  which, after taking into account the fact that Ll as well as L′ are positive semi-definite, implies that Ll = L′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Thus, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D8) and (D30), we obtain  Rl = |τl⟩⟨τl|E′ ⊗ 1E′′  (l = 0, 1, 2, 3)  (D34)  and,  R4 =  �  1 −  3  ∑  i=0  |τi⟩⟨τi|  �  E′  ⊗ 1E′′ = ΓE′ ⊗ 1E′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D35)  By virtue of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D9) we finally arrive at the desired forms of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D6) and (D7), completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  An interesting consequence of the above theorem is the certification of bound entangled state when Eve observes  the final outcome of her measurement E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' Assume that the states are certified as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B9) and Eve’s measurement E2 is certified as in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D6) and (D7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Consequently, when Eve observes the last outcome of her measurement, the post-measurement state with the external parties is  given by  U ρABC U† = 1  4ΓA′B′C′ ⊗ ˜ρA′′B′′C′′,  (D36)  where U = �  s Us and the unitaries Us are the same as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' The post-measurement state when Eve observes the last outcome of her measurement E2 is given by  ρABC =  1  P(4|e = 2)TrE  ��  1ABC ⊗ R4|2  �  �  s=A,B,C  |ψss⟩⟨ψss|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D37)  Now, substituting the states |ψss⟩ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (B9) and the measurement element R4|2 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' (D7) and then using the  fact that P(4|e = 2) = 1/2, we get that  U ρABC U† = 2 TrE  �  (1ABC ⊗ ΓE′ ⊗ 1E′′)  �  s=A,B,C  |φ+⟩⟨φ+|s′s′ ⊗ |ξs′′s′′⟩⟨ξs′′s′′|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content='  (D38)  Again using the identity (1 ⊗ Q)|φ+⟩ = (QT ⊗ 1)|φ+⟩, we get  U ρABC U† = 2 TrE  �  (ΓA′B′C′ ⊗ 1A′′B′′C′′ ⊗ 1E)  �  s=A,B,C  |φ+⟩⟨φ+|s′s′ ⊗ |ξs′′s′′⟩⟨ξs′′s′′|  �  ,  (D39)  where we also used the fact that ΓT = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
+page_content=' After tracing the E subsystem we arrive at  U ρABC U† = 1  4ΓA′B′C′ ⊗ ˜ρA′′B′′C′′,  (D40)  where ˜ρA′′B′′C′′ = TrE′′  ��  s=A,B,C |ξs′′s′′⟩⟨ξs′′s′′|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFIT4oBgHgl3EQf-Cuq/content/2301.11409v1.pdf'}
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+ 
+ 
+ © 2022 Md. Ahsan Habib, M. A. H. Akhand and Md. Abdus Samad Kamal. This open-access article is distributed under a 
+Creative Commons Attribution (CC-BY) 4.0 license. 
+ Journal of Computer Science 
+ 
+ 
+ 
+Original Research Paper  
+Emotion Recognition from Microblog Managing Emoticon 
+with Text and Classifying using 1D CNN 
+ 
+1Md. Ahsan Habib, 1M. A. H. Akhand and 2Md. Abdus Samad Kamal  
+ 
+1Department of Computer Science & Engineering, Khulna University of Engineering & Technology, Bangladesh 
+2Graduate School of Science and Technology, Gunma University, Japan 
+ 
+Article history 
+Received: 17-10-2022 
+Revised: 17-11-2022 
+Accepted: 24-11-2022 
+ 
+Corresponding Author:  
+M. A. H. Akhand 
+Department of Computer 
+Science & Engineering, Khulna 
+University of Engineering & 
+Technology, Bangladesh 
+Email: akhand@cse.kuet.ac.bd 
+Abstract: Microblog, an online-based broadcast medium, is a widely used 
+forum for people to share their thoughts and opinions. Recently, Emotion 
+Recognition (ER) from microblogs is an inspiring research topic in diverse 
+areas. In the machine learning domain, automatic emotion recognition from 
+microblogs is a challenging task, especially, for better outcomes considering 
+diverse content. Emoticon becomes very common in the text of microblogs 
+as it reinforces the meaning of content. This study proposes an emotion 
+recognition scheme considering both the texts and emoticons from microblog 
+data. Emoticons are considered unique expressions of the users' emotions and 
+can be changed by the proper emotional words. The succession of emoticons 
+appearing in the microblog data is preserved and a 1D Convolutional Neural 
+Network (CNN) is employed for emotion classification. The experimental 
+result shows that the proposed emotion recognition scheme outperforms the 
+other existing methods while tested on Twitter data. 
+ 
+Keywords: Deep Learning, CNN, Emotion Recognition, Emoticons 
+ 
+Introduction 
+Emotion often refers to a complex state of feeling such 
+as happiness, joy, anger, disgust, fear, love, and hatred 
+that occurs in physical and psychological changes and has 
+an impact on one's thinking and actions. Emotions can 
+have a significant impact on people’s lives. People's 
+emotions can be expressed in a variety of ways, including 
+speech, facial expressions, bodily gestures, verbal 
+expressions using text, and so on (Castellano et al., 2008; 
+Murugappan et al., 2021; Singh et al., 2020). Different 
+social media platforms like Facebook, Twitter, Instagram, 
+WhatsApp, Sina Weibo, etc. have grown in popularity in 
+recent years where people express their emotions and 
+thoughts (Che et al., 2021; Wang et al., 2016). Users share 
+millions of tweets and posts every day. Therefore, social 
+media contents are the most prospective sources for 
+understanding emotional states and human thoughts.  
+Microblog, an online-based broadcast medium, is a 
+widely used forum for people to share their thoughts and 
+opinions. 
+In 
+our 
+daily lives, 
+microblogs 
+make 
+communication easier. Twitter (Java et al., 2007), 
+LinkedIn, Facebook, Instagram, Snapchat, Tumblr, 
+WhatsApp, etc. are the most popular microblogs. 
+Globally, there are 3.5 billion users on social media, 
+according to 2019 social media projections which equate 
+to approximately 45% of the current population and this 
+figure is only increasing (Mohsin, 2020). Any post can be 
+hit immediately by a large number of people through 
+microblogs. People express their thoughts by sharing 
+posts that reflect one's sentiments.  
+Emotion Recognition (ER) from microblog data is the 
+most promising and challenging research finding in the 
+field of information and communication technology. 
+Unimodal, bimodal, and multimodal are the three possible 
+categories of ER. Unimodal emotion recognition uses 
+only one type of information, such as facial expression, 
+text, or speech, whereas bimodal emotion recognition uses 
+both speech and facial expression. This study introduces the 
+unimodal emotion recognition method to determine one’s 
+Emotion Category (EC). Machine Learning (ML) based 
+approach and knowledge-based approach (Chaffar and 
+Inkpen, 2011) are two major techniques for identifying or 
+recognizing emotions. The knowledge-based technique also 
+known as the lexicon-based technique uses a set of rules to 
+detect emotion from given data (Nirenburg and Mahesh, 
+1997) whereas the ML-based technique uses a model for 
+learning the patterns from features generated from microblog 
+data (Kotsiantis et al., 2006).  
+Deep Learning (DL) based approaches for emotion 
+recognition 
+from 
+microblog 
+data 
+have 
+emerged 
+remarkably and shown promising results nowadays. DL 
+employs different architectures to learn the patterns in 
+data. There are two major steps in DL-based emotion 
+
+Science
+PublicationsMd. Ahsan Habib et al. / Journal of Computer Science 2022, 18 (12): 1170.1178 
+DOI: 10.3844/jcssp.2022.1170.1178 
+ 
+1171 
+recognition from microblog data: (i) Processing data and 
+(ii) classifying them using the proper DL model. Firstly, 
+the collected data is processed, transformed, and 
+represented in the appropriate format for the envisioned 
+DL model. Secondly, a DL model is prepared or trained 
+with the data to classify emotion. Along with different 
+processing techniques, different DL models are investigated 
+in the last several years for emotion recognition 
+(Batbaatar et al., 2019; Guo et al., 2021a). These DL-based 
+methods run on preprocessed data and do not include explicit 
+functionality (Batbaatar et al., 2019; Yang et al., 2018). 
+Some experiments consider emoticons along with the text as 
+input data, while others only consider the text.  
+This research aims to develop an improved CNN 
+based emotion recognition model from microblogs 
+considering both text and emoticons. This study takes into 
+account emoticons as they have a substantial significance to 
+users’ emotions and represented them by corresponding 
+emotional words. The word and emoticon sequences that 
+appeared in the microblog are preserved in this study. 
+Considering all other pre-processing steps, CNN is applied 
+for emotion recognition since it is pertinent for sequential 
+data classification. Experiments using Twitter data on texts 
+with emoticons and texts-only showed the effectiveness of 
+the proposed CNN approach, with better classification 
+accuracy considering both emoticons and texts compared to 
+classification accuracy when only considering texts.  
+The remaining paper is structured as follows. At first, it 
+reviews different existing works related to emotion 
+recognition, and then the proposed CNN method is 
+explained. After that, the experimental studies and 
+results are demonstrated. Lastly, the paper concludes 
+with a few remarks. 
+Literature Review 
+Many DL-based approaches are examined in the last 
+several years for emotion recognition from microblog data 
+(Islam et al., 2020; Mehta et al., 2021). Attention model  
+(Wei et al., 2019; Yuan and Zhang, 2021), BERT RCNN 
+(Pan and Xu, 2021), CNN (Xu et al., 2020), GRU (Liu et al., 
+2021), LSTM (Arun et al., 2019; Batbaatar et al., 2019;      
+Guo et al., 2021b; Islam et al., 2021), graph convolution 
+network (Lai et al. 2020), etc., are most prominent 
+techniques employed in this research domain. Most of the 
+methods only consider textual data (Xu et al., 2020) and a 
+few consider both text and emoticons (Islam et al., 2020).  
+Yang et al. (2018) developed an enhanced CNN method 
+considering both emoticons and texts. They represented the 
+emoticons and words as two separate vectors and projected 
+into one emotional space. Then CNN is employed for 
+emotion classification. The proposed model is applied to the 
+Twitter dataset, NLPCC2013, and Weibo dataset.  
+The work proposed by Islam et al. (2020) also took 
+both the emoticons and texts as input where it applied 
+LSTM to classify emotions. The Twitter dataset was 
+used to measure the efficacy of the model; however, the 
+dataset was small enough. The work was then extended 
+by Islam et al. (2021) and achieved remarkable accuracy 
+with a relatively large dataset.  
+Batbaatar et al. (2019) introduced a Semantic Emotion 
+Neural Network (SENN) model that includes both CNN 
+and BiLSTM for ER. Here, the CNN model focused on 
+the emotional connectivity between words after extracting 
+emotional features, while the BiLSTM was employed to 
+build the semantic relationship after collecting contextual 
+information. The SENN used Twitter data and other social 
+media data (only text) without specifying whether or not 
+emoticons were used in the decision-making process.  
+Wei et al. (2019) developed an emotion recognition 
+approach by incorporating both the dual attention 
+mechanism and Bidirectional Long Short-term Memory 
+(BiLSTM). In their work, they first used the BiLSTM model 
+to semantically encode the microblog data and then 
+introduced the sentiment word attention and self-attention 
+into the BiLSTM model. Lastly, they used the Softmax 
+classifier to classify the sentiment of microblogs. Chinese 
+microblog 
+Sina 
+Weibo-based 
+NLPCC2013 
+and 
+NLPCC2014 datasets are used for the experimental purpose.  
+Another attention-based dual-channel microblog 
+emotion recognition model is developed by Yuan and 
+Zhang (2021). This study used RoBERTa-WWM and the 
+multi-head attention model for their work. They 
+constructed the emotional knowledge set of each sentence 
+extending the emotional resource library and used the pre-
+training model RoBERTa-WWM for feature representation. 
+After that, the Text CNN-BiGRU network and a Multi-
+Head Attention network took the sentence feature and the 
+emotional knowledge as input to obtain deeper semantics 
+features and attention features of emotional knowledge. 
+And finally, the semantic feature and the emotional 
+knowledge attention feature are combined to train the 
+model. Chinese microblog NLPCC2014 dataset is used to 
+show the efficacy of the model.  
+Pan and Xu (2021) developed a deep learning-based 
+sentiment analysis model employing BERT RCNN for 
+netizens during public health emergencies. They used 
+BERT which uses static masking for fine-tuning the input 
+data and trained them into vectors to represent it. Then it 
+took the trained vectors as the input features of the 
+upstream model and learned the microblog data features 
+through RCNN network. In contrast, the work proposed 
+by Yuan and Zhang (2021) used dynamic masking 
+incorporated in RoBERTa making the model more robust.  
+Another model, called Semantic Emoticon Emotion 
+Recognition (SEER) (Liu et al., 2021), used both the 
+attention mechanism and Bi-GRU (bidirectional gated 
+recurrent unit) network to classify emotion. Then they 
+constructed an emoticon distribution model to obtain 
+the emotion vectors.  
+Arun et al. (2019) developed EPUSAMCNN (Emotion-
+Prediction Using Semantic Analysis Multi-Dimensional 
+Convolutional Neural Network) model incorporating both 
+
+Md. Ahsan Habib et al. / Journal of Computer Science 2022, 18 (12): 1170.1178 
+DOI: 10.3844/jcssp.2022.1170.1178 
+ 
+1172 
+the LSTM (Long Short Term Memory Networks) and 
+MCNN 
+(Multi-Dimensional 
+Convolutional 
+Neural 
+Network). They used MCNN to increase their proficiency 
+in recognizing correct feelings for microblog data and 
+BiLSTM for classifying them.  
+Another model named ERNIE-BiLSTM is developed for 
+sentiment classification by Guo et al. (2021a). At first, they 
+used 
+ERNIE 
+(Knowledge 
+Enhanced 
+Semantic 
+Representation) pretrained model for word featuring. It 
+considered both the enhancement of the semantic 
+representation of words and preserves the contextual 
+information along with the polysemy of words also. After 
+training through ERNIE, they used BiLSTM for sentiment 
+classification. They experimented with their developed 
+model with the Chinese microblog Sina Weibo based 
+NLPCC2014 dataset.  
+An emotion classification model is developed in 2020 by 
+Lai et al. (2020). They used syntax based GCN (Graph 
+Convolution Network) model focusing on the diverse 
+grammatical structures. The accuracy of the model is 
+enhanced by a percentile-based pooling technique proposed 
+by them. They experimented with their developed model 
+with the Chinese microblog dataset on their own.  
+Emotion Recognition from Microblog 
+Managing Emoticon with Text  
+Social media has been the most common medium of 
+venting feelings in the era of globalization (Gräbner et al., 
+2012; Guo et al., 2021b; Wu et al., 2020; Xu et al., 2020). 
+People share their thoughts by posting videos, texts, 
+audio, photos, etc. to express emotions. Microblogs are 
+the most popular among them. Millions of words, images, 
+videos, audio, hashtags, and various signs and symbols 
+with various meanings can be found on microblogs. One 
+of the most widely used microblogging platforms is 
+Twitter. Since emoticons reinforce the meaning of 
+content, they should be given special consideration in 
+emotion recognition along with texts.  
+The emoticon and its interactions with texts are given 
+particular consideration in the proposed method. To 
+identify the true emotion of people, both emoticons and 
+text possess equal significance. Several pieces of research 
+in the literature have described emoticons as noisy inputs 
+that should be omitted during the pre-processing stage 
+(Hogenboom et al., 2013) but this should not be. The 
+proposed model did proper emotion analysis with the aid 
+of emotional words and other texts in the microblog. 
+The working procedure of the proposed emotion 
+recognition model is demonstrated in Fig. 1 for a sample 
+post with an emoticon. The proposed CNN scheme 
+contains four consecutive phases. A lookup table is used 
+in Task 1 to translate emoticons into corresponding 
+emotion words. In Task 2, Integer Encoding (IE), the Task 
+of converting words into a series of integers, is done. Then 
+in Task 3 padding is done to make an equal-length vector 
+sequence of integers. Finally, CNN is applied to recognize 
+specific emotions (Sad, Happy, Angry, or Love) in Task 4.  
+The whole methodology where four individual processes 
+are shown is described in Algorithm 1. Data processing and 
+CNN classification are two major tasks in this algorithm. The 
+first three processing steps are under data processing. In the 
+following subsections, the steps are described briefly.  
+ 
+Algorithm 1: Proposed ER scheme  
+ Input: Microblog Data D of Word Size N  
+ Output: Category of Emotion  
+ // Task 1: Replacing emoticon(s) to corresponding 
+meaning  
+ For t = 1 to N do  
+ 
+If (D[i] is emoticon(s)) then  
+ 
+D[i] ← Emoticon. meaning (D[i])  
+ 
+End If  
+ End For  
+ // Task 2: Integer Encoding (IE) using Tokenizer 
+ For t = 1 to N do  
+ 
+IE[i] = Tokenizer (D[i]) 
+ End For  
+ // Task 3: Zero padding at first to make fixed L length 
+ For t = 1 to L-N do 
+ 
+P[i] ← 0 // Considering 0 for initial values 
+ End For  
+ For i = L-N+1 to N do 
+ 
+P[i] ← IE[i] // Copy the rest values 
+ End For  
+ // Task 4: Emotion classification using CNN 
+ // Embedding integer to 2D vector 
+ For t = 1 to L do 
+ 
+U [x, y] = Embedding (P[i]) 
+ End For  
+ 
+Data Processing 
+One of the most critical phases in our developed 
+scheme is to process microblog data. Among several 
+social platforms' data, Twitter data is used including both 
+emoticons and text. Certain pre-processing steps are 
+required to eliminate unnecessary content and noisy input 
+from the data. Case conversion, user name removal, 
+hashtag, punctuation mark removal, and so on are all part 
+of the cleaning process. Then the clean microblog data 
+containing both texts and emoticons is processed into 
+three processes. Task 1 (emoticon conversion step) 
+searches microblog data for emoticon(s) and then uses the 
+Emoticon.meaning() function emoticons are replaced 
+with corresponding meaning. Individual emoticon word 
+meanings are stored in a lookup table used by the function. 
+Needless information (if any) is eliminated and IE is 
+achieved in the Tokenization step (Task 2) using the 
+function Text_to_sequence().  
+
+Md. Ahsan Habib et al. / Journal of Computer Science 2022, 18 (12): 1170.1178 
+DOI: 10.3844/jcssp.2022.1170.1178 
+ 
+1173 
+ 
+ 
+Fig. 1: The proposed ER architecture for a sample microblog data containing emoticons with texts 
+ 
+ 
+ 
+Fig. 2: 1D CNN architecture of the proposed ER 
+ 
+In Task 3, padding is done using the pad_sequence() 
+function to form equal length word vector sequence. Zero 
+padding at first is performed in this case. Lastly, in Task 
+4, the CNN model is applied to recognize particular 
+emotions (Happy, Love, Sad, or Angry).  
+ER using 1D CNN 
+CNN is a deep neural network very popular for 
+analyzing visual 2D images. CNN has multiple layers 
+having convolution and pooling operations. CNN's 
+convolutional layers provide a summary of an image's 
+features. By summarizing the presence of features of the 
+feature map, pooling layers down samples feature maps. 
+For images or image-like 2D inputs, the Conv2D layer 
+architecture is primarily used in standard CNN. Finally, a 
+fully connected layer is employed for classification 
+purposes called Dense Layer.  
+This study considers CNN architecture with 1D 
+convolutional operation on the blog text data mimicking 
+the idea from CNN operation on time series data. For 
+time-series data CNN uses Conv1D architecture        
+(Amo-Boateng, 2020). As text data is considered time-
+series data CNN employs the Conv1D architecture for 
+ER. The kernel in Conv1D slides in one dimensional way. 
+Two Conv1D layers along with two max-pooling layers, 
+a flattened layer, and two dense layers show promising 
+outcomes in emotion recognition from text data. 
+Generally, time-series data is used for forecasting or 
+single output prediction but in our proposed method the 
+output layer contains multiple nodes as emotion 
+recognition is considered a multiclass problem.  
+Figure 2 illustrates the CNN architecture of the 
+developed scheme for emotion recognition from microblog 
+data. CNN is popular for analyzing texts and recognizing 
+their features or patterns of them. The architecture of the 
+proposed scheme consists of an input layer, an embedding 
+layer, two convolutional layers, two maxpooling layers, a 
+flatten layer, two dense layers, and lastly the output layer. 
+The size of the words used in the proposed model is the input 
+dimension of the embedding layer and the output dimension 
+is 128. The first and second convolutional layers have 64 and 
+32 dimensionalities of output spaces, respectively. Both 
+convolutional layers use kernel size 3 and the relu activation 
+function. For both max-pooling layers, the max-pooling 
+window size is set to 2 and the pooling window moves 2 
+strides for each pooling step. The following flattened layer 
+flattens the input. The first dense layer contains 16 with a 
+relu activation function. As there are four possible class 
+labels, the proposed CNN architecture ends with a dense 
+layer with four nodes. To get the probability for each class 
+the softmax activation function is used. 
+
+My kids are smart
+My kids are
+and It makes me very
+Process 1:
+smart Grinning_facc and
+Process 2:
+happy
+Emoticon
+It makes me very happy
+Tokenization
+Convcrsion
+[nl,n2,..,n11]
+Detected Emotion:
+Process 4:
+[0, 0,...,nl, n2, .+., n10, nll
+Happy
+Classification
+Process 3:
+Padding
+using CNNO
+O
+InputLayer
+O
+Convolutional
+Max-pooling
+EmbcddingLaycr
+IDLayer1
+Laycr 1
+OutputLayer
+DenseLayer1
+Dense Layer 2
+Max-pooling
+Convolutional
+Flatten
+[Size32X16]
+[Size 16X4]
+Layer2
+IDLayer2
+LayerMd. Ahsan Habib et al. / Journal of Computer Science 2022, 18 (12): 1170.1178 
+DOI: 10.3844/jcssp.2022.1170.1178 
+ 
+1174 
+Results and Discussion  
+This 
+segment 
+demonstrates 
+microblog 
+data 
+preparation, experimental setup, experimental results, and 
+analysis of the proposed emotion recognition scheme.  
+Dataset Preparation 
+This study uses the Twitter dataset, a pool of English 
+tweets collected using the Twitter API. Firstly, tweeps, 
+a python library for downloading tweets from Twitter, 
+is used to collect tweets for this study purpose. The 
+language filtering is enabled by the Twitter API, which 
+permits the definition of the retrieved tweets' language. 
+To extract English Tweets, the optional language 
+parameter is set to 'en’ in the Twitter Search URL. 
+There are four emotion class labels in a total of 16011 
+tweets. The numeric notation of the class labels is as 
+follows: Sad, happy, love, and angry are represented by 
+1, 2, 3, and 4 respectively. 75% (12008 tweets) of the 
+collected data is used to train the proposed CNN 
+architecture and the rest 25% (4003 tweets) is used as 
+a test set in this study. Table 1 illustrates a snatch of 
+the tweets with their corresponding EC. Table 2 
+displays the 16 emoticons that are used in the proposed 
+scheme, along with their related word meaning.  
+Experimental Setup  
+Keras (Powerful Open Source Python Library) text 
+tokenization utility class is applied to convert the data 
+words into numerical entities. Besides the 'Out of 
+Vocabulary (OOV)' words can be handled here. The 
+softmax and relu are considered activation functions for 
+this 
+emotion 
+recognition 
+multiclass 
+classification 
+problem. For the loss function and optimizer, the 
+categorical-cross entropy and rmsprop are used, 
+respectively. The proposed CNN model and data 
+processing are implemented in the Python programming 
+language. A web-based data-science environment such as 
+"www.kaggle.com" is used to implement the experiment.  
+The proposed CNN approach is trained with batch 
+sizes 32, 64, and 128 per batch. This experiment is run 
+on a PC (Intel(R) Core (TM) i7-7700 CPU @ 3.60 
+GHz, RAM 16 GB, 64-bit OS) with Windows 10 
+environment OS. 
+Experimental Results and Performance Comparison 
+The proposed CNN model's main benefit is that it 
+considers emoticons in addition to emotion recognition 
+from real-life Twitter data. Only text data is directed to 
+the proposed CNN method without emoticons and the 
+influence of emoticons in emotion recognition is 
+observed. Figure 3 illustrates both training set and test set 
+accuracies varying CNN training epochs up to 200 for 
+several batch sizes. Compared to text-only accuracy, the 
+proposed method achieves higher accuracy for both 
+emoticon and text data as shown in the figure. 
+It is also worth mentioning that although the training 
+set accuracy of text-only is consistent with the proposed 
+CNN architecture, in terms of test accuracy, it obtains 
+higher results than the text-only case. At batch size 128, 
+the proposed CNN method achieves 39.9% test accuracy 
+for the text-only data within 10 epochs. In contrast, the 
+scheme achieves 88.0% test accuracy considering both 
+emoticon and text at batch size 32 within 10 epochs. In 
+any machine learning system, higher test set accuracy is 
+desired because it is the indication of the system's ability for 
+generalization. More accuracy in the test set specifies that 
+emoticon use in addition to text boosted the capability of 
+learning the emotion properly of the proposed CNN method. 
+ 
+Table 1: A Snatch of Tweets and Corresponding EC 
+Microblog data (emoticons with texts) 
+Emotion category 
+Good morning 
+  
+2  
+Today is not my day 
+  
+1  
+ I can't handle this  
+1  
+He looks ginger lol 
+  
+3  
+ I don't need the vaccine  
+3  
+Coming home to this period 
+  
+4 
+ 
+Table 2: Emoticons with corresponding word meanings 
+Emoticon 
+Word meaning 
+Emoticon 
+Word meaning 
+ 
+Grinning face 
+ 
+Angry face 
+ 
+Grinning face with smiling eyes 
+ 
+Pouting face 
+ 
+Beaming face with smiling eyes 
+ 
+Face with steam from nose 
+ 
+Smiling face 
+ 
+Face with symbols on the mouth 
+ 
+Loudly crying face 
+ 
+Smiling face with heart-eyes 
+ 
+Crying face 
+ 
+Smiling face with hearts 
+ 
+Pleading face 
+ 
+Face blowing a kiss 
+ 
+Frowning face 
+ 
+Kissing face with closed eyes 
+ 
+ 
+
+商Md. Ahsan Habib et al. / Journal of Computer Science 2022, 18 (12): 1170.1178 
+DOI: 10.3844/jcssp.2022.1170.1178 
+ 
+1175 
+Tables 3 and 4 demonstrate the confusion matrices of the 
+developed emotion recognition model for emoticons with 
+text and text-only cases, respectively. Figure 3(b) depicts the 
+best test case accuracy for both cases. The differences 
+between the predicted emotions and labeled emotions are 
+shown in confusion matrices. These matrices depict four 
+category-wise emotion recognition for emoticons with text 
+and text-only cases. In terms of the test set, every category of 
+emotion holds around 1000 tweets and the proposed CNN 
+method accurately classifies 890 cases for the ‘Angry' 
+category which is the best performance using both text and 
+emoticon data. The text-only case, on the other hand, 
+performed best for the ‘Sad' category, correctly classifying 
+429 out of 1000 cases. The other performance evaluation 
+metric can be attained from the mentioned confusion 
+matrices of the proposed CNN scheme for both cases.
+ 
+Table 3: Confusion matrix of proposed CNN Scheme considering emoticon and text data 
+ 
+Predicted emotion category 
+ 
+--------------------------------------------------------------------------------------- 
+  
+Actual emotion category 
+Sad  
+Happy  
+Angry  
+Love  
+Total 
+Sad  
+875  
+26  
+61  
+39  
+1001 
+Happy  
+31  
+882  
+45  
+43  
+1001 
+Angry  
+47  
+37  
+890  
+27  
+1001 
+Love  
+45  
+42  
+34  
+879  
+1000 
+ 
+Table 4: Confusion matrix for text data only 
+ 
+Predicted emotion category  
+ 
+--------------------------------------------------------------------------------------- 
+Actual emotion category 
+Sad  
+Happy  
+Angry  
+Love  
+Total  
+Sad  
+429  
+148  
+247  
+177  
+1001  
+Happy  
+187  
+412  
+208  
+194  
+1001  
+Angry  
+257  
+220  
+387  
+137  
+1001  
+Love  
+225  
+240  
+185  
+350  
+1000  
+ 
+Table 5:  Comparison of the proposed CNN method with other existing methods on Twitter data 
+Sl.  
+Work Ref, (Authors, Year)  
+Methodology  
+Sample size (Training + Test)  
+Test set accuracy 
+1  
+Wikarsa and Thahir (2015)  
+Naive Bayes  
+268 (116+152)  
+71.30%  
+2  
+Yang et al. (2018)  
+CNN  
+7200 (4600+2600)  
+72.60%  
+3  
+Batbaatar et al. (2019)  
+CNN + BiLSTM  
+19,678 (Not Mentioned)  
+61.30%  
+4  
+Islam et al. (2020) 
+LSTM  
+3085 (2313+772)  
+82.10%  
+5  
+Liu et al. (2021) 
+Bi-GRU  
+10,000 (8000+2000)  
+85.76%  
+ 
+ 
+ 
+15,000 (12,000+3000)  
+86.35%  
+6  
+Proposed Method  
+CNN  
+16,012 (12,009 + 4003)  
+88.00%  
+ 
+ 
+ 
+Fig. 3: Performance of ER for proposed CNN method (considering both emoticon and text) and for text-only for different Batch Sizes 
+(BS)
+
+0.9
+1
+0.8
+TextOnly(BS32)
+TextOnly(BS32)
+0.95
+TextOnly(BS64)
+-TextOnly(BS64)
+0.7
+60
+Training Set
+TextOnly(BS128)
+-TextOnly(BS128)
+Text&Emoticon(BS32)
+0.6
+TestSet
+XText&Emoticon(BS32)
+Text&Emoticon (BS64)
+0.85
+米Text&Emoticon(BS64)
+Text&Emoticon(BS128)
+-Text&Emoticon(BS128)
+0.4
+0.8
+0.3
+0.75
+0.2
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+0.7
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+Epoch
+Epoch
+(a) Training Set Accuracy vs. Epochs
+(b) Test Set Accuracy vs.EpochsMd. Ahsan Habib et al. / Journal of Computer Science 2022, 18 (12): 1170.1178 
+DOI: 10.3844/jcssp.2022.1170.1178 
+ 
+1176 
+Table 5 compares the proposed CNN method's 
+emotion classification accuracy to other existing methods 
+using Twitter data. The table also includes the 
+methodology used in various studies and dataset sizes. 
+The proposed method contains 16012 samples where 
+12009 tweets are for training and the rest 4003 tweets are 
+for testing purposes. Wikarsa and Thahir (2015) used 268 
+tweets only 116 of them were used for training, which is 
+not feasible. They applied Naive Bayes algorithm for 
+classification and achieved an accuracy of 72.3%.       
+Yang et al. (2018) used 4600 tweets for training purposes 
+and applied CNN architecture to their model. Their model 
+achieved 72.6% accuracy. Batbaatar et al. (2019) used 
+19,678 tweets for their study but didn't mention the train-
+test split ratio. They employed both CNN and BiLSTM 
+and achieved an accuracy of 61.3%. For training purposes, 
+Islam et al. (2020) used 2313 tweets which is not 
+promising. They used LSTM based model and achieved 
+82.1% accuracy. On the other hand, Liu et al. (2021) 
+employed two Chinese microblogs Sina Weibo based 
+NLPCC2013 and NLPCC2014 datasets containing 10,000 
+and 15,000 sentences respectively. The model employed 
+Bi-GRU architecture and achieved 85.76% accuracy for 
+the NLPCC2013 dataset and 86.35% accuracy for the 
+NLPCC2014 dataset. However, both datasets hold only 5400 
+sentences containing emoticons which might not produce a 
+significant effect on classification. However, the proposed 
+emotion recognition model with an 88.0% test set accuracy 
+outperformed all other methods except (Islam et al., 2021). 
+Showing competitive performance with (Islam et al., 2021), 
+the proposed method has a significant contribution. For the 
+emotion recognition model, CNN uses Conv1D architecture 
+rather than the standard CNN Conv2D architecture that is 
+most commonly used for imagery data. The proposed 
+method successfully classifies emotion using CNN Conv1D 
+architecture because text data is considered time series data. 
+Finally, classification with CNN considering emoticons with 
+texts has been revealed as a promising ER technique from 
+microblog data. 
+Conclusion 
+Nowadays, social media has become the most 
+prevalent podium to express one's feelings & emotions 
+and for a better kind, emoticons are commonly used with 
+texts. In the ML domain, emotion recognition from 
+microblog data has emerged as a challenging and 
+promising research finding. For emotion recognition, 
+most of the existing methods consider only text data for 
+simplicity which is not sufficient. In this study, an 
+emotion recognition model using CNN is developed 
+considering emoticons in addition to text. As emoticons 
+can have a significant role in the emotional behavior of 
+human beings using microblog data, the proposed CNN 
+technique 
+outperforms 
+other 
+emotion 
+recognition 
+methods considering emoticons in addition to text using 
+real-life Twitter data. In summary, this research 
+developed an emotion classification technique and the 
+effectiveness of emoticon consideration in emotion 
+recognition from microblog data.  
+This study opens some future research scopes in this 
+area. This study added a significant wing in the field of 
+emotion recognition from English microblogs. A similar 
+concept may be suitable for other language microblogs. 
+Moreover, the recognition system brings more realistic if 
+emotional states like surprise and disgust can be 
+considered, which remained for further research. 
+Moreover, with a large dataset, the proposed CNN 
+approach could produce a more realistic result.  
+Acknowledgment 
+The authors are thankful to Juyana Islam, Sr. Software 
+Engineer at Samsung R&D Institute Bangladesh for 
+sharing twitter dataset of their pilot study on emotion 
+recognition (Islam et al. 2021). 
+Funding Information 
+This research received no specific grant from any 
+funding agency in the public, commercial, or not-for-
+profit sectors. 
+Author’s Contributions 
+Md. Ahsan Habib: Participated in design, conducted 
+experiments, performed result analysis and contributed to 
+the writing of the manuscript. 
+M. A. H. Akhand: Designed the research plan and 
+organized the study, analyzed and interpreted results and 
+prepared the manuscript. 
+M. A. S. Kamal: Participated in design, contributed to 
+model illustration and reviewed the manuscript. 
+Ethics 
+It has been testified by the authors that this article has 
+not been submitted to be published in any other journal 
+and contains no ethical issues. 
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+page_content=' Abdus Samad Kamal  1Department of Computer Science & Engineering, Khulna University of Engineering & Technology, Bangladesh 2Graduate School of Science and Technology, Gunma University, Japan  Article history  Received: 17 10 2022  Revised: 17 11 2022  Accepted: 24 11 2022  Corresponding Author: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
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+page_content=' Akhand Department of Computer Science & Engineering, Khulna University of Engineering & Technology, Bangladesh Email: akhand@cse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='kuet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='bd Abstract: Microblog, an online-based broadcast medium, is a widely used forum for people to share their thoughts and opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Recently, Emotion Recognition (ER) from microblogs is an inspiring research topic in diverse areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' In the machine learning domain, automatic emotion recognition from microblogs is a challenging task, especially, for better outcomes considering diverse content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Emoticon becomes very common in the text of microblogs as it reinforces the meaning of content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' This study proposes an emotion recognition scheme considering both the texts and emoticons from microblog data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=" Emoticons are considered unique expressions of the users' emotions and can be changed by the proper emotional words." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The succession of emoticons appearing in the microblog data is preserved and a 1D Convolutional Neural Network (CNN) is employed for emotion classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The experimental result shows that the proposed emotion recognition scheme outperforms the other existing methods while tested on Twitter data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content="  Keywords: Deep Learning, CNN, Emotion Recognition, Emoticons  Introduction Emotion often refers to a complex state of feeling such as happiness, joy, anger, disgust, fear, love, and hatred that occurs in physical and psychological changes and has an impact on one's thinking and actions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Emotions can have a significant impact on people’s lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=" People's emotions can be expressed in a variety of ways, including speech, facial expressions, bodily gestures, verbal expressions using text, and so on (Castellano et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Murugappan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Different social media platforms like Facebook, Twitter, Instagram, WhatsApp, Sina Weibo, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' have grown in popularity in recent years where people express their emotions and thoughts (Che et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Users share millions of tweets and posts every day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Therefore, social media contents are the most prospective sources for understanding emotional states and human thoughts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Microblog, an online-based broadcast medium, is a widely used forum for people to share their thoughts and opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' In our daily lives, microblogs make communication easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Twitter (Java et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2007), LinkedIn, Facebook, Instagram, Snapchat, Tumblr, WhatsApp, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' are the most popular microblogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Globally, there are 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='5 billion users on social media, according to 2019 social media projections which equate to approximately 45% of the current population and this figure is only increasing (Mohsin, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Any post can be hit immediately by a large number of people through microblogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content="  People express their thoughts by sharing posts that reflect one's sentiments." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Emotion Recognition (ER) from microblog data is the most promising and challenging research finding in the field of information and communication technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Unimodal, bimodal, and multimodal are the three possible categories of ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Unimodal emotion recognition uses only one type of information, such as facial expression, text, or speech, whereas bimodal emotion recognition uses both speech and facial expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' This study introduces the unimodal emotion recognition method to determine one’s Emotion Category (EC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Machine Learning (ML) based approach and knowledge-based approach (Chaffar and Inkpen, 2011) are two major techniques for identifying or recognizing emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The knowledge-based technique also known as the lexicon-based technique uses a set of rules to detect emotion from given data (Nirenburg and Mahesh, 1997) whereas the ML-based technique uses a model for learning the patterns from features generated from microblog data (Kotsiantis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Deep Learning (DL) based approaches for emotion recognition from microblog data have emerged remarkably and shown promising results nowadays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' DL employs different architectures to learn the patterns in data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' There are two major steps in DL-based emotion  Science PublicationsMd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Ahsan Habib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' / Journal of Computer Science 2022, 18 (12): 1170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1178 DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='3844/jcssp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1178  1171 recognition from microblog data: (i) Processing data and (ii) classifying them using the proper DL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Firstly, the collected data is processed, transformed, and represented in the appropriate format for the envisioned DL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Secondly, a DL model is prepared or trained with the data to classify emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Along with different processing techniques, different DL models are investigated in the last several years for emotion recognition (Batbaatar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' These DL-based methods run on preprocessed data and do not include explicit functionality (Batbaatar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Some experiments consider emoticons along with the text as input data, while others only consider the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' This research aims to develop an improved CNN based emotion recognition model from microblogs considering both text and emoticons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' This study takes into account emoticons as they have a substantial significance to users’ emotions and represented them by corresponding emotional words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The word and emoticon sequences that appeared in the microblog are preserved in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Considering all other pre-processing steps, CNN is applied for emotion recognition since it is pertinent for sequential data classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='  Experiments using Twitter data on texts with emoticons and texts-only showed the effectiveness of the proposed CNN approach, with better classification accuracy considering both emoticons and texts compared to classification accuracy when only considering texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The remaining paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' At first, it reviews different existing works related to emotion recognition, and then the proposed CNN method is explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' After that, the experimental studies and results are demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Lastly, the paper concludes with a few remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Literature Review Many DL-based approaches are examined in the last several years for emotion recognition from microblog data (Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Mehta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Attention model (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Yuan and Zhang, 2021), BERT RCNN (Pan and Xu, 2021), CNN (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2020), GRU (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2021), LSTM (Arun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Batbaatar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2021), graph convolution network (Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' 2020), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', are most prominent techniques employed in this research domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Most of the methods only consider textual data (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2020) and a few consider both text and emoticons (Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' (2018) developed an enhanced CNN method considering both emoticons and texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' They represented the emoticons and words as two separate vectors and projected into one emotional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Then CNN is employed for emotion classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='  The proposed model is applied to the Twitter dataset, NLPCC2013, and Weibo dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The work proposed by Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' (2020) also took both the emoticons and texts as input where it applied LSTM to classify emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The Twitter dataset was used to measure the efficacy of the model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' however, the dataset was small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The work was then extended by Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' (2021) and achieved remarkable accuracy with a relatively large dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Batbaatar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' (2019) introduced a Semantic Emotion Neural Network (SENN) model that includes both CNN and BiLSTM for ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Here, the CNN model focused on the emotional connectivity between words after extracting emotional features, while the BiLSTM was employed to build the semantic relationship after collecting contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The SENN used Twitter data and other social media data (only text) without specifying whether or not emoticons were used in the decision-making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' (2019) developed an emotion recognition approach by incorporating both the dual attention mechanism and Bidirectional Long Short-term Memory (BiLSTM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' In their work, they first used the BiLSTM model to semantically encode the microblog data and then introduced the sentiment word attention and self-attention into the BiLSTM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Lastly, they used the Softmax classifier to classify the sentiment of microblogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Chinese microblog Sina Weibo-based NLPCC2013 and NLPCC2014 datasets are used for the experimental purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='  Another attention-based dual-channel microblog emotion recognition model is developed by Yuan and Zhang (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' This study used RoBERTa-WWM and the multi-head attention model for their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' They constructed the emotional knowledge set of each sentence extending the emotional resource library and used the pre- training model RoBERTa-WWM for feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' After that, the Text CNN-BiGRU network and a Multi- Head Attention network took the sentence feature and the emotional knowledge as input to obtain deeper semantics features and attention features of emotional knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' And finally, the semantic feature and the emotional knowledge attention feature are combined to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Chinese microblog NLPCC2014 dataset is used to show the efficacy of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Pan and Xu (2021) developed a deep learning-based sentiment analysis model employing BERT RCNN for netizens during public health emergencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' They used BERT which uses static masking for fine-tuning the input data and trained them into vectors to represent it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Then it took the trained vectors as the input features of the upstream model and learned the microblog data features through RCNN network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' In contrast, the work proposed by Yuan and Zhang (2021) used dynamic masking incorporated in RoBERTa making the model more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='  Another model, called Semantic Emoticon Emotion Recognition (SEER) (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2021), used both the attention mechanism and Bi-GRU (bidirectional gated recurrent unit) network to classify emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Then they constructed an emoticon distribution model to obtain the emotion vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Arun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' (2019) developed EPUSAMCNN (Emotion- Prediction Using Semantic Analysis Multi-Dimensional Convolutional Neural Network) model incorporating both  Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Ahsan Habib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' / Journal of Computer Science 2022, 18 (12): 1170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1178 DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='3844/jcssp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1178  1172 the LSTM (Long Short Term Memory Networks) and MCNN (Multi-Dimensional Convolutional Neural Network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' They used MCNN to increase their proficiency in recognizing correct feelings for microblog data and BiLSTM for classifying them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Another model named ERNIE-BiLSTM is developed for sentiment classification by Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' At first, they used ERNIE (Knowledge Enhanced Semantic Representation) pretrained model for word featuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' It considered both the enhancement of the semantic representation of words and preserves the contextual information along with the polysemy of words also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' After training through ERNIE, they used BiLSTM for sentiment classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' They experimented with their developed model with the Chinese microblog Sina Weibo based NLPCC2014 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' An emotion classification model is developed in 2020 by Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' They used syntax based GCN (Graph Convolution Network) model focusing on the diverse grammatical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The accuracy of the model is enhanced by a percentile-based pooling technique proposed by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' They experimented with their developed model with the Chinese microblog dataset on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Emotion Recognition from Microblog Managing Emoticon with Text Social media has been the most common medium of venting feelings in the era of globalization (Gräbner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' People share their thoughts by posting videos, texts, audio, photos, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' to express emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='  Microblogs are the most popular among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Millions of words, images, videos, audio, hashtags, and various signs and symbols with various meanings can be found on microblogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' One of the most widely used microblogging platforms is Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Since emoticons reinforce the meaning of content, they should be given special consideration in emotion recognition along with texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The emoticon and its interactions with texts are given particular consideration in the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' To identify the true emotion of people, both emoticons and text possess equal significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Several pieces of research in the literature have described emoticons as noisy inputs that should be omitted during the pre-processing stage (Hogenboom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2013) but this should not be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The proposed model did proper emotion analysis with the aid of emotional words and other texts in the microblog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The working procedure of the proposed emotion recognition model is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' 1 for a sample post with an emoticon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The proposed CNN scheme contains four consecutive phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' A lookup table is used in Task 1 to translate emoticons into corresponding emotion words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' In Task 2, Integer Encoding (IE), the Task of converting words into a series of integers, is done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Then in Task 3 padding is done to make an equal-length vector sequence of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Finally, CNN is applied to recognize specific emotions (Sad, Happy, Angry, or Love) in Task 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='  The whole methodology where four individual processes are shown is described in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Data processing and CNN classification are two major tasks in this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The first three processing steps are under data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' In the following subsections, the steps are described briefly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='  Algorithm 1: Proposed ER scheme Input: Microblog Data D of Word Size N Output: Category of Emotion // Task 1: Replacing emoticon(s) to corresponding meaning For t = 1 to N do  If (D[i] is emoticon(s)) then  D[i] ← Emoticon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' meaning (D[i])  End If End For // Task 2: Integer Encoding (IE) using Tokenizer For t = 1 to N do  IE[i] = Tokenizer (D[i]) End For // Task 3: Zero padding at first to make fixed L length For t = 1 to L-N do  P[i] ← 0 // Considering 0 for initial values End For For i = L-N+1 to N do  P[i] ← IE[i] // Copy the rest values End For // Task 4: Emotion classification using CNN // Embedding integer to 2D vector For t = 1 to L do  U [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' y] = Embedding (P[i]) End For  Data Processing One of the most critical phases in our developed scheme is to process microblog data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=" Among several social platforms' data, Twitter data is used including both emoticons and text." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Certain pre-processing steps are required to eliminate unnecessary content and noisy input from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Case conversion, user name removal, hashtag, punctuation mark removal, and so on are all part of the cleaning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Then the clean microblog data containing both texts and emoticons is processed into three processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Task 1 (emoticon conversion step) searches microblog data for emoticon(s) and then uses the Emoticon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='meaning() function emoticons are replaced with corresponding meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Individual emoticon word meanings are stored in a lookup table used by the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Needless information (if any) is eliminated and IE is achieved in the Tokenization step (Task 2) using the function Text_to_sequence().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='  Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Ahsan Habib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' / Journal of Computer Science 2022, 18 (12): 1170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1178 DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='3844/jcssp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1178  1173  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' 1: The proposed ER architecture for a sample microblog data containing emoticons with texts  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' 2: 1D CNN architecture of the proposed ER  In Task 3, padding is done using the pad_sequence() function to form equal length word vector sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Zero padding at first is performed in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Lastly, in Task 4, the CNN model is applied to recognize particular emotions (Happy, Love, Sad, or Angry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' ER using 1D CNN CNN is a deep neural network very popular for analyzing visual 2D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' CNN has multiple layers having convolution and pooling operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=" CNN's convolutional layers provide a summary of an image's features." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' By summarizing the presence of features of the feature map, pooling layers down samples feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' For images or image-like 2D inputs, the Conv2D layer architecture is primarily used in standard CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Finally, a fully connected layer is employed for classification purposes called Dense Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' This study considers CNN architecture with 1D convolutional operation on the blog text data mimicking the idea from CNN operation on time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' For time-series data CNN uses Conv1D architecture (Amo-Boateng, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' As text data is considered time- series data CNN employs the Conv1D architecture for ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The kernel in Conv1D slides in one dimensional way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Two Conv1D layers along with two max-pooling layers, a flattened layer, and two dense layers show promising outcomes in emotion recognition from text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='  Generally, time-series data is used for forecasting or single output prediction but in our proposed method the output layer contains multiple nodes as emotion recognition is considered a multiclass problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Figure 2 illustrates the CNN architecture of the developed scheme for emotion recognition from microblog data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' CNN is popular for analyzing texts and recognizing their features or patterns of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The architecture of the proposed scheme consists of an input layer, an embedding layer, two convolutional layers, two maxpooling layers, a flatten layer, two dense layers, and lastly the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The size of the words used in the proposed model is the input dimension of the embedding layer and the output dimension is 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The first and second convolutional layers have 64 and 32 dimensionalities of output spaces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Both convolutional layers use kernel size 3 and the relu activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' For both max-pooling layers, the max-pooling window size is set to 2 and the pooling window moves 2 strides for each pooling step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The following flattened layer flattens the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The first dense layer contains 16 with a relu activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' As there are four possible class labels, the proposed CNN architecture ends with a dense layer with four nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' To get the probability for each class the softmax activation function is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='  My kids are smart My kids are and It makes me very Process 1: smart Grinning_facc and Process 2: happy Emoticon It makes me very happy Tokenization Convcrsion [nl,n2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='.,n11] Detected Emotion: Process 4: [0, 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=',nl, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', n10, nll Happy Classification Process 3: Padding using CNNO O InputLayer O Convolutional Max-pooling EmbcddingLaycr IDLayer1 Laycr 1 OutputLayer DenseLayer1 Dense Layer 2 Max-pooling Convolutional Flatten [Size32X16] [Size 16X4] Layer2 IDLayer2 LayerMd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Ahsan Habib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' / Journal of Computer Science 2022, 18 (12): 1170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1178 DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='3844/jcssp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1178  1174 Results and Discussion This segment demonstrates microblog data preparation, experimental setup, experimental results, and analysis of the proposed emotion recognition scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Dataset Preparation This study uses the Twitter dataset, a pool of English tweets collected using the Twitter API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Firstly, tweeps, a python library for downloading tweets from Twitter, is used to collect tweets for this study purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=" The language filtering is enabled by the Twitter API, which permits the definition of the retrieved tweets' language." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=" To extract English Tweets, the optional language parameter is set to 'en’ in the Twitter Search URL." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' There are four emotion class labels in a total of 16011 tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The numeric notation of the class labels is as follows: Sad, happy, love, and angry are represented by 1, 2, 3, and 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' 75% (12008 tweets) of the collected data is used to train the proposed CNN architecture and the rest 25% (4003 tweets) is used as a test set in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Table 1 illustrates a snatch of the tweets with their corresponding EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Table 2 displays the 16 emoticons that are used in the proposed scheme, along with their related word meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Experimental Setup Keras (Powerful Open Source Python Library) text tokenization utility class is applied to convert the data words into numerical entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=" Besides the 'Out of Vocabulary (OOV)' words can be handled here." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='  The softmax and relu are considered activation functions for this emotion recognition multiclass classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' For the loss function and optimizer, the categorical-cross entropy and rmsprop are used, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The proposed CNN model and data processing are implemented in the Python programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' A web-based data-science environment such as "www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='com" is used to implement the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The proposed CNN approach is trained with batch sizes 32, 64, and 128 per batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' This experiment is run on a PC (Intel(R) Core (TM) i7-7700 CPU @ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='60 GHz, RAM 16 GB, 64-bit OS) with Windows 10 environment OS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=" Experimental Results and Performance Comparison The proposed CNN model's main benefit is that it considers emoticons in addition to emotion recognition from real-life Twitter data." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Only text data is directed to the proposed CNN method without emoticons and the influence of emoticons in emotion recognition is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Figure 3 illustrates both training set and test set accuracies varying CNN training epochs up to 200 for several batch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Compared to text-only accuracy, the proposed method achieves higher accuracy for both emoticon and text data as shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' It is also worth mentioning that although the training set accuracy of text-only is consistent with the proposed CNN architecture, in terms of test accuracy, it obtains higher results than the text-only case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='  At batch size 128, the proposed CNN method achieves 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='9% test accuracy for the text-only data within 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' In contrast, the scheme achieves 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='0% test accuracy considering both emoticon and text at batch size 32 within 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=" In any machine learning system, higher test set accuracy is desired because it is the indication of the system's ability for generalization." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' More accuracy in the test set specifies that emoticon use in addition to text boosted the capability of learning the emotion properly of the proposed CNN method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Table 1: A Snatch of Tweets and Corresponding EC Microblog data (emoticons with texts) Emotion category Good morning  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='2 Today is not my day  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content="I can't handle this  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='He looks ginger lol  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content="3 I don't need the vaccine 3 Coming home to this period  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Table 2: Emoticons with corresponding word meanings Emoticon Word meaning Emoticon Word meaning  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Grinning face  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Angry face  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Grinning face with smiling eyes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Pouting face  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Beaming face with smiling eyes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Face with steam from nose  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Smiling face  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Face with symbols on the mouth  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Loudly crying face  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Smiling face with heart eyes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Crying face  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Smiling face with hearts  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Pleading face  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Face blowing a kiss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Frowning face  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Kissing face with closed eyes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='商Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Ahsan Habib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' / Journal of Computer Science 2022, 18 (12): 1170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1178 DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='3844/jcssp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1178  1175 Tables 3 and 4 demonstrate the confusion matrices of the developed emotion recognition model for emoticons with text and text-only cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Figure 3(b) depicts the best test case accuracy for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The differences between the predicted emotions and labeled emotions are shown in confusion matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' These matrices depict four category-wise emotion recognition for emoticons with text and text-only cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=" In terms of the test set, every category of emotion holds around 1000 tweets and the proposed CNN method accurately classifies 890 cases for the ‘Angry' category which is the best performance using both text and emoticon data." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=" The text-only case, on the other hand, performed best for the ‘Sad' category, correctly classifying 429 out of 1000 cases." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The other performance evaluation metric can be attained from the mentioned confusion matrices of the proposed CNN scheme for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Table 3: Confusion matrix of proposed CNN Scheme considering emoticon and text data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Predicted emotion category  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='---------------------------------------------------------------------------------------  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Actual emotion category  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Sad  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Happy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Angry  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Love  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Total  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Sad  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='875  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='26  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='61  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='39  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1001  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Happy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='31  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='882  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='45  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='43  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1001  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Angry  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='47  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='37  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='890  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='27  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1001  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Love  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='45  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='42  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='879  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Table 4: Confusion matrix for text data only  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Predicted emotion category  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='-------------------------------------------------------------------------------------- Actual emotion category  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Sad  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Happy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Angry  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Love  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Total  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Sad  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='429  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='148  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='247  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='177  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1001  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Happy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='187  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='412  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='208  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='194  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1001  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Angry  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
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+page_content='Table 5:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='Comparison of the proposed CNN method with other existing methods on Twitter data Sl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Work Ref, (Authors, Year) Methodology Sample size (Training + Test) Test set accuracy 1 Wikarsa and Thahir (2015) Naive Bayes 268 (116+152) 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='30% 2 Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' (2018) CNN 7200 (4600+2600) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='60% 3 Batbaatar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' (2019) CNN + BiLSTM 19,678 (Not Mentioned) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='30% 4 Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' (2020) LSTM 3085 (2313+772) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='10% 5 Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' (2021) Bi-GRU 10,000 (8000+2000) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='76%  15,000 (12,000+3000)  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='35%  6  Proposed Method  CNN  16,012 (12,009 + 4003)  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='00%  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' 3: Performance of ER for proposed CNN method (considering both emoticon and text) and for text-only for different Batch Sizes (BS)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='9 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='8 TextOnly(BS32) TextOnly(BS32) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='95 TextOnly(BS64) -TextOnly(BS64) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='7 60 Training Set TextOnly(BS128) -TextOnly(BS128) Text&Emoticon(BS32) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='6 TestSet XText&Emoticon(BS32) Text&Emoticon (BS64) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='85 米Text&Emoticon(BS64) Text&Emoticon(BS128) -Text&Emoticon(BS128) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='2 0 20 40 60 80 100 120 140 160 180 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='7 0 20 40 60 80 100 120 140 160 180 200 Epoch Epoch (a) Training Set Accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Epochs (b) Test Set Accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='EpochsMd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Ahsan Habib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' / Journal of Computer Science 2022, 18 (12): 1170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1178 DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='3844/jcssp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content="1178  1176 Table 5 compares the proposed CNN method's emotion classification accuracy to other existing methods using Twitter data." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The table also includes the methodology used in various studies and dataset sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The proposed method contains 16012 samples where 12009 tweets are for training and the rest 4003 tweets are for testing purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Wikarsa and Thahir (2015) used 268 tweets only 116 of them were used for training, which is not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' They applied Naive Bayes algorithm for classification and achieved an accuracy of 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' (2018) used 4600 tweets for training purposes and applied CNN architecture to their model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Their model achieved 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='6% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Batbaatar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=" (2019) used 19,678 tweets for their study but didn't mention the train- test split ratio." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' They employed both CNN and BiLSTM and achieved an accuracy of 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' For training purposes, Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' (2020) used 2313 tweets which is not promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' They used LSTM based model and achieved 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' On the other hand, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' (2021) employed two Chinese microblogs Sina Weibo based NLPCC2013 and NLPCC2014 datasets containing 10,000 and 15,000 sentences respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The model employed Bi-GRU architecture and achieved 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='76% accuracy for the NLPCC2013 dataset and 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='35% accuracy for the NLPCC2014 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' However, both datasets hold only 5400 sentences containing emoticons which might not produce a significant effect on classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='  However, the proposed emotion recognition model with an 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='0% test set accuracy outperformed all other methods except (Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Showing competitive performance with (Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', 2021), the proposed method has a significant contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' For the emotion recognition model, CNN uses Conv1D architecture rather than the standard CNN Conv2D architecture that is most commonly used for imagery data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' The proposed method successfully classifies emotion using CNN Conv1D architecture because text data is considered time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Finally, classification with CNN considering emoticons with texts has been revealed as a promising ER technique from microblog data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=" Conclusion Nowadays, social media has become the most prevalent podium to express one's feelings & emotions and for a better kind, emoticons are commonly used with texts." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' In the ML domain, emotion recognition from microblog data has emerged as a challenging and promising research finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' For emotion recognition, most of the existing methods consider only text data for simplicity which is not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' In this study, an emotion recognition model using CNN is developed considering emoticons in addition to text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' As emoticons can have a significant role in the emotional behavior of human beings using microblog data, the proposed CNN technique outperforms other emotion recognition methods considering emoticons in addition to text using real-life Twitter data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='  In summary, this research developed an emotion classification technique and the effectiveness of emoticon consideration in emotion recognition from microblog data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' This study opens some future research scopes in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' This study added a significant wing in the field of emotion recognition from English microblogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' A similar concept may be suitable for other language microblogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Moreover, the recognition system brings more realistic if emotional states like surprise and disgust can be considered, which remained for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Moreover, with a large dataset, the proposed CNN approach could produce a more realistic result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Acknowledgment The authors are thankful to Juyana Islam, Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Software Engineer at Samsung R&D Institute Bangladesh for sharing twitter dataset of their pilot study on emotion recognition (Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Funding Information This research received no specific grant from any funding agency in the public, commercial, or not-for- profit sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Author’s Contributions Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Ahsan Habib: Participated in design, conducted experiments, performed result analysis and contributed to the writing of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Akhand: Designed the research plan and organized the study, analyzed and interpreted results and prepared the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Kamal: Participated in design, contributed to model illustration and reviewed the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='  Ethics It has been testified by the authors that this article has not been submitted to be published in any other journal and contains no ethical issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' References Arun, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', Vineeth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=', & Prudhvi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Emotion prediction using semantic analysis neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Journal of Advanced Research in Dynamical and Control Systems, 11(4), 1184-1191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
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+page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='org/abstract/document/8573804 Yuan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
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+page_content=' (2021, July).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Dual-Channel Microblog Emotion Analysis Based on RoBERTa- WWM and Multi-Head Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' In The International Conference on Natural Computation, Fuzzy Systems and Knowledge Discovery (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' 486- 495).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' Springer, Cham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
+page_content='1007/978-3- 030-89698-0_50' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE1T4oBgHgl3EQfLANo/content/2301.02971v1.pdf'}
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+ 
+Jameel, Samer Kais, Sezgin Aydin, Nebras H. Ghaeb, Jafar Majidpour, Tarik A. Rashid, Sinan Q. Salih, and Poh Soon JosephNg. 2022. "Exploiting the 
+Generative Adversarial Network  Approach to Create a Synthetic Topography Corneal Image" Biomolecules 12, no. 12: 1888. 
+https://doi.org/10.3390/biom12121888 
+1 
+ 
+Exploiting the Generative Adversarial Network Approach to 
+Create a Synthetic Topography Corneal Image 
+Samer Kais Jameel 1, Sezgin Aydin 2, Nebras H. Ghaeb 3, Jafar Majidpour 1, Tarik A. Rashid 4,*, Sinan Q. Salih 5 and 
+P. S. JosephNg 6,* 
+1 Computer Science Department, University of Raparin, Rania 46012, Iraq. samer.kais@uor.edu.krd  
+2  Department of Natural and Mathematical Sciences, Engineer Faculty, Tarsus University, Tarsus 33402, Tur-
+key. sezginaydin@tarsus.edu.tr  
+3 Biomedical Engineering Department, Al-Khawarezmi Eng. College, University of Baghdad, Baghdad 1001, 
+Iraq. nebras@kecbu.uobaghdad.edu.iq; jafar.majidpoor@uor.edu.krd  
+4  Computer Science and Engineering Department, University of Kurdistan Hewlêr, Erbil 44001, Iraq. tarik.ah-
+med@ukh.edu.krd  
+5  Department of Communication Technology Engineering, College of Information Technology, Imam Ja'afar 
+Al-Sadiq University, Baghdad 10011, Iraq. sinan.salih@sadiq.edu.iq   
+6  Faculty of Data Science & Information Technology, INTI International University, Persiaran Perdana BBN, 
+Nilai 71800, Negeri Sembilan, Malaysia. joseph.ng@newinti.edu.my  
+* Correspondence: tarik.ahmed@ukh.edu.krd (T.A.R,); joseph.ng@newinti.edu.my (p.S.J.) 
+Abstract: Corneal diseases are the most common eye disorders. Deep learning techniques are used 
+to perform automated diagnoses of cornea. Deep learning networks require large-scale annotated 
+datasets, which is conceded as a weakness of deep learning. In this work, a method for synthesizing 
+medical images using conditional generative adversarial networks (CGANs), is presented. It also 
+illustrates how produced medical images may be utilized to enrich medical data, improve clinical 
+decisions, and boost the performance of the conventional neural network (CNN) for medical image 
+diagnosis. The study includes using corneal topography captured using a Pentacam device from 
+patients with corneal diseases. The dataset contained 3448 different corneal images. Furthermore, it 
+shows how an unbalanced dataset affects the performance of classifiers, where the data are balanced 
+using the resampling approach. Finally, the results obtained from CNN networks trained on the 
+balanced dataset are compared to those obtained from CNN networks trained on the imbalanced 
+dataset. For performance, the system estimated the diagnosis accuracy, precision, and F1-score met-
+rics. Lastly, some generated images were shown to an expert for evaluation and to see how well 
+experts could identify the type of image and its condition. The expert recognized the image as useful 
+for medical diagnosis and for determining the severity class according to the shape and values, by 
+generating images based on real cases that could be used as new different stages of illness between 
+healthy and unhealthy patients. 
+Keywords: conditional generative adversarial networks, transfer learning, synthesize images, cor-
+neal diseases, data augmentation 
+ 
+1. Introduction 
+Medical image datasets are one of the most important problems facing researchers in 
+the field of machine learning [1]. The limited amount of medical data comes from the dif-
+ficulty of capturing it [2]. With the problem of final ethical approval, the acquisition and 
+labelling of medical images are time-consuming, and considerable effort needs to be spent 
+by both researchers and specialists [3,4]. Several studies tried to overcome the dataset 
+scarcity challenge through the famous task in computer vision, a method called data aug-
+mentation [5]. Using classic data augmentation can give a simple extra feature where it 
+involves simple modifications, such as rotation, translation, scaling, and flipping [6]. On 
+the other hand, some researchers employed innovative techniques for data augmentation 
+
+ 
+ 
+to improve the system training process, based on synthesizing high-quality sample im-
+ages using a generative model known as generative adversarial networks (GANs) [7–9]. 
+The GANs involved two networks; the first generates a real image from the input 
+with the help of the noise, and the other discriminates between real and fake (generated 
+by the first network) images. This model has been used in many studies hoping to gener-
+ate realistic images, especially for medical imaging applications, such as image-to-image 
+translation [10], image inpainting [11], segmentation-to-image translation [12], medical 
+cross-modality translations [13], and label-to-segmentation translation [14]. 
+Exploiting the GAN models by researchers led to the creation of cross-modality im-
+ages, such as a PET scan, which was generated from a CT scan of the abdomen to show 
+the presence of liver lesions. The GAN model of image inpainting has served as inspira-
+tion for many studies. Costa et al. [15] used a fully convolutional network to learn retinal 
+vessel segmentation images. The binary vessel tree was then translated into a new retinal 
+image. By using chest X-ray images, Dai et al. [16] generated lung and heart image seg-
+mentation by training a GAN model. Xu et al. [17] trained a model to translate brain MRI 
+images into binary segmentation maps for brain tumour images. Nie et al. [18] trained a 
+patch-based GAN to translate between brain CT and MRI images. As a step of image re-
+finement, they recommended using an auto-context model. Schlegl et al. [19] trained a 
+GAN model on normal retinal. To detect anomalies in retinal images, the model was tested 
+on normal and abnormal data. 
+Based on what was mentioned above, the scarcity of data needs to be resolved so that 
+researchers can use it more freely to analyze that data and produce results that serve the 
+scientific process. The latter motivated the authors of this paper to use GAN models with 
+the ability to synthesize real images, increase the existing data, and overcome the problem 
+of lacking data. In this work, high-quality corneal images based on GAN models are syn-
+thesized for a specific task of corneal disease diagnosis to improve the clinical decision by 
+introducing different stages and predicted shapes for images with illness. As an illustrated 
+sample of manipulation for the imaging in the cornea, the different stages of keratoconus 
+are, in most cases, unclear in borderlines. From a clinical perspective, overlapping features 
+between stages of keratoconus lead to a controversial approach to treatment. To decide 
+the severity and clinical or surgical procedure of work per patient clinically, considerable 
+evidence is collected from different images per case to reach the final approach. The pos-
+sibility of studying the effect and weight of this evidence per case is an attractive medical 
+training to produce a final highly medical sensation and observation for the trained phy-
+sician. In more detail, thinning in pachymetry images with its location, steepening in the 
+inferior or superior position of the tangential mapping, and the isolated land or tongue 
+shape that may appear in elevation front and back maps, with the astigmatism axis and 
+obliqueness of the bowtie, would improve the effectiveness of the final diagnosis. 
+The cornea, which protects the eye from external substances and helps to control vis-
+ual focus, is stiff but very sensitive to touch [20]. There are many corneal disorders, for 
+instance, bullous keratopathy, Cogan syndrome, corneal ulcer, herpes simplex keratitis, 
+herpes zoster ophthalmicus, etc. [21]. Any disorders in the cornea may cause ripping, dis-
+comfort, and dwindling vision clarity and, finally, may lead to blindness. On the other 
+hand, any action on the cornea, such as vision correction, requires a diagnosis of the cor-
+nea’s health before treatment [22]. Clinical decisions on the human cornea require review-
+ing numerous aspects, and ophthalmologists must handle this revision. Corneal topo-
+graphical parameters are so extensive that it is difficult for surgeons or ophthalmologists 
+to remember them all and make decisions [23]. As a consequence, based on deep learning 
+models, we also proposed to build a sophisticated medical system using the original and 
+the generated images (using the GAN model) for diagnosing corneal cases, to aid clini-
+cians in the interpretation of medical images and improve clinical decision-making. 
+Many researchers used a variety of complex and diverse medical devices to collect 
+data, as well as a variety of diagnostic approaches. Salih and Hussein (2018) used 732 
+submaps as inputs to the deep learning network; a kind of deep learning technology called 
+the VGG-16 network was utilized to predict corneal abnormalities and normality [24]. The 
+
+ 
+ 
+detection of the keratoconus eyes dataset and recognition of the normal cornea was the 
+focus of a group of authors who used 145 normal cornea cases and 312 keratoconus cases 
+from a database of photographs. As a classification tool, they used support vector machine 
+(SVM) and multilayer perceptron methods. The features were extracted from the input 
+images, then passed to the classifiers [25]. A group of researchers used a compilation of 
+data from both Placido and Scheimpug as a feature vector. The prototype was tested with 
+and without a posterior corneal surface, and it performed well in both situations. The 
+thickness and posterior characteristics were found to be critical tools for predicting cor-
+neal keratoconus and avoiding corneal ectasia surgery in patients with early corneal ecta-
+sia disease [26]. Researchers employed machine learning techniques, such as ANN, SVM, 
+regression analysis, and decision tree algorithms, to identify the disease. The information 
+was gathered from a group of patients; in total, 23 cases of ectasia after LASIK were dis-
+covered, as well as 266 stable post-LASIK cases with over a year of follow-up. They con-
+cluded that this study method still needed to be validated [27]. Samer et al. presented a 
+method known as SWFT for diagnosing the corneal image by extracting features from the 
+corneal image using a Wavelet and diagnosing it using an SVM classifier [28]. In 2021, 
+Samer and his participants designed an LIP algorithm to extract corneal image features, 
+and they evaluated their method using many classifiers. Thus, they could train a system 
+capable of automatically classifying corneal diseases [22]. We used deep learning tech-
+niques in the current study to diagnose corneal diseases. GAN networks were used as a 
+tool to generate realistic corneal images. On the other hand, pre-trained convolutional 
+neural networks (CNN) [29–31] are employed in diagnosing corneal diseases, which have 
+recently been used in many medical imaging studies and have been reported to improve 
+performance for a broad range of medical tasks. 
+This paper has made the following contributions: 
+(1) Using the GAN model for creating high-quality corneal images from topograph-
+ical images to solve the scarcity of the cornea dataset. 
+(2) Examining various transfer learning methods as a based solution for the corneal 
+diagnosis task. 
+(3) Augmentation of the dataset to be used in training the networks, using the gener-
+ated synthetic data for improved clinical decisions. 
+(4) Solving the issue of time consumption that is suffered by deep learning networks. 
+2. Corneal Diseases Diagnosis 
+This section begins by describing the data and its features. The architecture of the 
+GAN model for cornea image creation is discussed after that. Due to the restricted quan-
+tity of data available for training transfer learning networks, we have presented a method 
+for augmenting synthesized images. 
+2.1. Dataset 
+The dataset is made up of images taken by scanning the cornea with a device called 
+Pentacam, which generates various images and parameters known as corneal topography. 
+Ophthalmologists use corneal topography to check eye conditions in clinics. Each pa-
+tient’s eye data, which includes four corneal maps (sagittal, corneal thickness (CT), eleva-
+tion front (EF), and elevation back maps (EB)) with a set of parameters, are saved inde-
+pendently [32] (see Figure 1). The data were gathered using a Pentacam (OCULUS, Ger-
+many), an image Scheimpflug instrument. The camera scans the eye from many angles in 
+a circular pattern, producing maps with information about the anterior and posterior parts 
+
+ 
+ 
+of the cornea and a quick screening report. The Pentacam can be upgraded and altered to 
+meet the user’s requirements [33]. 
+Figure 1. The four corneal maps: (a) sagittal, (b) elevation front, (c) corneal thickness, and (d) eleva-
+tion back maps. 
+It is worth noting that the data were obtained from the Al-Amal center in Baghdad, 
+Iraq, and the data were labelled with the help of eye specialists, Dr. Nebras H. Gareb, an 
+Ophthalmic Consultant, and Dr. Sohaib A. Mohammed and Dr. Ali A. Al-Razaq, Senior 
+Specialist Ophthalmologists. The images were categorized based on all four corneal maps, 
+and each map was treated separately and labelled as normal or abnormal. As such, we 
+have eight categories of cornea cases. The collected data contains 3448 images of the four 
+maps that have been scientifically collected and classified. The number of images for each 
+class is 248 Normal_Sagittal, 460 Abnormal_Sagittal, 338 Normal_Corneal Thickness, 548 
+Abnormal_Corneal Thickness, 765 Normal_Elevation Front, 167 Abnormal_Elevation 
+Front, 693 Normal_Elevation Back, and 229 Abnormal_ Elevation Back maps. 
+2.2.  Transfer Learning Models 
+There are numerous common transfer learning models available in computer vision 
+that are typically utilized as a tool for the categorization of medical images; however, in 
+this study, the MobileNetv2 [34], Resnet50 [35], Xception [36], Vision Transformer (ViT) 
+[37], Co-scale conv-attentional image Transformers (CoaT) [38], and Swin transformer 
+(Swin-T) [39] models have been used, which are trained by the original and synthesized 
+images to evaluate the system’s effectiveness for diagnosing corneal instances. The mod-
+els demonstrate the influence of synthesized and imbalanced datasets on the corneal di-
+agnosis task; the data were manipulated, and varied numbers of data were used for train-
+ing and testing. To be balanced, the data were processed using the resample method (over-
+sampling and downsampling). After training each transfer learning model, the results are 
+compared to the results of other approaches see tables 3 and 4. 
+ 
+ 
+
+SinsOrdasPiot
+EevetonFrort
+TO.O
+OD
+50
+OD
+110
+ICO
+42:0
+10
+-10
+-30
+340
+200
+10.0
+toredTstine
+BFS-653PatDe-100
+333
+OD
+OD
+国中
+10
+AI
+Sn
+C
+d 
+ 
+The Resnet50 forecasts the delta required to get from one layer to the next and arrive 
+at the final prediction. It addresses the vanishing gradient problem by enabling the gradi-
+ent to flow through an additional shortcut path. It enables the model to skip over a CNN 
+weight layer if it is not required. This helps to avoid the difficulty of overfitting the train-
+ing set. ResNet50 is a 50-layer network [36]. The MobileNetv2 is a convolutional architec-
+ture built for usage with mobile or low-cost devices that minimizes network cost and size 
+[40]. Segmentation, classification, and object recognition may all be performed with the 
+MobileNetV2 model. In comparison to its predecessor, MobileNetV2 includes two new 
+features occurring linearly between layers, and bottleneck shortcuts are established [41]. 
+Xception is a depthwise separable convolutions-based deep convolutional neural network 
+architecture; Google researchers came up with the idea. Xception has three different flows: 
+entry, middle, and exit. The data initially pass via the entering flow, then eight times 
+through the middle flow, and finally through the exit flow. Batch normalization is applied 
+to all convolution and separable convolution layers [36]. 
+[37] have investigated the possibility of using transformers for straightforward image 
+recognition. Apart from the initial patch extraction step, this architecture does not have 
+any image-specific inductive biases, which sets it apart from previous research leveraging 
+self-attention in computer vision. Instead, [37,42] employs a standard transformer encoder 
+seen in natural language processing to decode an image as a sequence of patches. With 
+pre-training on massive datasets, this straightforward approach scales remarkably well. 
+Therefore, vision transformer competes with or outperforms the state-of-the-art on many 
+picture classification datasets, while only requiring a little initial investment. CoaT, an 
+image classifier based on the transformer, features cross-scale attention and efficient conv-
+attention operations, and is given in [38]. CoaT models achieve strong classification results 
+on ImageNet, and their utility for subsequent computer vision tasks, such as object detec-
+tion and instance segmentation, has been established. In [39], a novel vision Transformer 
+called Swin-T is introduced; it generates a hierarchical feature representation and scales 
+computationally linearly with the size of the input image. 
+For all models, corneal images were fed into the networks to train the models and 
+extract the weights. For 20 epochs, we used a batch size of 32. Moreover, we employed the 
+Adam optimization approach, with a learning rate of 0.001, to iteratively modify network 
+weights. Table 1 displays all of the parameter values utilized by the various classifiers. 
+Table 1. Values of the parameters used in the classifiers. 
+Method. 
+Image size 
+Parameters 
+MobilenetV2 
+224×224 
+3.5M 
+Resnet50 
+224×224 
+25.6 
+Xception 
+299×299 
+22.9 
+ViT 
+128×128 
+36.3 
+CoaT 
+224×224 
+22M 
+Swin-T 
+224×224 
+29M 
+2.3.  Generating Synthetic Cornea Images 
+The diagnostic ratio is negatively affected by a lack of data [43], and this is the fun-
+damental challenge with model training [44]. We synthesized new examples that were 
+learned from existing data examples using a new way of producing synthetic corneal im-
+ages using generative adversarial networks (GANs) to expand the training data and en-
+hance diagnostic rates. GANs are deep CNN networks that generate new data from pre-
+viously trained data such as images [45]. For synthesizing labeled images of the cornea, 
+we employed conditional GANs [46]. The structure of the CGAN model used in this work 
+(see Figure 2) is two networks that compete against one another to achieve a common 
+goal, which is to learn the distribution of data 𝑝𝑑𝑎𝑡𝑎 from samples (images in our work). 
+Whereas in the first network, called the generator G network, an image G(x) is generated, 
+
+ 
+ 
+usually from noise shaped by the uniform distribution 𝑃𝑧, which is close to the target im-
+age, as it produces an image representing the class you want to generate, in addition to 
+noise, to function as an assistant factor that aids the model in synthesizing images that are 
+close to reality. On the other hand, the second network, dubbed Discriminator 𝐷, tries to 
+discern between real and fake images entered into the network; in other words, the input 
+is 𝑥, whereas the output is D(x). It compares the image created by the rival network to the 
+actual image. The loss function, shown in equation (1), is optimized to train adversarial 
+networks [47]. 
+𝑚𝑖𝑛𝐺𝑚𝑎𝑥𝐷=𝐸𝑥~𝑃𝑑𝑎𝑡𝑎𝑙𝑜𝑔𝐷(𝑥) + 𝐸𝑧~𝑃𝑧[log (1 − 𝐷(𝐺(𝑧)))] (1) 
+where the 𝐷 is trained to maximize 𝐷(𝑥) for images derived from real data and 
+minimize the D(x) that is derived from not real data. On the other hand, the Generator 
+seeks to trick the Discriminator by generating an image 𝐺(𝑧), which calls for maximizing 
+the value of 𝐷(𝐺(𝑧)). These two networks are still in competition during the training 
+phase, with the Generator attempting to improve its performance to deceive the Discrim-
+inator, while the latter distinguishes between the real and fake images. 
+Figure 2. Structure of the proposed system. 
+The generator accepts a vector of random numbers with a size of 100 created by uni-
+form distribution, and this vector reshapes into 4×4×1024. The architecture involved four 
+deconvolution layers to up-sample the image using a 5×5 filter size. Finally, the output is 
+the image with a size of 64×64×3. Except for the last layer, batch normalization and ReLU 
+activation functions are used. The Discriminator-issued class label, in addition to the real 
+or fake decision, derives from a corneal image with size 64×64×3 using a filter with size 
+5x5 with four convolutional layers. To reduce the spatial dimensionality, stride convolu-
+tion is used in each layer. Batch normalization and ReLU were also applied in each layer 
+(except the fully connected layer). 
+The training of CGAN was conducted separately to generate every corneal image 
+category, as well as conducted iteratively for the Discriminator and Generator. The noise 
+sample 𝑍1. … 𝑍𝑛 derives from a uniform distribution in the range [–11], 𝑛 = 100. The 
+slope of the leak of ReLU was equal to 0.2. The zero-centered center normal distribution 
+was employed to initialize the weights with a standard deviation of 0.02. Moreover, for 20 
+epochs, we used the Adam optimizer, and the learning rate was equal to 0.0001. Figure 2 
+illustrates the structure of the proposed system. 
+
+Loss Function
+Noise
+Classifiers
+Gcncrator
+MobileNet V2
+Encoder
+Decoder
+Gcnerated Imagcs
+Discriminator
+Labels
+ResNet 50
+Synthesize Images
+Encoder
+Encoder Nctwork
+Dccoder Network
+Xccption
+Results
+Encodcr Nctwork
+ViT
+Abnormal
+Ground Truth
+CoaT
+Swin-T
+an 
+ 
+3. Results 
+The goal of this research, in which all of the steps have been outlined in detail in 
+Table 2, is to find out to what extent generated data affect the diagnosis of corneal diseases, 
+and how well classifiers can classify them. Therefore, the CGAN model has been trained 
+to deal with data disparities; in other words, each corneal disease’s image generated is 
+separated with high-quality topographical images by using fine-tuning parameters to dis-
+band the scarcity of the cornea dataset. For clinical decision transfer, learning methods 
+have been exploited, where the augmented dataset is used in training the networks. 
+Table 2. Proposed Method: 
+ 
+Inputs: D: Dataset, img: a cornea’s image which is selected from the D; 
+1 
+GI = Build a model M which generate images from noise and targeting D  
+2 
+For I = 1: CNN classifieres // (MobilenetV2, Resnet50, Xception, ViT, CoaT, and Swin-T) 
+3 
+[accuracy, precision, recall, f1-score] = Calculate metrics [Accuracy, Precision, 
+Recall, F1-score] from GI  
+4 
+End for 
+5 
+[SSIM, MSE, PSNR, FID] = Calculate [SSIM, MSE, PSNR, FID] between an image from 
+GI and D  
+6 
+End 
+ 
+The results of diagnosing corneal diseases are reported using different types of trans-
+fer learning models, such as MobileNetv2, Resnet50, and Xception. 
+To detect the importance of data generation, as well as its effect on classification tasks, 
+we used the original dataset to train and test each classifier with and without corneal-
+generated images. 
+On the other hand, to assess the strength of the synthesis model and its ability to 
+synthesize convergent data in a particular category and divergent from other categories, 
+each classifier was trained on the synthesized data without using the original data. We 
+employed eight-fold cross-validation with case separation at the patient level in all of our 
+experiments and evaluations. The used examples contained the corneal cases (normal or 
+abnormal for each corneal map). 
+For each batch of data images, we trained the network and assessed the outcomes 
+individually. The CGAN architecture is used to train each corneal-case class separately, 
+utilizing the same eight-fold cross-validation method and data split. Following training, 
+the generator is capable of creating realistic corneal case images separately using a vector 
+of noise formed by uniform distributions (see Figure 3). Accordingly, the model synthe-
+sized eight different cases of corneal images: normal and abnormal cases for sagittal, cor-
+neal thickness, elevation front, and elevation back images. 
+ 
+We employed two main kinds of metrics in our research. First, we used observational 
+error metrics such as accuracy, precision, recall, and F1-score metrics to evaluate classifi-
+cation accuracy (equations 2, 3, 4, and 5, respectively). Second, we used equations 6 and 7 
+to evaluate the synthesized image’s quality with the original images via the structural 
+similarity index method (SSIM) [48] and the peak signal-to-noise ratio (PSNR) [49]. 
+𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = 
+𝑇𝑃+𝑇𝑁
+𝑇𝑁+𝑇𝑃+𝐹𝑁+𝐹𝑃 (2) 
+𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 =
+𝑇𝑃
+𝑇𝑃+𝐹𝑃 (3) 
+𝑅𝑒𝑐𝑎𝑙𝑙 =
+𝑇𝑃
+𝑇𝑃+𝐹𝑁 (4) 
+𝐹1_𝑆𝑐𝑜𝑟𝑒 =
+2𝑇𝑃
+2𝑇𝑃+𝐹𝑃+𝐹𝑁 (5) 
+where TP = true positives, TN = true negatives, FP = false positives, and FN = false 
+negatives. 
+
+ 
+ 
+Structural similarity (SSIM) [48] is an image quality measurement based on equation 
+(6) between the approximated image 𝑦𝑒
+𝐿 and the ground truth image 𝑦𝑡
+𝐿. 
+𝑆𝑆𝐼𝑀(𝑦𝑡
+𝐿, 𝑦𝑒
+𝐿) =
+1
+𝑀 ∑
+(2𝜇𝑗𝑡𝜇𝑗𝑒+𝑐1)(2𝜎𝑗𝑡𝑒+𝑐2)
+(𝜇2𝑗𝑡+𝜇2𝑗𝑒+𝑐1)(𝜎2𝑗𝑡+𝜎2𝑗𝑒+𝑐2
+𝑀
+𝑗=1
+ (6) 
+In contrast, peak signal-to-noise ratio (PSNR) [49] is an objective assessment based 
+on comparisons using particular numerical criteria [50,51]; a higher PSNR value indicates 
+better image quality. Images generated by equation (7) have significant numerical differ-
+ences at the low end of the PSNR scale [52,53]. 
+𝑃𝑆𝑁𝑅(𝑓, 𝑔) = 10𝑙𝑜𝑔10(
+2552
+𝑀𝑆𝐸(𝑓,𝑔))) (7) 
+MATLAB2020b is used for the implementation of corneal diagnosing. All training 
+processes were performed using an NVIDIA GeForce GTX 1660 GPU. 
+Using the above-mentioned metrics for different classifiers, few results were rec-
+orded when no synthesized data were used; this might be due to overfitting over the 
+smaller number of training images. Conversely, using the CGAN model, the results im-
+proved as the number of training instances grew (see Table 3). 
+Table 3. Performance comparison for classification of corneal conditions among obstetric models 
+(%). 
+Classifier 
+Data 
+Accuracy 
+Precision 
+Recall 
+F1-score 
+MobilenetV2 
+Original 
+75.2 
+72.4 
+73.2 
+72.3 
+Synthesized 
+88.6 
+86.5 
+89.8 
+87.5 
+Resnet50 
+Original 
+77.13 
+74.6 
+74.6 
+74.3 
+Synthesized 
+90.5 
+90 
+90.4 
+90.1 
+Xception 
+Original 
+78.9 
+75.6 
+75.7 
+75.1 
+Synthesized 
+90.7 
+90 
+90.6 
+90.2 
+ViT 
+Original 
+71.2 
+68.2 
+68.1 
+67 
+Synthesized 
+88.7 
+90.7 
+84.4 
+86.2 
+CoaT 
+Original 
+65.6 
+64.9 
+65.2 
+65.1 
+Synthesized 
+69.3 
+68.1 
+68.4 
+68.2 
+Swin-T 
+Original 
+58.4 
+56.3 
+57.5 
+56.9 
+Synthesized 
+63.4 
+62.5 
+62.7 
+62.6 
+Since our data images are unbalanced, we suggested revealing how the corneal diag-
+nosis would be affected if a balanced dataset was available. Therefore, we used the tradi-
+tional data balancing methods, where we conducted data resampling using both ap-
+proaches to make a balanced dataset out of an imbalanced one. The first approach was 
+undersampling (keeping all samples in the rare class and randomly selecting an equal 
+number of samples in the abundant class); the second approach was oversampling (in-
+creasing the size of rare samples using repetition). These two approaches were applied to 
+the data before and after generating images. 
+Results reported that, generally, when applying data resampling on the original data 
+(before using the CGAN model), the classifiers achieved a moral performance, while the 
+data were balanced. Moreover, training by oversampling synthesized data for all classifi-
+ers outperforms training by underdamped synthesized data. On the other hand, applying 
+oversampled data on the generated image (after implementing the CGAN model) will not 
+affect the classifier results since the data are vast enough to train the models correctly. In 
+contrast, undersampling negatively affected the achievement of classifiers due to the data 
+being decreased again (see Table 4). 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+ 
+Table 4. Performance comparison for classifying corneal conditions among obstetric models after 
+balancing data (%). 
+OVS: oversampling, UNS: undersampling. 
+This issue of whether the set of images generated was sufficiently distinct to allow 
+classification between the corneal case categories was investigated with the help of an 
+expert ophthalmologist. We provided him with 500 randomly generated images with var-
+ious categories to classify and diagnose. Table 5 summarizes the findings, and Table 6 
+shows the average of SSIM and PSNR for a random selection of 100 images. 
+ 
+Figure 3. Samples of the generated image using the Conditional Generative Adversarial Network 
+(CGAN) model. 
+Table 5. The results from the expert (%). 
+ 
+Sagittal Images 
+CT Images 
+EF and EB Images 
+Diagnosis by an Expert 
+0.94 
+0.98 
+0.93 
+Table 6. Average of Structural Similarity Index (SSIM) and peak signal-to-noise ratio (PSNR) for 
+100 random images. 
+SSIM 
+PSNR 
+0.872 
+33.221 
+The SSIM and PSNR have been calculated before and after training the CGAN model 
+on a random sample of 100 images. Table 6 shows that the model can generate synthetic 
+Classifier 
+Data 
+Accuracy 
+Precision 
+Recall 
+F1-score 
+OVS 
+UNS 
+OVS 
+UNS 
+OVS 
+UNS 
+OVS 
+UNS 
+MobilenetV2 
+Original 
+85.5 
+75.4 
+85.7 
+76.2 
+85.5 
+75.4 
+85.3 
+75.4 
+Synthesized 
+88.5 
+81.1 
+88.8 
+81.6 
+88.4 
+81.1 
+88.4 
+81 
+Resnet50 
+Original 
+86.36 
+75.7 
+86.3 
+76 
+86.6 
+75.7 
+86.3 
+75.5 
+Synthesized 
+90.2 
+82.8 
+90.8 
+83 
+90.8 
+82.8 
+90.8 
+82.7 
+Xception 
+Original 
+86 
+77.3 
+86.2 
+78 
+86 
+77.3 
+85.9 
+76.7 
+Synthesized 
+90 
+82.7 
+90.3 
+82.7 
+90 
+82.6 
+90 
+82.6 
+ViT 
+Original 
+74.5 
+70.9 
+73.2 
+68.4 
+72.8 
+69.5 
+73 
+68.9 
+Synthesized 
+89.8 
+86.1 
+88.2 
+85.5 
+88.9 
+85.6 
+88.5 
+85.6 
+CoaT 
+Original 
+69.4 
+63.7 
+69.1 
+62.6 
+68.9 
+62.8 
+69 
+62.7 
+Synthesized 
+73.8 
+66.8 
+72.6 
+65.7 
+72.9 
+65.9 
+72.7 
+65.8 
+Swin-T 
+Original 
+60.2 
+56.9 
+59.8 
+56.5 
+58.9 
+56.5 
+59.3 
+56.5 
+Synthesized 
+65.6 
+61.7 
+64.6 
+60.8 
+64 
+60 
+64.3 
+60.4 
+
+32. 
+ 
+images very close to the original. Therefore, we can consider those images to be legitimate 
+for training CNNs models, and ophthalmologists can use them in clinical research. 
+The CNN classifiers are repeatedly tested in this work to determine the testing pro-
+cess. The suggested model can be applied in real-time, where testing images only takes a 
+few moments, according to Table 7. While the CoaT model requires the longest ATT, the 
+ATT for ViT beats the other classifiers. 
+Table 7.  Convolutional neural network (CNN) classifier’s average time test (ATT) (sec.). 
+MobilenetV2 
+Resnet50 
+Xception 
+ViT 
+CoaT 
+Swin-T 
+0.0258 
+0.0187 
+0.0152 
+0.0108 
+0.0342 
+0.0203 
+ 
+The high quality of the images can be seen in the images synthesized from the test 
+images using the CGAN model, which are displayed in Figure 4. It is also possible to no-
+tice the stability of the structures and morphologies of the images. 
+Figure 4. Example of original and synthesis images. 
+4. Discussion 
+The objectives of this work were to apply the CGAN model to generate synthetic 
+medical images for data augmentation to expand limited datasets and improve clinical 
+decision-making for corneal diseases. Thus, we investigated the extent to which synthetic 
+corneal images help another system perform better behind the scenes. The study used a 
+small dataset comprising the sagittal, corneal thickness, elevation front, and elevation 
+back of corneal images. Each class has its distinct characteristics, although there is consid-
+erable intra-class variation. Our diagnosis was based on the four maps, each of which was 
+examined to determine whether it was normal or diseased. To identify corneal disorders, 
+a variety of transfer learning architectures were employed. We discovered that by utiliz-
+ing the CGAN model to synthesize extra realistic images, we could increase the size of the 
+training data groups, thus boosting the clinical decision. The diagnostic outcomes for mo-
+bilenetV2, Resnet50, Xception, ViT, CoaT, and Swin-T classifiers improved from 75.2 % to 
+88.6 %, 77.13% to 90.5%, 78.9% to 90.7 %, 71.2% to 88.7%, 65.6% to 69.3%, and 58.4% to 
+63.4%, respectively. Results from Table 3 show that the synthetic data samples generated 
+can increase the variability of the input dataset, resulting in more accurate clinical deci-
+sions. 
+The scores demonstrate that the synthesized images have useful visuals and, more 
+crucially, useful characteristics that may be used in computer-aided diagnosis. The other 
+aspect of this research is to test the effect of data balance on diagnostic results, where we 
+used the resampling method to make the dataset balanced. The results showed that train-
+ing the model before generating a new set of data on a balanced dataset is very important, 
+especially in circumstances where data are scarce. On the contrary, we did not notice a 
+significant impact on the performance of the classifiers when using the data resampling 
+Original 
+images  
+Synthesis 
+images  
+
+ 
+ 
+on the generated data because the data was sufficient and suitable for training the models 
+without the need to balance them using data balancing methods. This is clear evidence of 
+the importance of the model proposed in this paper. In a final experiment, we compared 
+the performance of the classifiers-based systems employed in this study for clinical deci-
+sion-making (Table 4). The highest performance was derived from synthesized data in the 
+Xception classifier, whereas the best performance came from using balance data in Res-
+net50 when using the oversampling approach, but the ViT model while using the under-
+sampling approach. 
+This work has several limitations. For example, the training complexity was en-
+hanced by training distinct GANs for each corneal case class. It might be useful to look 
+into GAN designs that produce multi-class samples at the same time. Another type of 
+GAN learning process might increase the quality of the corneal image. It is also possible 
+to do more research to improve the training loss function by adding regularization terms. 
+Because the human factor is critical in evaluating the proposed model’s outputs, an 
+expert opinion was obtained after providing him with a set of generated corneal images 
+containing a randomly selected set of normal and abnormal corneal images. The following 
+was the expert’s opinion: “Creating a new template for the corneal topographical of four 
+refractive maps is considered an interesting subject as it enriched the overall expected 
+shapes that could be seen during the daily clinic. These new images which created based 
+on real cases collected previously and diagnosed that the new images are still inside the 
+reality borderlines. Gain good experience with the new shapes and specify the further 
+required steps of a diagnosis other than the topographical maps that could be specified 
+advanced for predicted out-of-skim cases. In such a way, offline training for the new oph-
+thalmologists and improving the skill of diagnosis with the preparation for new unseen 
+cases could be done.” In the future, we look to develop our research to exploit other GANs 
+that might benefit from corneal image synthesis for better achievement. 
+5. Conclusion 
+In conclusion, we proposed a strategy for improving performance in a medical issue 
+with little data by generating synthetic medical images for data augmentation. On a cor-
+neal diseases diagnosis task, we discovered that synthetic data augmentation beat tradi-
+tional data augmentation in accuracy by roughly 13%. Additionally, we investigated the 
+performance of the classifiers in different conditions, and we found that while working 
+with cornea images to diagnose diseases, the Xcepton classifier is more responsive than 
+the rest of the used classifiers. We anticipate that synthetic augmentation can help with a 
+variety of medical issues and that the method we have outlined can lead to more powerful 
+and reliable support systems. 
+Author Contributions: “Conceptualization, S. K. J. and S. A.; methodology, S. K. J., S. A., N. H. G., 
+and J. M.; software, S. K. J. and S. A.; validation, S. A., N. H. G., and J. M.; formal analysis, S. K. J., 
+and S. A.; investigation, N. H. G., and J. M.; resources, S. K. J., S. A., N. H. G., and J. M.; data curation, 
+S. K. J., S. A., N. H. G., and J. M.; writing—original draft preparation, S. K. J., S. A., N. H. G., and J. 
+M.; writing—review and editing, T. A. R. and S. Q. S.; visualization, T. A. R.; supervision, T. A. R.; 
+project administration; funding acquisition, S. Q. S. and P. S. J. All authors have read and agreed to 
+the published version of the manuscript.” 
+Funding: Dr.P. S. JosephNg, Faculty of Data Science & Information Technology, INTI International 
+University, Persiaran Perdana BBN, 71800 Nilai, Negeri Sembilan, Malaysia. 
+Institutional Review Board Statement: The manuscript is conducted within the ethical manner ad-
+vised by the targeted journal. 
+Informed Consent Statement: Not applicable. 
+Data Availability Statement: Data can be shared upon request from the corresponding author. 
+Acknowledgments: None. 
+Conflicts of Interest: The authors declare no conflict of interest to any party. 
+
+ 
+ 
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+page_content='my  Correspondence: tarik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content='R,);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' joseph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='ng@newinti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='my (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=')  Abstract: Corneal diseases are the most common eye disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Deep learning techniques are used  to perform automated diagnoses of cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Deep learning networks require large-scale annotated  datasets, which is conceded as a weakness of deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' In this work, a method for synthesizing  medical images using conditional generative adversarial networks (CGANs), is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' It also  illustrates how produced medical images may be utilized to enrich medical data, improve clinical  decisions, and boost the performance of the conventional neural network (CNN) for medical image  diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The study includes using corneal topography captured using a Pentacam device from  patients with corneal diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The dataset contained 3448 different corneal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Furthermore, it  shows how an unbalanced dataset affects the performance of classifiers, where the data are balanced  using the resampling approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Finally, the results obtained from CNN networks trained on the  balanced dataset are compared to those obtained from CNN networks trained on the imbalanced  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' For performance, the system estimated the diagnosis accuracy, precision, and F1-score met-  rics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Lastly, some generated images were shown to an expert for evaluation and to see how well  experts could identify the type of image and its condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The expert recognized the image as useful  for medical diagnosis and for determining the severity class according to the shape and values, by  generating images based on real cases that could be used as new different stages of illness between  healthy and unhealthy patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Keywords: conditional generative adversarial networks, transfer learning, synthesize images, cor-  neal diseases, data augmentation  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Introduction  Medical image datasets are one of the most important problems facing researchers in  the field of machine learning [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The limited amount of medical data comes from the dif-  ficulty of capturing it [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' With the problem of final ethical approval, the acquisition and  labelling of medical images are time-consuming, and considerable effort needs to be spent  by both researchers and specialists [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Several studies tried to overcome the dataset  scarcity challenge through the famous task in computer vision, a method called data aug-  mentation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Using classic data augmentation can give a simple extra feature where it  involves simple modifications, such as rotation, translation, scaling, and flipping [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' On  the other hand, some researchers employed innovative techniques for data augmentation  to improve the system training process, based on synthesizing high-quality sample im-  ages using a generative model known as generative adversarial networks (GANs) [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  The GANs involved two networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' the first generates a real image from the input  with the help of the noise, and the other discriminates between real and fake (generated  by the first network) images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' This model has been used in many studies hoping to gener-  ate realistic images, especially for medical imaging applications, such as image-to-image  translation [10], image inpainting [11], segmentation-to-image translation [12], medical  cross-modality translations [13], and label-to-segmentation translation [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Exploiting the GAN models by researchers led to the creation of cross-modality im-  ages, such as a PET scan, which was generated from a CT scan of the abdomen to show  the presence of liver lesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The GAN model of image inpainting has served as inspira-  tion for many studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' [15] used a fully convolutional network to learn retinal  vessel segmentation images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The binary vessel tree was then translated into a new retinal  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' By using chest X-ray images, Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' [16] generated lung and heart image seg-  mentation by training a GAN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' [17] trained a model to translate brain MRI  images into binary segmentation maps for brain tumour images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Nie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' [18] trained a  patch-based GAN to translate between brain CT and MRI images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' As a step of image re-  finement, they recommended using an auto-context model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Schlegl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' [19] trained a  GAN model on normal retinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' To detect anomalies in retinal images, the model was tested  on normal and abnormal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Based on what was mentioned above, the scarcity of data needs to be resolved so that  researchers can use it more freely to analyze that data and produce results that serve the  scientific process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The latter motivated the authors of this paper to use GAN models with  the ability to synthesize real images, increase the existing data, and overcome the problem  of lacking data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' In this work, high-quality corneal images based on GAN models are syn-  thesized for a specific task of corneal disease diagnosis to improve the clinical decision by  introducing different stages and predicted shapes for images with illness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' As an illustrated  sample of manipulation for the imaging in the cornea, the different stages of keratoconus  are, in most cases, unclear in borderlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' From a clinical perspective, overlapping features  between stages of keratoconus lead to a controversial approach to treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' To decide  the severity and clinical or surgical procedure of work per patient clinically, considerable  evidence is collected from different images per case to reach the final approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The pos-  sibility of studying the effect and weight of this evidence per case is an attractive medical  training to produce a final highly medical sensation and observation for the trained phy-  sician.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' In more detail, thinning in pachymetry images with its location, steepening in the  inferior or superior position of the tangential mapping, and the isolated land or tongue  shape that may appear in elevation front and back maps, with the astigmatism axis and  obliqueness of the bowtie, would improve the effectiveness of the final diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  The cornea, which protects the eye from external substances and helps to control vis-  ual focus, is stiff but very sensitive to touch [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' There are many corneal disorders, for  instance, bullous keratopathy, Cogan syndrome, corneal ulcer, herpes simplex keratitis,  herpes zoster ophthalmicus, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Any disorders in the cornea may cause ripping, dis-  comfort, and dwindling vision clarity and, finally, may lead to blindness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' On the other  hand, any action on the cornea, such as vision correction, requires a diagnosis of the cor-  nea’s health before treatment [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Clinical decisions on the human cornea require review-  ing numerous aspects, and ophthalmologists must handle this revision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Corneal topo-  graphical parameters are so extensive that it is difficult for surgeons or ophthalmologists  to remember them all and make decisions [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' As a consequence, based on deep learning  models, we also proposed to build a sophisticated medical system using the original and  the generated images (using the GAN model) for diagnosing corneal cases, to aid clini-  cians in the interpretation of medical images and improve clinical decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Many researchers used a variety of complex and diverse medical devices to collect  data, as well as a variety of diagnostic approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Salih and Hussein (2018) used 732  submaps as inputs to the deep learning network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' a kind of deep learning technology called  the VGG-16 network was utilized to predict corneal abnormalities and normality [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The  detection of the keratoconus eyes dataset and recognition of the normal cornea was the  focus of a group of authors who used 145 normal cornea cases and 312 keratoconus cases  from a database of photographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' As a classification tool, they used support vector machine  (SVM) and multilayer perceptron methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The features were extracted from the input  images, then passed to the classifiers [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' A group of researchers used a compilation of  data from both Placido and Scheimpug as a feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The prototype was tested with  and without a posterior corneal surface, and it performed well in both situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The  thickness and posterior characteristics were found to be critical tools for predicting cor-  neal keratoconus and avoiding corneal ectasia surgery in patients with early corneal ecta-  sia disease [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Researchers employed machine learning techniques, such as ANN, SVM,  regression analysis, and decision tree algorithms, to identify the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The information  was gathered from a group of patients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' in total, 23 cases of ectasia after LASIK were dis-  covered, as well as 266 stable post-LASIK cases with over a year of follow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' They con-  cluded that this study method still needed to be validated [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Samer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' presented a  method known as SWFT for diagnosing the corneal image by extracting features from the  corneal image using a Wavelet and diagnosing it using an SVM classifier [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' In 2021,  Samer and his participants designed an LIP algorithm to extract corneal image features,  and they evaluated their method using many classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Thus, they could train a system  capable of automatically classifying corneal diseases [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' We used deep learning tech-  niques in the current study to diagnose corneal diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' GAN networks were used as a  tool to generate realistic corneal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' On the other hand, pre-trained convolutional  neural networks (CNN) [29–31] are employed in diagnosing corneal diseases, which have  recently been used in many medical imaging studies and have been reported to improve  performance for a broad range of medical tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  This paper has made the following contributions:  (1) Using the GAN model for creating high-quality corneal images from topograph-  ical images to solve the scarcity of the cornea dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  (2) Examining various transfer learning methods as a based solution for the corneal  diagnosis task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  (3) Augmentation of the dataset to be used in training the networks, using the gener-  ated synthetic data for improved clinical decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  (4) Solving the issue of time consumption that is suffered by deep learning networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Corneal Diseases Diagnosis  This section begins by describing the data and its features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The architecture of the  GAN model for cornea image creation is discussed after that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Due to the restricted quan-  tity of data available for training transfer learning networks, we have presented a method  for augmenting synthesized images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Dataset  The dataset is made up of images taken by scanning the cornea with a device called  Pentacam, which generates various images and parameters known as corneal topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Ophthalmologists use corneal topography to check eye conditions in clinics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Each pa-  tient’s eye data, which includes four corneal maps (sagittal, corneal thickness (CT), eleva-  tion front (EF), and elevation back maps (EB)) with a set of parameters, are saved inde-  pendently [32] (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The data were gathered using a Pentacam (OCULUS, Ger-  many), an image Scheimpflug instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The camera scans the eye from many angles in  a circular pattern, producing maps with information about the anterior and posterior parts  of the cornea and a quick screening report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The Pentacam can be upgraded and altered to  meet the user’s requirements [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The four corneal maps: (a) sagittal, (b) elevation front, (c) corneal thickness, and (d) eleva-  tion back maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  It is worth noting that the data were obtained from the Al-Amal center in Baghdad,  Iraq, and the data were labelled with the help of eye specialists, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Nebras H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Gareb, an  Ophthalmic Consultant, and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Sohaib A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Mohammed and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Ali A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Al-Razaq, Senior  Specialist Ophthalmologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The images were categorized based on all four corneal maps,  and each map was treated separately and labelled as normal or abnormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' As such, we  have eight categories of cornea cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The collected data contains 3448 images of the four  maps that have been scientifically collected and classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The number of images for each  class is 248 Normal_Sagittal, 460 Abnormal_Sagittal, 338 Normal_Corneal Thickness, 548  Abnormal_Corneal Thickness, 765 Normal_Elevation Front, 167 Abnormal_Elevation  Front, 693 Normal_Elevation Back, and 229 Abnormal_ Elevation Back maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Transfer Learning Models  There are numerous common transfer learning models available in computer vision  that are typically utilized as a tool for the categorization of medical images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' however, in  this study, the MobileNetv2 [34], Resnet50 [35], Xception [36], Vision Transformer (ViT)  [37], Co-scale conv-attentional image Transformers (CoaT) [38], and Swin transformer  (Swin-T) [39] models have been used, which are trained by the original and synthesized  images to evaluate the system’s effectiveness for diagnosing corneal instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The mod-  els demonstrate the influence of synthesized and imbalanced datasets on the corneal di-  agnosis task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' the data were manipulated, and varied numbers of data were used for train-  ing and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' To be balanced, the data were processed using the resample method (over-  sampling and downsampling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' After training each transfer learning model, the results are  compared to the results of other approaches see tables 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  SinsOrdasPiot  EevetonFrort  TO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='O  OD  50  OD  110  ICO  42:0  10  10  30  340  200  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='0  toredTstine  BFS-653PatDe-100  333  OD  OD  国中  10  AI  Sn  C  d  The Resnet50 forecasts the delta required to get from one layer to the next and arrive  at the final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' It addresses the vanishing gradient problem by enabling the gradi-  ent to flow through an additional shortcut path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' It enables the model to skip over a CNN  weight layer if it is not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' This helps to avoid the difficulty of overfitting the train-  ing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' ResNet50 is a 50-layer network [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The MobileNetv2 is a convolutional architec-  ture built for usage with mobile or low-cost devices that minimizes network cost and size  [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Segmentation, classification, and object recognition may all be performed with the  MobileNetV2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' In comparison to its predecessor, MobileNetV2 includes two new  features occurring linearly between layers, and bottleneck shortcuts are established [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Xception is a depthwise separable convolutions-based deep convolutional neural network  architecture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Google researchers came up with the idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Xception has three different flows:  entry, middle, and exit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The data initially pass via the entering flow, then eight times  through the middle flow, and finally through the exit flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Batch normalization is applied  to all convolution and separable convolution layers [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  [37] have investigated the possibility of using transformers for straightforward image  recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Apart from the initial patch extraction step, this architecture does not have  any image-specific inductive biases, which sets it apart from previous research leveraging  self-attention in computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Instead, [37,42] employs a standard transformer encoder  seen in natural language processing to decode an image as a sequence of patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' With  pre-training on massive datasets, this straightforward approach scales remarkably well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Therefore, vision transformer competes with or outperforms the state-of-the-art on many  picture classification datasets, while only requiring a little initial investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' CoaT, an  image classifier based on the transformer, features cross-scale attention and efficient conv-  attention operations, and is given in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' CoaT models achieve strong classification results  on ImageNet, and their utility for subsequent computer vision tasks, such as object detec-  tion and instance segmentation, has been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' In [39], a novel vision Transformer  called Swin-T is introduced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' it generates a hierarchical feature representation and scales  computationally linearly with the size of the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  For all models, corneal images were fed into the networks to train the models and  extract the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' For 20 epochs, we used a batch size of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Moreover, we employed the  Adam optimization approach, with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='001, to iteratively modify network  weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Table 1 displays all of the parameter values utilized by the various classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Values of the parameters used in the classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Image size  Parameters  MobilenetV2  224×224  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='5M  Resnet50  224×224  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6  Xception  299×299  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='9  ViT  128×128  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='3  CoaT  224×224  22M  Swin-T  224×224  29M  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Generating Synthetic Cornea Images  The diagnostic ratio is negatively affected by a lack of data [43], and this is the fun-  damental challenge with model training [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' We synthesized new examples that were  learned from existing data examples using a new way of producing synthetic corneal im-  ages using generative adversarial networks (GANs) to expand the training data and en-  hance diagnostic rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' GANs are deep CNN networks that generate new data from pre-  viously trained data such as images [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' For synthesizing labeled images of the cornea,  we employed conditional GANs [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The structure of the CGAN model used in this work  (see Figure 2) is two networks that compete against one another to achieve a common  goal, which is to learn the distribution of data 𝑝𝑑𝑎𝑡𝑎 from samples (images in our work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Whereas in the first network, called the generator G network, an image G(x) is generated,  usually from noise shaped by the uniform distribution 𝑃𝑧, which is close to the target im-  age, as it produces an image representing the class you want to generate, in addition to  noise, to function as an assistant factor that aids the model in synthesizing images that are  close to reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' On the other hand, the second network, dubbed Discriminator 𝐷, tries to  discern between real and fake images entered into the network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' in other words, the input  is 𝑥, whereas the output is D(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' It compares the image created by the rival network to the  actual image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The loss function, shown in equation (1), is optimized to train adversarial  networks [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  𝑚𝑖𝑛𝐺𝑚𝑎𝑥𝐷=𝐸𝑥~𝑃𝑑𝑎𝑡𝑎𝑙𝑜𝑔𝐷(𝑥) + 𝐸𝑧~𝑃𝑧[log (1 − 𝐷(𝐺(𝑧)))] (1)  where the 𝐷 is trained to maximize 𝐷(𝑥) for images derived from real data and  minimize the D(x) that is derived from not real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' On the other hand, the Generator  seeks to trick the Discriminator by generating an image 𝐺(𝑧), which calls for maximizing  the value of 𝐷(𝐺(𝑧)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' These two networks are still in competition during the training  phase, with the Generator attempting to improve its performance to deceive the Discrim-  inator, while the latter distinguishes between the real and fake images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Structure of the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  The generator accepts a vector of random numbers with a size of 100 created by uni-  form distribution, and this vector reshapes into 4×4×1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The architecture involved four  deconvolution layers to up-sample the image using a 5×5 filter size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Finally, the output is  the image with a size of 64×64×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Except for the last layer, batch normalization and ReLU  activation functions are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The Discriminator-issued class label, in addition to the real  or fake decision, derives from a corneal image with size 64×64×3 using a filter with size  5x5 with four convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' To reduce the spatial dimensionality, stride convolu-  tion is used in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Batch normalization and ReLU were also applied in each layer  (except the fully connected layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  The training of CGAN was conducted separately to generate every corneal image  category, as well as conducted iteratively for the Discriminator and Generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The noise  sample 𝑍1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' … 𝑍𝑛 derives from a uniform distribution in the range [–11], 𝑛 = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The  slope of the leak of ReLU was equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The zero-centered center normal distribution  was employed to initialize the weights with a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Moreover, for 20  epochs, we used the Adam optimizer, and the learning rate was equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Figure 2  illustrates the structure of the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Loss Function  Noise  Classifiers  Gcncrator  MobileNet V2  Encoder  Decoder  Gcnerated Imagcs  Discriminator  Labels  ResNet 50  Synthesize Images  Encoder  Encoder Nctwork  Dccoder Network  Xccption  Results  Encodcr Nctwork  ViT  Abnormal  Ground Truth  CoaT  Swin-T  an  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Results  The goal of this research, in which all of the steps have been outlined in detail in  Table 2, is to find out to what extent generated data affect the diagnosis of corneal diseases,  and how well classifiers can classify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Therefore, the CGAN model has been trained  to deal with data disparities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' in other words, each corneal disease’s image generated is  separated with high-quality topographical images by using fine-tuning parameters to dis-  band the scarcity of the cornea dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' For clinical decision transfer, learning methods  have been exploited, where the augmented dataset is used in training the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Proposed Method:  Inputs: D: Dataset, img: a cornea’s image which is selected from the D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  1  GI = Build a model M which generate images from noise and targeting D  2  For I = 1: CNN classifieres // (MobilenetV2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Resnet50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Xception,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' ViT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' CoaT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' and Swin-T)  3  [accuracy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' precision,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' recall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' f1-score] = Calculate metrics [Accuracy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Precision,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Recall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' F1-score] from GI  4  End for  5  [SSIM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' MSE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' PSNR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' FID] = Calculate [SSIM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' MSE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' PSNR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' FID] between an image from  GI and D  6  End  The results of diagnosing corneal diseases are reported using different types of trans-  fer learning models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' such as MobileNetv2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Resnet50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' and Xception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  To detect the importance of data generation, as well as its effect on classification tasks,  we used the original dataset to train and test each classifier with and without corneal-  generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  On the other hand, to assess the strength of the synthesis model and its ability to  synthesize convergent data in a particular category and divergent from other categories,  each classifier was trained on the synthesized data without using the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' We  employed eight-fold cross-validation with case separation at the patient level in all of our  experiments and evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The used examples contained the corneal cases (normal or  abnormal for each corneal map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  For each batch of data images, we trained the network and assessed the outcomes  individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The CGAN architecture is used to train each corneal-case class separately,  utilizing the same eight-fold cross-validation method and data split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Following training,  the generator is capable of creating realistic corneal case images separately using a vector  of noise formed by uniform distributions (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Accordingly, the model synthe-  sized eight different cases of corneal images: normal and abnormal cases for sagittal, cor-  neal thickness, elevation front, and elevation back images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  We employed two main kinds of metrics in our research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' First, we used observational  error metrics such as accuracy, precision, recall, and F1-score metrics to evaluate classifi-  cation accuracy (equations 2, 3, 4, and 5, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Second, we used equations 6 and 7  to evaluate the synthesized image’s quality with the original images via the structural  similarity index method (SSIM) [48] and the peak signal-to-noise ratio (PSNR) [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 =  𝑇𝑃+𝑇𝑁  𝑇𝑁+𝑇𝑃+𝐹𝑁+𝐹𝑃 (2)  𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 =  𝑇𝑃  𝑇𝑃+𝐹𝑃 (3)  𝑅𝑒𝑐𝑎𝑙𝑙 =  𝑇𝑃  𝑇𝑃+𝐹𝑁 (4)  𝐹1_𝑆𝑐𝑜𝑟𝑒 =  2𝑇𝑃  2𝑇𝑃+𝐹𝑃+𝐹𝑁 (5)  where TP = true positives, TN = true negatives, FP = false positives, and FN = false  negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Structural similarity (SSIM) [48] is an image quality measurement based on equation  (6) between the approximated image 𝑦𝑒  𝐿 and the ground truth image 𝑦𝑡  𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  𝑆𝑆𝐼𝑀(𝑦𝑡  𝐿, 𝑦𝑒  𝐿) =  1  𝑀 ∑  (2𝜇𝑗𝑡𝜇𝑗𝑒+𝑐1)(2𝜎𝑗𝑡𝑒+𝑐2)  (𝜇2𝑗𝑡+𝜇2𝑗𝑒+𝑐1)(𝜎2𝑗𝑡+𝜎2𝑗𝑒+𝑐2  𝑀  𝑗=1  (6)  In contrast, peak signal-to-noise ratio (PSNR) [49] is an objective assessment based  on comparisons using particular numerical criteria [50,51];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' a higher PSNR value indicates  better image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Images generated by equation (7) have significant numerical differ-  ences at the low end of the PSNR scale [52,53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  𝑃𝑆𝑁𝑅(𝑓, 𝑔) = 10𝑙𝑜𝑔10(  2552  𝑀𝑆𝐸(𝑓,𝑔))) (7)  MATLAB2020b is used for the implementation of corneal diagnosing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' All training  processes were performed using an NVIDIA GeForce GTX 1660 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Using the above-mentioned metrics for different classifiers, few results were rec-  orded when no synthesized data were used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' this might be due to overfitting over the  smaller number of training images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Conversely, using the CGAN model, the results im-  proved as the number of training instances grew (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Performance comparison for classification of corneal conditions among obstetric models  (%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Classifier  Data  Accuracy  Precision  Recall  F1-score  MobilenetV2  Original  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='4  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='3  Synthesized  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='5  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='8  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='5  Resnet50  Original  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='13  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='3  Synthesized  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='5  90  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='4  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='1  Xception  Original  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='9  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='1  Synthesized  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7  90  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2  ViT  Original  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='1  67  Synthesized  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='4  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2  CoaT  Original  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='9  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='1  Synthesized  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='3  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='1  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='4  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2  Swin-T  Original  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='4  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='3  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='5  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='9  Synthesized  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='4  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='5  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6  Since our data images are unbalanced, we suggested revealing how the corneal diag-  nosis would be affected if a balanced dataset was available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Therefore, we used the tradi-  tional data balancing methods, where we conducted data resampling using both ap-  proaches to make a balanced dataset out of an imbalanced one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The first approach was  undersampling (keeping all samples in the rare class and randomly selecting an equal  number of samples in the abundant class);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' the second approach was oversampling (in-  creasing the size of rare samples using repetition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' These two approaches were applied to  the data before and after generating images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Results reported that, generally, when applying data resampling on the original data  (before using the CGAN model), the classifiers achieved a moral performance, while the  data were balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Moreover, training by oversampling synthesized data for all classifi-  ers outperforms training by underdamped synthesized data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' On the other hand, applying  oversampled data on the generated image (after implementing the CGAN model) will not  affect the classifier results since the data are vast enough to train the models correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' In  contrast, undersampling negatively affected the achievement of classifiers due to the data  being decreased again (see Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Performance comparison for classifying corneal conditions among obstetric models after  balancing data (%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  OVS: oversampling, UNS: undersampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  This issue of whether the set of images generated was sufficiently distinct to allow  classification between the corneal case categories was investigated with the help of an  expert ophthalmologist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' We provided him with 500 randomly generated images with var-  ious categories to classify and diagnose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Table 5 summarizes the findings, and Table 6  shows the average of SSIM and PSNR for a random selection of 100 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Samples of the generated image using the Conditional Generative Adversarial Network  (CGAN) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The results from the expert (%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Sagittal Images  CT Images  EF and EB Images  Diagnosis by an Expert  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='93  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Average of Structural Similarity Index (SSIM) and peak signal-to-noise ratio (PSNR) for  100 random images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  SSIM  PSNR  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='872  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='221  The SSIM and PSNR have been calculated before and after training the CGAN model  on a random sample of 100 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Table 6 shows that the model can generate synthetic  Classifier  Data  Accuracy  Precision  Recall  F1-score  OVS  UNS  OVS  UNS  OVS  UNS  OVS  UNS  MobilenetV2  Original  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='5  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='4  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='5  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='4  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='3  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='4  Synthesized  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='5  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='1  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='8  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='4  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='1  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='4  81  Resnet50  Original  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='36  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='3  76  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='3  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='5  Synthesized  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='8  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='8  83  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='8  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='8  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='8  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7  Xception  Original  86  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='3  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2  78  86  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='3  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='9  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7  Synthesized  90  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='3  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7  90  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6  90  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6  ViT  Original  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='5  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='9  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='4  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='8  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='5  73  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='9  Synthesized  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='8  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='1  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='5  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='9  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='5  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6  CoaT  Original  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='4  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='1  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='9  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='8  69  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7  Synthesized  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='8  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='8  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='9  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='9  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='8  Swin-T  Original  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='9  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='8  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='5  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='9  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='5  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='3  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='5  Synthesized  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='8  64  60  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='3  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='4  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  images very close to the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Therefore, we can consider those images to be legitimate  for training CNNs models, and ophthalmologists can use them in clinical research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  The CNN classifiers are repeatedly tested in this work to determine the testing pro-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The suggested model can be applied in real-time, where testing images only takes a  few moments, according to Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' While the CoaT model requires the longest ATT, the  ATT for ViT beats the other classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Convolutional neural network (CNN) classifier’s average time test (ATT) (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  MobilenetV2  Resnet50  Xception  ViT  CoaT  Swin-T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='0258  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='0187  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='0152  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='0108  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='0342  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='0203  The high quality of the images can be seen in the images synthesized from the test  images using the CGAN model, which are displayed in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' It is also possible to no-  tice the stability of the structures and morphologies of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Example of original and synthesis images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Discussion  The objectives of this work were to apply the CGAN model to generate synthetic  medical images for data augmentation to expand limited datasets and improve clinical  decision-making for corneal diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Thus, we investigated the extent to which synthetic  corneal images help another system perform better behind the scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The study used a  small dataset comprising the sagittal, corneal thickness, elevation front, and elevation  back of corneal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Each class has its distinct characteristics, although there is consid-  erable intra-class variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Our diagnosis was based on the four maps, each of which was  examined to determine whether it was normal or diseased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' To identify corneal disorders,  a variety of transfer learning architectures were employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' We discovered that by utiliz-  ing the CGAN model to synthesize extra realistic images, we could increase the size of the  training data groups, thus boosting the clinical decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The diagnostic outcomes for mo-  bilenetV2, Resnet50, Xception, ViT, CoaT, and Swin-T classifiers improved from 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2 % to  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6 %, 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='13% to 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='5%, 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='9% to 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7 %, 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2% to 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='7%, 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='6% to 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='3%, and 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='4% to  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='4%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Results from Table 3 show that the synthetic data samples generated  can increase the variability of the input dataset, resulting in more accurate clinical deci-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  The scores demonstrate that the synthesized images have useful visuals and, more  crucially, useful characteristics that may be used in computer-aided diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The other  aspect of this research is to test the effect of data balance on diagnostic results, where we  used the resampling method to make the dataset balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The results showed that train-  ing the model before generating a new set of data on a balanced dataset is very important,  especially in circumstances where data are scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' On the contrary, we did not notice a  significant impact on the performance of the classifiers when using the data resampling  Original  images  Synthesis  images  on the generated data because the data was sufficient and suitable for training the models  without the need to balance them using data balancing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' This is clear evidence of  the importance of the model proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' In a final experiment, we compared  the performance of the classifiers-based systems employed in this study for clinical deci-  sion-making (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The highest performance was derived from synthesized data in the  Xception classifier, whereas the best performance came from using balance data in Res-  net50 when using the oversampling approach, but the ViT model while using the under-  sampling approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  This work has several limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' For example, the training complexity was en-  hanced by training distinct GANs for each corneal case class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' It might be useful to look  into GAN designs that produce multi-class samples at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Another type of  GAN learning process might increase the quality of the corneal image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' It is also possible  to do more research to improve the training loss function by adding regularization terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Because the human factor is critical in evaluating the proposed model’s outputs, an  expert opinion was obtained after providing him with a set of generated corneal images  containing a randomly selected set of normal and abnormal corneal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' The following  was the expert’s opinion: “Creating a new template for the corneal topographical of four  refractive maps is considered an interesting subject as it enriched the overall expected  shapes that could be seen during the daily clinic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' These new images which created based  on real cases collected previously and diagnosed that the new images are still inside the  reality borderlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Gain good experience with the new shapes and specify the further  required steps of a diagnosis other than the topographical maps that could be specified  advanced for predicted out-of-skim cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' In such a way, offline training for the new oph-  thalmologists and improving the skill of diagnosis with the preparation for new unseen  cases could be done.” In the future, we look to develop our research to exploit other GANs  that might benefit from corneal image synthesis for better achievement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Conclusion  In conclusion, we proposed a strategy for improving performance in a medical issue  with little data by generating synthetic medical images for data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' On a cor-  neal diseases diagnosis task, we discovered that synthetic data augmentation beat tradi-  tional data augmentation in accuracy by roughly 13%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Additionally, we investigated the  performance of the classifiers in different conditions, and we found that while working  with cornea images to diagnose diseases, the Xcepton classifier is more responsive than  the rest of the used classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' We anticipate that synthetic augmentation can help with a  variety of medical issues and that the method we have outlined can lead to more powerful  and reliable support systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Author Contributions: “Conceptualization, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' methodology, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=',  and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' validation, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' funding acquisition, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' JosephNg, Faculty of Data Science & Information Technology, INTI International  University, Persiaran Perdana BBN, 71800 Nilai, Negeri Sembilan, Malaysia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Institutional Review Board Statement: The manuscript is conducted within the ethical manner ad-  vised by the targeted journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Informed Consent Statement: Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Data Availability Statement: Data can be shared upon request from the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Acknowledgments: None.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Conflicts of Interest: The authors declare no conflict of interest to any party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Zhong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' Kang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' Volume 34, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' Energy-based generative adversarial network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' Long, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content='  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' Lian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' Ruan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Wang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Shen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Medical image synthesis with context-aware generative  adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' In International Conference on Medical Image Computing and Computer-Assisted Intervention, Proceedings of the 20th  International Conference, Quebec City, QC, Canada, 11–13 September 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content='  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Schmidt-Erfurth, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Langs, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Unsupervised anomaly detection with generative adversarial net-  works to guide marker discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' In International Conference on Information Processing in Medical Imaging, Proceedings of the 25th Inter-  national Conference, IPMI 2017, Boone, NC, USA, 25–30 June 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content='  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Jameel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' Aydin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Ghaeb, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Machine Learning Techniques for Corneal Diseases Diagnosis: A Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content='  2021, 21, 2150016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Ruchi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Amador, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Tormanen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Saghizadeh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Arumugaswami, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' Kumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Kramerov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Ljubimov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Systemic diseases and the cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Eye Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' 2021, 204, 108455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Jameel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Aydin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Ghaeb, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Local information pattern descriptor for corneal diseases diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Electr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content='  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Shanthi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Aruljyothi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Balasundaram, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Janakiraman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Nirmaladevi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Pyingkodi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Artificial intelligence applications  in different imaging modalities for corneal topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Ophthalmol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' 2021, 67, 801–816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Nazar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Hussein, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' vector machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Curr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' 2018, 10, 75461–75467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Ikram, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' Rozema, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Consejo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Corneal modeling and Keratoconus identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Biomath Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' 2018, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Arbelaez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' Savini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Use of a support vector machine for keratoconus and subclinical kera-  toconus detection by topographic and tomographic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='ophtha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content='005  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Lopes, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+page_content=' Kamel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' An information theoretic image-quality measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' In Conference Proceedings, IEEE Canadian  Conference on Electrical and Computer Engineering (Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' 98TH8341), Toronto, ON, Canada, 25–28 May 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' IEEE: Bellevue, WA, USA,  1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Volume 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' 169–172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='  Dosselmann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' Existing and emerging image quality metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' In Proceedings of the Canadian Conference on Electrical  and Computer Engineering, Saskatoon, SK, Canada, 1–4 May 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' IEEE: Bellevue, WA, USA, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
+page_content=' 1906–1913.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFKT4oBgHgl3EQfpy4G/content/2301.11871v1.pdf'}
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+Relativistic corrections to exclusive decays ψ(n) → ρπ and possible
+solution to the “ρπ-puzzle”
+Nikolay Kivel
+Physik-Department, Technische Universit¨at M¨unchen,
+James-Franck-Str. 1, 85748 Garching, Germany
+January 11, 2023
+Abstract
+We study relativistic corrections to exclusive S-wave charmonium decays into ρπ and
+γπ final states. The contribution of relative order v2 and the set of associated higher order
+corrections are calculated using NRQCD and collinear factorisation framework. Numerical
+estimates show that the dominant effect is provided by the corrections of relative order
+v2. The numerical values of these contributions are of the same order as the leading-order
+ones. These results suggest that the sum of relativistic and radiative QCD corrections can
+potentially explain the “ρπ-puzzle”.
+1
+arXiv:2301.03884v1  [hep-ph]  10 Jan 2023
+
+1
+Introduction
+A description of S-wave charmonium decays into ρπ final state is already a long-standing problem
+in QCD phenomenology.
+The branching ratios for J/ψ and excited state ψ(3686) ≡ ψ′ are
+measured sufficiently accurately and their ratio is found to be very small [1]
+Qρπ =
+Br[ψ′ → ρπ]
+Br[J/ψ → ρπ] ≈ 0.20 × 10−2.
+(1)
+This corresponds to a strong violation of the “13%-rule”, which suggests that Qρπ ≈ Qe+e− ≃
+0.13. The latter is valid only if the decay amplitudes of S-wave charmonium are dominated
+by the leading-order contribution in the QCD factorisation framework (pQCD). Therefore the
+disagreement between the data and qualitative theoretical expectation indicates about large
+dynamical effects, which are not accounted by the leading-order approximation of pQCD.
+The problem has attracted a lot of attention and many different qualitative ideas and phe-
+nomenological models have been proposed in order to understand the small value of Qρπ. Almost
+all of proposed explanations use different ideas about long distance QCD dynamics; a compre-
+hensive overview of the topic can be found in Refs. [2,3].
+The dominant role of some nonperturbative dynamics is related to the fact that the QCD
+helicity selection rule suppresses the valence contribution to the decay amplitude. Therefore, it
+is necessary to take into account for the one of outgoing mesons a non-valence component of
+the wave functions, which is suppressed by additional power Λ/mc. However, already long ago
+in Refs. [4, 5] it was found that pQCD framework yields a reliable leading-order estimate for
+the J/ψ branching ratio. In Ref. [4] the non-valence contributions are described by the three-
+particles twist-3 light-cone distribution amplitudes (LCDAs). These nonperturbative functions
+are process independent and the first few moments of these functions can be estimated using
+QCD sum rules. At present time corresponding matrix elements were studied and revised for
+various mesons, see updates in Refs. [6–8]. Therefore it is reasonable to believe that pQCD
+description is a good starting point in order to develop a systematic description of the process
+within the effective field theory framework.
+Following this way one faces with the problem in the description of ψ′ → ρπ, which must
+be strongly suppressed relative J/ψ → ρπ in order to get the small ratio (1). There are various
+assumptions about possible dynamical origins for this suppression. Often they are related to
+the fact that the mass of excited state ψ′ is close to the open charm threshold and this can
+lead to dynamical effects, which provide the crucial difference between J/ψ and ψ′ decays.
+The possible scenarios include: destructive interference of the large non-valence and valence
+contributions [4, 9]; suppression of the colour-singlet c¯c-wave function at the origin for ψ′ and
+the dominance of the colour-octet state [10]; cancellation between c¯c and D ¯D components of
+ψ′ [11]; cancellation between S- and D-wave components of ψ′ [12] and others [2].
+On the other hand, the potential of the effective field theory framework to study the problem
+was not been fully exploited yet. Especially it is interesting to study the higher order corrections,
+which are different for J/ψ and ψ′. In this way the natural violation of the “13% rule” can be
+related to relativistic corrections in NRQCD [13].
+In fact, already an order v2 nonrelativistic QCD matrix elements ⟨0| χ†σ · ϵ ∇2ψ |ψ(n, ϵ)⟩
+have very different values for J/ψ and ψ′, that was noticed already long time ago [14]. Recently,
+the relativistic corrections to exclusive ψ(n) → p¯p decays have been studied in Ref. [15]. It
+is found that corrections of relative order v2 are large and comparable with the leading-order
+contribution.
+This effect is closely related to the structure of the integrand in the collinear
+convolution integral describing the decay amplitude.
+This observation holds for both states
+2
+
+Figure 1:
+a) Typical diagrams describing the subprocess Q ¯Q → V P, where V = ρ, γ. The blobs
+denote the light-cone matrix elements, see explanation in the text. b) An example of diagrams,
+describing the contribution with the perturbative photon coupling.
+J/ψ and ψ′ but for excited state the absolute effect is larger because the corresponding matrix
+element is larger. The similar mechanism may also be relevant for other hadronic decay channels
+including ψ(n) → ρπ decays.
+Therefore, the main purpose of this paper is to calculate the relativistic corrections to ψ(n) →
+ρπ and to study their numerical effect. As a first step in this direction we will calculate the
+correction of relative order v2 combining NRQCD expansion with the leading-order collinear
+expansion. We will use the NRQCD projection technique developed in Refs. [16–19], which is
+also effective for calculations of exclusive amplitudes. This technique also allows one to resum
+a part of higher order corrections, which are related to the corrections to quark-antiquark wave
+function in the potential model [19]. Such consideration is also useful providing an estimate of
+possible effects from higher order contributions.
+2
+Relativistic corrections to ψ(n) → ρπ and ψ(n) → γπ decays
+To describe the J/ψ(P) → ρ(p)π(p′) decay amplitude we use the charmonium rest frame and
+assume that outgoing momenta are directed along z-axis. The amplitude is defined as
+�
+ρ(p)π(p′)
+�� iT |ψ(n)⟩ = i(2π)4δ(p + p′ − P)iϵαβµνϵαe∗β p′µpν
+(pp′) Aρπ,
+(2)
+where ϵ and e∗ denotes polarisation vectors of ψ and ρ-meson, respectively. The amplitude
+Aρπ can be described as a superposition of a hard kernel with nonperturbative matrix elements
+describing the long distance coupling with hadronic states. In order to calculate the hard kernel,
+we perform an NRQCD matching, which is combined with the collinear light-cone expansion for
+the light quarks. This technique allows one to perform the matching at the amplitude level and
+to find the hard kernels for corrections associated with the specific set of higher order NRQCD
+matrix elements [19]
+�
+v2n�
+=
+⟨0| χ†σ · ϵ
+�
+− i
+2
+←→
+D
+�2n
+ψ |ψ(n, ϵ)⟩
+m2n
+c
+⟨0| χ†σ · ϵψ |ψ(n, ϵ)⟩
+≃
+�
+v2�n ,
+(3)
+where the last equality is valid up to corrections O(v2) [20].
+The diagrams, which describe the decay amplitude are schematically shown in Fig.1. The
+long distance hadronisation dynamics of outgoing mesons is described by the twist-2 and twist-3
+light-cone distribution amplitudes (LCDAs). Various properties and models for required LCDAs
+3
+
+can be found in Refs. [7,8]. The twist-2 light-cone matrix elements read 1
+�
+π+(p′)
+�� ¯u(z1+)/¯nγ5d(z2+) |0⟩ = −ifπ
+�
+p′¯n
+� � 1
+0
+du eiu(p′¯n)(z1n)/2+i(1−u)(p′¯n)(z2n)/2 φ2π(u),
+(4)
+�
+ρ−(p)
+�� ¯d(z1−)γµ
+⊥/nu(z2−) |0⟩ = if⊥
+ρ e∗µ
+⊥ (pn)
+� 1
+0
+dy eiy(pn)(z1¯n)/2+i(1−y)(pn)(z2¯n)/2 φ⊥
+2ρ(y),
+(5)
+where we use auxiliary light-cone vectors
+n = (1, 0, 0, −1), ¯n = (1, 0, 0, 1), gµν
+⊥ = gµν − 1
+2(nµ¯nν + nν¯nµ),
+(6)
+p′ = (p′¯n)n
+2 + m2
+π
+(p′¯n)
+¯n
+2 , p = (pn) ¯n
+2 + m2
+ρ
+(pn)
+n
+2 , (p′¯n) ∼ (pn) ∼ mc.
+(7)
+and the short notation for the arguments of quark fields
+q(zi+) ≡ q((zin)¯n/2), q(zi−) ≡ q((zi¯n)n/2).
+(8)
+The required twist-3 three-particles LCDAs are defined as
+�
+π+(p′)
+�� ¯u(z1+)/¯nγµ
+⊥γ5gG¯nµ(z3+)d(z2+) |0⟩ = −2f3π
+�
+p′¯n
+�2 FT [φ3π(ui)] ,
+(9)
+�
+ρ−(p, e)
+�� ¯d(z1−)/ngGµn(z2−)u(z3−) |0⟩ = − fρmρ(pz)2e∗µ
+⊥ FT [φ3ρ(yi)] ,
+(10)
+�
+ρ−(p, e)
+�� ¯d(z1)/nγ5g ˜Gµn(λz)u(−z) |0⟩ = −ifρmρζ3(pz)2e∗µ
+⊥ FT
+�
+˜φ3ρ(yi)
+�
+,
+(11)
+where Gµn ≡ Gµνnν and
+FT [f(ui)] =
+�
+Dui eiu1(p′¯n)(z1n)/2+iu2(p′¯n)(z2n)/2+iu3(p′¯n)(z3n)/2φ3π(u1, u2, u3),
+(12)
+with
+Dui = du1du2du3δ(1 − u1 − u2 − u3).
+(13)
+The FT[φ3ρ(yi)] is defined analogously but with yi(pn)(zi¯n) in the Fourier transformation. The
+normalisation constants fπ,ρ, ζ3, f3π and models for various LCDAs will be discussed below.
+The expression for the amplitude can be written as
+Aρπ = ⟨0| χ†σ · ϵψ |ψ(n, ϵ)⟩
+�2Mψ
+2E
+1
+4π
+�
+dΩ Tr
+�
+Π1 ˆAQ
+�
+,
+(14)
+where ˆAQ describes subprocess Q ¯Q → ρπ with the quark-antiquark pair in the initial state. The
+heavy quark projector on the triplet spin state Π1 reads [19]
+Π1 =
+−1
+2
+√
+2E(E + m)
+�1
+2 /P + m + q/
+� /P + 2E
+4E
+ϵ/
+�1
+2 /P − m − q/
+�
+⊗
+1
+√Nc
+,
+(15)
+and is normalised as
+Tr
+�
+Π1Π†
+1
+�
+= 4E2,
+(16)
+1For simplicity, we do not explicitly show the gauge links in the light-cone operators assuming the appropriate
+light-cone gauge.
+4
+
+where E is the heavy quark energy pQ = (E, q), p ¯Q = (E, −q) and E =
+�
+m2c + q2.
+The
+integration dΩ over the angles of the relative momentum q in Eq.(14) is used to get the state
+with L = 0. Therefore the relevant amplitude ˆAQ is the function of relative momentum square
+q2 only, which is substituted q2 → m2
+c
+�
+v2�
+in the final expression (14), various technical details
+concerning NRQCD matching can be found in Refs. [17,19].
+Calculation of the diagrams as in Fig.1 gives
+Aρ−π+ = − ⟨0| χ†σ · ϵψ
+��ψ(3S1)
+� �
+2Mψ (παs)2 10
+27
+�
+1 + mc
+E
+� f⊥
+ρ f3π
+[4E2]2
+�
+Jπ + fρmρζ3 fπ
+fρ⊥f3π
+Jρ
+�
+,
+(17)
+where the dimensionless collinear convolution integrals Jπ and Jρ describe contributions with
+twist-3 π- and ρ-LCDAs, respectively. These integrals also depend on the NRQCD parameter
+�
+v2�
+. In the leading-order limit
+�
+v2�
+→ 0, E → m2
+c Eq.(17) reproduces the result from Ref. [4]
+Alo
+ρ−π+ = − ⟨0| χ†σ · ϵψ |ψ(n, ϵ)⟩
+�
+2Mψ (παs)2 20
+27
+f⊥
+ρ f3π
+[4m2c]2
+�
+Jlo
+π + fρmρζ3fπ
+fρ⊥f3π
+Jlo
+ρ
+�
+,
+(18)
+where
+Jlo
+π =
+�
+Dui
+φ3π(ui)
+u1u2u3
+� 1
+0
+dy φ⊥
+2ρ(y)
+1 − y
+2u1
+(y¯u2 + u2¯y) (yu1 + ¯y¯u1),
+(19)
+Jlo
+ρ =
+� 1
+0
+du φ2π(u)
+1 − u
+�
+Dyi
+�
+φ3ρ + ˜φ3ρ
+�
+(yi)
+y1y2y3
+1
+y2¯u + u¯y2
+,
+(20)
+with ¯x = 1 − x.
+The analytical expressions for the integrals Jπ,ρ in Eq.(17) are somewhat
+lengthy and presented in Appendix.
+In order to estimate these integrals we use the following models of LCDAs
+φ⊥
+2ρ(y) = 6y(1 − y)
+�
+1 + a2ρ C3/2
+2
+(2y − 1)
+�
+,
+(21)
+φ2π(u) = 6u(1 − u)
+�
+1 + aπ
+2C3/2
+2
+(2u − 1)
+�
+,
+(22)
+φ3ρ(yi) = 360y1y2y2
+3(y1 − y2)ω3ρ,
+(23)
+˜φ3ρ(yi) = 360y1y2y2
+3
+�
+1 + ˜ω3ρ
+ζ3
+1
+2 (7y3 − 3)
+�
+,
+(24)
+φ3π(ui) = 360u1u2u2
+3
+�
+1 + ω3π
+1
+2(7α3 − 3)
+�
+.
+(25)
+The different nonperturbative moments, which enter in the definitions (4)-(11) and (21)-(25),
+were estimated in Refs. [4, 7, 8]. Their values are summarised in Table 1. In the numerical
+estimates we fix for the factorisation scale the value µ = 2 GeV and use αs ≃ 0.30.
+All the convolution integrals calculated with the models (21)-(25) are well defined, which
+confirms that collinear factorisation is also valid beyond the leading-order approximation.
+As a first step of the numerical analysis let us consider the the leading-order estimate for the
+branching ratio of J/ψ. For that purpose we use the estimates for the NRQCD matrix element
+obtained in Ref. [18]
+���⟨0| χ†σ · ϵψ |J/ψ⟩
+���
+2
+≃ 0.440.
+(26)
+5
+
+Table 1:
+The values of the moments, which parametrise the hadronic LCDAs. All values are
+given at the scale µ = 2 GeV. For the pion moments, the values are taken from Ref. [7], for the
+ρ-meson from Ref. [8].
+fπ,MeV
+fρ, MeV
+f⊥
+ρ , MeV
+a2π
+a2ρ
+f3π, GeV2
+ζ3ρ
+ω3ρ
+˜ω3ρ
+ω3π
+131
+216
+143
+0.19
+0.11
+0.31 × 10−2
+0.02
+0.09
+−0.04
+−1.1
+For the various masses in Eq.(18) we use Mψ = 3.1 GeV, mρ = 775 MeV, for the pole c-quark
+mass mc = 1.4 GeV and for the total width ΓJ/ψ = 93 KeV [1]. Then for the sum of all final
+states ρ±π∓ and ρ0π0 we obtain
+Br[J/ψ → ρπ]lo ≃ 1.0%,
+(27)
+which is somewhat smaller then the corresponding experimental value 1.69(15)%. This updated
+result confirm the conclusion of Ref. [4], that the LO NRQCD approximation works sufficiently
+well for the J/ψ decay.2 On the other hand this approximation can not describe branching ratio
+ψ′ → ρπ.
+Consider now the effect provided by the relativistic corrections in Eq.(17). The one part
+is provided by the resummation of relativistic corrections in the factor E = m2
+c
+�
+1 + ⟨v2⟩ in
+Eq.(17). This effect can be understood as transition from the scale 4m2
+c to the scale M2
+ψ ≃
+4m2
+c(1 +
+�
+v2�
+). These corrections reduce the ratio Qρπ due to the factor (1 +
+�
+v2�
+J/ψ)2/(1 +
+�
+v2�
+ψ′)2 ∼ M4
+ψ/M 4
+ψ′ ∼ 0.50. However, this can not explain the very small value Qρπ in Eq.(1).
+The second effect of the relativistic corrections is associated with the modification of the
+hard kernels in the convolution integrals Jρ,π. Because these integrals depend on meson LCDAs,
+the resulting effect of the relativistic corrections is also sensitive to hadronic nonperturbative
+structure.
+For the numerical calculation we use for J/ψ the estimate from Ref. [18]
+�
+v2�
+J/ψ ≈ 0.225,
+(28)
+and for the excite state ψ′ we apply the following estimate
+�
+v2�
+ψ′ = Mψ′ − MJ/ψ + E1
+mc
+≈ 0.64,
+(29)
+where E1 =
+�
+v2�
+J/ψ mc ≃ 315 MeV is the binding energy for J/ψ. The resulting value of
+�
+v2�
+ψ′
+is much larger than
+�
+v2�
+J/ψ, which can have a significant numerical effect and, therefore, affect
+the value of Qρπ.
+The given calculation of the relativistic corrections is complete at the relative order v2 only.
+The resummation of higher orders
+�
+v2�n with n > 2 describes the part of the relativistic correc-
+tions associated with the quark-antiquark wave function only [19]. We use this approximation in
+order to study a possible effect from higher-order contributions. Therefore, for the comparison,
+we present the values of the integrals in Eq.(17) obtained in the leading-order approximation
+Jlo (
+�
+v2�
+→ 0), in the next-to-leading approximation Jnlo, which takes into accont the next-to-
+leading correction Jnlo = Jlo +
+�
+v2�
+J(1), and the integral J, which includes all powers
+�
+v2�n :
+J = Jlo + � �
+v2�n J(n).
+2We assume that the difference about factor two is not a large discrepancy taking into account various uncer-
+tainties from scale setting, pole mass mc, etc., which we do not consider now.
+6
+
+The total integral in Eq.(17) is described by the sum of two contributions with the different
+LCDAs
+Jρπ = Jπ + fρmρζ3 fπ
+fρ⊥f3π
+Jρ,
+(30)
+where, schematically, Jπ = φ3π ∗ Tπ ∗ φ⊥
+2ρ and Jρ = φ3ρ ∗ Tρ ∗ φ2π + ˜φ3ρ ∗ ˜Tρ ∗ φ2π (the asterisk
+denotes the convolution integrals, Tπ,ρ are the hard kernels). Using parameters from Table 1
+one finds
+fρmρζ3 fπ
+fρ⊥f3π
+≈ 0.99.
+(31)
+Therefore the normalisation couplings in the definitions (4)-(11) do not provide any numerical
+difference between the two terms in Eq.(30).
+The results for convolution integrals (30) are
+presented in Table 2
+Table 2:
+Numerical result for the convolution integrals Jρπ
+Jlo
+ρπ
+Jnlo
+ρπ /Jlo
+ρπ
+Jρπ/Jlo
+ρπ
+J/ψ
+630
+0.53
+0.45
+ψ′
+630
+−0.46
+−0.65
+The effect of the relativistic corrections is negative and the values of the LO integrals are
+substantially reduced. Notice that neglecting the higher-order corrections in v2 in the square of
+the integral, one gets in case of J/ψ the strong cancellation
+|Jnlo
+ρπ |2 = (Jlo
+ρπ)2(2Jnlo
+ρπ /Jlo
+ρπ − 1) + O(v4) ≃ 0.06(Jlo
+ρπ)2.
+(32)
+Therefore we assume that it is better to take the large NLO correction exactly, i.e. do not
+expanding the square of the integral in powers of v2. At the same time the numerical effect from
+other higher order corrections is already much smaller.
+For ψ′ → ρπ the numerical effect is bigger because
+�
+v2�
+ψ′ is larger. One can also see that
+the dominant part of the correction is also provided by the contribution of relative order v2,
+which is obtained exactly in this calculation. The numerical dominance of this correction can
+be explained by the numerical enhancement of the corresponding convolution integrals in the
+same way as for the baryon decays [15].
+Let us assume that the relativistic correction of order v2 provides the dominant numerical
+effect for J/ψ and ψ′ states. Then, this allows one to suggest a possible explanation of the small
+ψ′ → ρπ width, which explains the ”ρπ-puzzle”. The NRQCD description of decay amplitudes
+also involves the O(αs) NLO QCD radiative correction, which can also provide substantial
+numerical effect. Usually this contribution is considered to be of the same order as relativistic
+corrections of relative order v2. The value of radiative corrections for J/ψ and ψ′ states is the
+same except the NRQCD matrix element. Therefore, if the radiative O(αs) correction is positive
+and large enough in order to compensate the negative contribution Jnlo for ψ′ then this naturally
+explains the small width for ψ′. On the other hand, such positive contribution will improve the
+description of J/ψ → ρπ increasing the value of the convolution integral, i.e. in this case the
+O(αs) also compensates the negative effect of the relativistic correction. This is in agreement
+with the observation that the leading-order description J/ψ → ρπ provides qualitatively good
+estimate.
+7
+
+Potentially this analysis can also be applied for other meson decays. The good feature of
+the collinear factorisation is that the hadronic nonperturbative content is described in terms
+of universal process independent LCDAs. Many of these functions were already studied in the
+literature. Even if the hard kernels are the same the differences in the models for LCDAs can
+affect the numerical balance and change the value Qhh′. Consider, for example, the decay of
+S-wave charmonia into γπ0 final state. In this case the decay amplitude is described by the same
+diagrams as in Fig.1(a) but with the photon LCDAs instead of ρ-meson. These diagrams describe
+the photon as a hadron, i.e. such contributions are sensitive to the nonperturbative components
+of the photon wave function. The contribution with the perturbative photon coupling appear
+from the diagrams Fig.1(b) only and therefore they are suppressed by electromagnetic coupling
+α or by additional QCD coupling αs. In such situation the contributions with nonperturbative
+photon can provide a sizeable impact, see e.g. discussion in Ref. [21].
+The data for the branching fractions ψ(n) → γπ are known [1]
+Br[J/ψ → γπ0] = 3.56(17) × 10−5,
+Br[ψ′ → γπ0] = 1.04(22) × 10−6,
+(33)
+which yields
+Qγπ ≃ 0.03.
+(34)
+The width Γ[J/ψ → γπ0] can be well estimated using data for Γ[J/ψ → ρπ0] and VDM
+model [4].3. This indirectly support the picture with the dominant contribution from the non-
+perturbative photon coupling. However, the ratio Qγπ is about an order of magnitude larger
+than Qρπ.
+We will use the models for the photon LCDAs from Ref. [21]. The twist-2 light-cone matrix
+element reads
+⟨γ(q, e)| ¯q(z1−)/¯nγµ
+⊥q(z2−) |0⟩ = ieqe fγe∗µ
+⊥ (qn)
+� 1
+0
+dy eiy(pn)(z1¯n)/2+i(1−y)(pn)(z2¯n)/2 φ⊥
+2γ(y), (35)
+where eu = 2/3, ed = −1/3, electric charge e =
+√
+4πα. The model for φ⊥
+2γ reads [21]
+φ⊥
+2γ(y) ≃ 6y(1 − y),
+fγ(2GeV) ≃ −47 MeV.
+(36)
+Twist-3 DAs matrix elements are defined as
+⟨γ(p)| ¯q(z1−)/ngGµn(z3−)q(z2−) |0⟩ = − eqe f3γ(qn)2ε∗µ
+⊥ FT [φ3γ(yi)] ,
+(37)
+⟨γ(p)| q(z1−)/nγ5g ˜Gµn(z2−)q(z3−) |0⟩ = −ieqe f3γ(qn)2ε∗µ
+⊥ FT
+�
+˜φ3γ(yi)
+�
+,
+(38)
+where Fourier transformation is the same as in Eq.(12). The models for twist-3 LCDAs read [21]
+φ3γ(yi) = 360y1y2y2
+3 (y1 − y2) ω3γ,
+(39)
+˜φ3γ(yi) = 360y1y2y2
+3
+�
+1 + ˜ω3γ
+1
+2 (7y3 − 3)
+�
+,
+˜ω3γ ≈ ˜ω3ρ/ζ3.
+(40)
+where
+f3γ(2GeV) = −0.32 × 10−2GeV2,
+ω3γ ≈ ω3ρ,
+˜ω3γ ≈ ˜ω3ρ/ζ3.
+(41)
+3We assume that the contribution of J/ψ → γ∗ → γπ is subleading, in contrast to the analysis in Ref. [4]. We
+guess that this contribution is overestimated in [4] due to the specific model of the pion LCDA .
+8
+
+The γπ-decay amplitude can be obtained from Eq.(17) substituting photon LCDAs instead of
+ρ-meson ones. The LO amplitude reads
+Alo
+γπ0 = − ⟨0| χ†σ · ϵψ |ψ(n, ϵ)⟩
+�
+2Mψ (παs)2 √
+2πα20
+27
+fγ f3π
+[4m2c]2
+�
+Jlo
+γπ + f3γ fπ
+f3πfγ
+Jlo
+γπ
+�
+.
+(42)
+The ratio of the normalisation couplings in Eq.(42) yields (µ = 2GeV)
+f3γ fπ
+f3πfγ
+≈ 2.92,
+(43)
+which is different from the analogous ratio for the ρ-meson (31).
+The leading-order numerical
+estimates gives
+Br[J/ψ → γπ0]lo ≃ 3.82 × 10−5,
+(44)
+which very well agrees with the data. The results for the total convolution integral Jγπ are
+presented in Table 3. Comparing with the analogous results for the ρπ-channel one finds that
+Table 3: Numerical result for the convolution integrals Jγπ
+Jlo
+γπ
+Jnlo
+γπ /Jlo
+γπ
+Jγπ/Jlo
+γπ
+J/ψ
+932
+0.62
+0.45
+ψ′
+932
+−0.49
+−0.74
+the both descriptions are qualitatively similar despite the different ratio (43) and the differences
+between the LCDAs φ2γ and φ⊥
+2ρ. Therefore one can again assume that O(αs) radiative correc-
+tions also play a crucial role in the understanding of the value of the decay width γπ. Moreover,
+the contributions with a perturbative photon, as in Fig. 1(b) can probably explain the larger
+value of Qγπ.
+3
+Conclusions
+In conclusion, we calculated and investigated relativistic corrections to the decay amplitudes
+ψ(n) → ρπ and ψ(n) → γπ within the pQCD (NRQCD and collinear factorisation) framework.
+This calculation includes the exact correction of relative order v2 and subset of the higher order
+corrections associated with the quark-antiquarks wave function. Numerical estimates show that
+an order v2 correction is large and give the dominant numerical effect, which can be related
+to the structure of the collinear integrals. If this observation is not affected by other higher
+order relativistic corrections, then one has to consider the relative v2-contribution as a special
+case. The obtained relativistic corrections are negative and large. In case ψ′ → ρπ the relative
+v2 contribution is much larger than the leading-order one. The different effects of relativistic
+corrections for J/ψ → ρπ and ψ′ → ρπ suggest a possible explanation for the ρπ-puzzle. If the
+QCD radiative correction is positive and large enough then it interferes destructively with the
+relativistic correction for ψ′ → ρπ giving the small branching fraction. At the same time such
+radiative correction will improve the description of J/ψ → ρπ reducing the negative effect of
+the relativistic correction. Therefore, we believe that further investigation of relative order v4
+corrections and QCD radiative corrections can help to clarify this scenario.
+We also expect that the same approach can be be used for an analysis other hadronic decay
+channels. As a simplest example, the decay ψ(n) → γπ was also considered. We studied the
+9
+
+contribution, which is given by similar diagrams but with nonperturbative photon instead of ρ-
+meson. Despite the difference between twist-2 LCDAs for photon and ρ−meson, the qualitative
+effect from the relativistic corrections is quite similar to ψ → ρπ, they are also large and negative.
+Therefore one can guess that the similar scenario with radiative corrections is also applicable
+here. The only difference with ψ(n) → ρπ is provided by the contributions with perturbative
+photon coupling, which are suppressed by O(αs) or O(α). Therefore, it can be that these expects
+are responsible for the larger value of Qγπ.
+4
+Appendix
+Here we provide the analytical expressions for the intergrals Jπ and Jρ introduced in Eq.(17).
+In order to simplify notations we use
+�
+v2�
+≡ v2,
+δ = 1 − 1/
+�
+1 + v2.
+(45)
+The first integral in Eq.(17) reads
+Jπ
+�
+v2�
+=
+�
+Dui
+φ3π(ui)
+u1u2u3
+� 1
+0
+dy φ⊥
+2 (y)
+y¯y
+� 2Aπ
+D1D3
++
+Bπ
+D1D2
+�
+,
+¯y = 1 − y,
+(46)
+where
+Di = δi1 (y1¯u2 + ¯y1u2) + δi2 (y2¯u1 + ¯y2u1) + δi3 u3,
+(47)
+with
+y1 = y, y2 = ¯y.
+(48)
+The symbol δik denotes the Kronecker delta.
+The numerators Aπ and Bπ are given by the sums
+Aπ =
+4
+�
+k=0
+fA
+k Ik[13],
+Bπ =
+4
+�
+k=0
+fB
+k Ik[12],
+(49)
+where
+Ik[ij] = 1
+2
+� 1
+−1
+dη
+vkηk
+(1 + vη ai) (1 − vη aj) =
+vk
+ai + aj
+∞
+�
+n=0
+vn an+1
+j
++ (−1)nan+1
+i
+n + 1 + k
+1
+2
+�
+1 + (−1)n+k�
+.
+(50)
+with
+aj = δ1j (1 − δ)
+y1 − u2
+y1¯u2 + ¯y1u2
++ δ2j (1 − δ)
+y2 − u1
+y2¯u1 + ¯y2u1
+− δ3j (1 − δ) .
+(51)
+The coefficients fA,B
+k
+≡ fA,B
+k
+(ui, y; δ) in Eq.(49) read
+fA
+0 = δ
+2 (3u3 − 2 − δ) ,
+fA
+1 = δ
+2
+(1 − δ)
+(2 − δ)u3,
+(52)
+fA
+2 = 1
+2
+(1 − δ)2
+(2 − δ)2 (4 − 3(2 − δ)u3 + 2δ(1 − δ)) ,
+(53)
+fA
+3 = −1
+2
+(1 − δ)3
+(2 − δ)2 u3,
+fA
+4 = −1
+2
+(1 − δ)4
+(2 − δ)2 .
+(54)
+10
+
+fB
+0 = u1y1 + u1y2 − δ
+2(u1 + u2 + y1 + y2 − δ)
+(55)
++
+δ
+2 − δ
+�
+u1y1 + u1y2 − (2 − δ) (u1 + u2 + y1 + y2) + (2 − δ)2�
+,
+(56)
+fB
+1 = 1
+2
+(1 − δ)
+(2 − δ) {4(u1y1 − u2y2) + δ(u1 + y1 − u2 − y2)} ,
+(57)
+fB
+2 = 1
+2
+(1 − δ)2
+(2 − δ)2 {6 (u1 + u2 + y1 + y2) − 2 (u1y1 + u1y2) − 8
+(58)
++δ (4 − 3 (u1 + u2 + y1 + y2)) + 2δ(2 − δ)} ,
+(59)
+fB
+3 = −1
+2
+(1 − δ)3
+(2 − δ)2 (u1 + y1 − u2 − y2) ,
+fB
+4 = −1
+2
+(1 − δ)4
+(2 − δ)2 .
+(60)
+The ρ-meson integral in Eq.(17) reads
+Jρ =
+� 1
+0
+du φ2π(u)
+u¯u
+�
+Dyi
+1
+y1y2y3
+� 2Aρ
+y3D2
++
+Bρ
+D1D2
+�
+.
+(61)
+The numerators Aρ and Bρ can be written as
+Aρ = φ3ρ(yi)
+4
+�
+k=0
+�
+fA
+k
+�
+Ik[23] + ˜φ3ρ(yi)
+4
+�
+k=0
+�
+˜fA
+k
+�
+Ik[23],
+(62)
+Bρ = φ3ρ(yi)
+4
+�
+k=0
+�
+fB
+k
+�
+Ik[12] + ˜φ3ρ(yi)
+4
+�
+k=0
+�
+˜fB
+k
+�
+Ik[12],
+(63)
+where the integrals Ik are defined in Eq.(50) with a bit different combination aj
+aj = δ1j (1 − δ)
+y1 − u2
+y1¯u2 + ¯y1u2
++ δ2j (1 − δ)
+y2 − u1
+y2¯u1 + ¯y2u1
++ δ3j (1 − δ) ,
+(64)
+and we again use for the two-particle LCDA u1 = u, u2 = 1 − u.
+The coefficients fA,B
+k
+and ˜fA,B
+k
+defined in Eqs.(62) and (63) read
+fA
+0 = 1
+4 (u1(2y3 − δ) + δ (δ − y2 − y3)) + δ2
+2
+(65)
++ 1
+4
+δ
+(2 − δ) (u1 (6 + 4y3 − 3δ) + (2 − δ) (3y2 + 2y3 − 2 − 3δ)) ,
+(66)
+fA
+1 = −1
+4
+1 − δ
+2 − δ (4u1 + δ) y3,
+(67)
+fA
+2 = −1
+4
+(1 − δ)2
+(2 − δ)2 (2u1y3 + (2 − δ) (2u1 + 2y2 + y3) − 2(1 − δ)(2 − δ)) ,
+(68)
+fA
+3 = 1
+4
+(1 − δ)3
+(2 − δ)2 y3,
+fA
+4 = 0.
+(69)
+fB
+0 = −fB
+2 = δ
+4 (u1 − u2 − y1 + y2) ,
+(70)
+fB
+1 = −fB
+3 = −δ
+4(1 − δ) (u1 + u2 − y1 − y2) ,
+fB
+4 = 0.
+(71)
+11
+
+˜fA
+0 = 1
+4 (u1(2y3 − δ) + δ (δ − y2 − y3))
+(72)
++ 1
+4
+δ
+(2 − δ) (u1 (2 + 4y3 − δ) + (2 − δ) (2 + y2 − 2y3 − 3δ)) ,
+(73)
+˜fA
+1 = −1
+4
+1 − δ
+2 − δ (4u1y3 + δ (2u1 − 2y2 + y3)) ,
+(74)
+˜fA
+2 = −1
+4
+(1 − δ)2
+(2 − δ)2 (2u1y3 − 3y3(2 − δ) + δ(2 − δ)) ,
+(75)
+˜fA
+3 = 1
+4
+(1 − δ)3
+(2 − δ)2 (2u1 − 2y2 + y3) ,
+˜fA
+4 = −1
+2
+(1 − δ)4
+(2 − δ)2 .
+(76)
+˜fB
+0 = δ
+4 (3 (u1 + u2 + y1 + y2) − 4 − 2δ) ,
+˜fB
+1 = −δ
+4
+1 − δ
+2 − δ (u1 − u2 + y1 − y2) ,
+(77)
+˜fB
+2 = 1
+4
+(1 − δ)2
+(2 − δ)2 (8 − 3 (2 − δ) (u1 + u2 + y1 + y2) + 4δ(1 − δ)) ,
+(78)
+˜fB
+3 = 1
+4
+(1 − δ)3
+(2 − δ)2 (u1 − u2 + y1 − y2) ,
+˜fB
+4 = −1
+2
+(1 − δ)4
+(2 − δ)2 .
+(79)
+References
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+12
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+[15] N. Kivel, “A study of relativistic corrections to J/ψ → p¯p decay,” [arXiv:2211.13603 [hep-
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+13
+
diff --git a/etE2T4oBgHgl3EQfbQfw/content/tmp_files/load_file.txt b/etE2T4oBgHgl3EQfbQfw/content/tmp_files/load_file.txt
new file mode 100644
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+++ b/etE2T4oBgHgl3EQfbQfw/content/tmp_files/load_file.txt
@@ -0,0 +1,417 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf,len=416
+page_content='Relativistic corrections to exclusive decays ψ(n) → ρπ and possible  solution to the “ρπ-puzzle”  Nikolay Kivel  Physik-Department, Technische Universit¨at M¨unchen,  James-Franck-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' 1, 85748 Garching, Germany  January 11, 2023  Abstract  We study relativistic corrections to exclusive S-wave charmonium decays into ρπ and  γπ final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The contribution of relative order v2 and the set of associated higher order  corrections are calculated using NRQCD and collinear factorisation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Numerical  estimates show that the dominant effect is provided by the corrections of relative order  v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The numerical values of these contributions are of the same order as the leading-order  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' These results suggest that the sum of relativistic and radiative QCD corrections can  potentially explain the “ρπ-puzzle”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='03884v1  [hep-ph]  10 Jan 2023  1  Introduction  A description of S-wave charmonium decays into ρπ final state is already a long-standing problem  in QCD phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  The branching ratios for J/ψ and excited state ψ(3686) ≡ ψ′ are  measured sufficiently accurately and their ratio is found to be very small [1]  Qρπ =  Br[ψ′ → ρπ]  Br[J/ψ → ρπ] ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='20 × 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (1)  This corresponds to a strong violation of the “13%-rule”, which suggests that Qρπ ≈ Qe+e− ≃  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The latter is valid only if the decay amplitudes of S-wave charmonium are dominated  by the leading-order contribution in the QCD factorisation framework (pQCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Therefore the  disagreement between the data and qualitative theoretical expectation indicates about large  dynamical effects, which are not accounted by the leading-order approximation of pQCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  The problem has attracted a lot of attention and many different qualitative ideas and phe-  nomenological models have been proposed in order to understand the small value of Qρπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Almost  all of proposed explanations use different ideas about long distance QCD dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' a compre-  hensive overview of the topic can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' [2,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  The dominant role of some nonperturbative dynamics is related to the fact that the QCD  helicity selection rule suppresses the valence contribution to the decay amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Therefore, it  is necessary to take into account for the one of outgoing mesons a non-valence component of  the wave functions, which is suppressed by additional power Λ/mc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' However, already long ago  in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' [4, 5] it was found that pQCD framework yields a reliable leading-order estimate for  the J/ψ branching ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' [4] the non-valence contributions are described by the three-  particles twist-3 light-cone distribution amplitudes (LCDAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' These nonperturbative functions  are process independent and the first few moments of these functions can be estimated using  QCD sum rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' At present time corresponding matrix elements were studied and revised for  various mesons, see updates in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Therefore it is reasonable to believe that pQCD  description is a good starting point in order to develop a systematic description of the process  within the effective field theory framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  Following this way one faces with the problem in the description of ψ′ → ρπ, which must  be strongly suppressed relative J/ψ → ρπ in order to get the small ratio (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' There are various  assumptions about possible dynamical origins for this suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Often they are related to  the fact that the mass of excited state ψ′ is close to the open charm threshold and this can  lead to dynamical effects, which provide the crucial difference between J/ψ and ψ′ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  The possible scenarios include: destructive interference of the large non-valence and valence  contributions [4, 9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' suppression of the colour-singlet c¯c-wave function at the origin for ψ′ and  the dominance of the colour-octet state [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' cancellation between c¯c and D ¯D components of  ψ′ [11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' cancellation between S- and D-wave components of ψ′ [12] and others [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  On the other hand, the potential of the effective field theory framework to study the problem  was not been fully exploited yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Especially it is interesting to study the higher order corrections,  which are different for J/ψ and ψ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' In this way the natural violation of the “13% rule” can be  related to relativistic corrections in NRQCD [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  In fact, already an order v2 nonrelativistic QCD matrix elements ⟨0| χ†σ · ϵ ∇2ψ |ψ(n, ϵ)⟩  have very different values for J/ψ and ψ′, that was noticed already long time ago [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Recently,  the relativistic corrections to exclusive ψ(n) → p¯p decays have been studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' It  is found that corrections of relative order v2 are large and comparable with the leading-order  contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  This effect is closely related to the structure of the integrand in the collinear  convolution integral describing the decay amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  This observation holds for both states  2  Figure 1:  a) Typical diagrams describing the subprocess Q ¯Q → V P, where V = ρ, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The blobs  denote the light-cone matrix elements, see explanation in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' b) An example of diagrams,  describing the contribution with the perturbative photon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  J/ψ and ψ′ but for excited state the absolute effect is larger because the corresponding matrix  element is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The similar mechanism may also be relevant for other hadronic decay channels  including ψ(n) → ρπ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  Therefore, the main purpose of this paper is to calculate the relativistic corrections to ψ(n) →  ρπ and to study their numerical effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' As a first step in this direction we will calculate the  correction of relative order v2 combining NRQCD expansion with the leading-order collinear  expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' We will use the NRQCD projection technique developed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' [16–19], which is  also effective for calculations of exclusive amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' This technique also allows one to resum  a part of higher order corrections, which are related to the corrections to quark-antiquark wave  function in the potential model [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Such consideration is also useful providing an estimate of  possible effects from higher order contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  2  Relativistic corrections to ψ(n) → ρπ and ψ(n) → γπ decays  To describe the J/ψ(P) → ρ(p)π(p′) decay amplitude we use the charmonium rest frame and  assume that outgoing momenta are directed along z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The amplitude is defined as  �  ρ(p)π(p′)  �� iT |ψ(n)⟩ = i(2π)4δ(p + p′ − P)iϵαβµνϵαe∗β p′µpν  (pp′) Aρπ,  (2)  where ϵ and e∗ denotes polarisation vectors of ψ and ρ-meson, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The amplitude  Aρπ can be described as a superposition of a hard kernel with nonperturbative matrix elements  describing the long distance coupling with hadronic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' In order to calculate the hard kernel,  we perform an NRQCD matching, which is combined with the collinear light-cone expansion for  the light quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' This technique allows one to perform the matching at the amplitude level and  to find the hard kernels for corrections associated with the specific set of higher order NRQCD  matrix elements [19]  �  v2n�  =  ⟨0| χ†σ · ϵ  �  − i  2  ←→  D  �2n  ψ |ψ(n, ϵ)⟩  m2n  c  ⟨0| χ†σ · ϵψ |ψ(n, ϵ)⟩  ≃  �  v2�n ,  (3)  where the last equality is valid up to corrections O(v2) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  The diagrams, which describe the decay amplitude are schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The  long distance hadronisation dynamics of outgoing mesons is described by the twist-2 and twist-3  light-cone distribution amplitudes (LCDAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Various properties and models for required LCDAs  3  can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' [7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The twist-2 light-cone matrix elements read 1  �  π+(p′)  �� ¯u(z1+)/¯nγ5d(z2+) |0⟩ = −ifπ  �  p′¯n  � � 1  0  du eiu(p′¯n)(z1n)/2+i(1−u)(p′¯n)(z2n)/2 φ2π(u),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (4)  �  ρ−(p)  �� ¯d(z1−)γµ  ⊥/nu(z2−) |0⟩ = if⊥  ρ e∗µ  ⊥ (pn)  � 1  0  dy eiy(pn)(z1¯n)/2+i(1−y)(pn)(z2¯n)/2 φ⊥  2ρ(y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (5)  where we use auxiliary light-cone vectors  n = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' −1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' ¯n = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' gµν  ⊥ = gµν − 1  2(nµ¯nν + nν¯nµ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (6)  p′ = (p′¯n)n  2 + m2  π  (p′¯n)  ¯n  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' p = (pn) ¯n  2 + m2  ρ  (pn)  n  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' (p′¯n) ∼ (pn) ∼ mc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (7)  and the short notation for the arguments of quark fields  q(zi+) ≡ q((zin)¯n/2), q(zi−) ≡ q((zi¯n)n/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (8)  The required twist-3 three-particles LCDAs are defined as  �  π+(p′)  �� ¯u(z1+)/¯nγµ  ⊥γ5gG¯nµ(z3+)d(z2+) |0⟩ = −2f3π  �  p′¯n  �2 FT [φ3π(ui)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (9)  �  ρ−(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' e)  �� ¯d(z1−)/ngGµn(z2−)u(z3−) |0⟩ = − fρmρ(pz)2e∗µ  ⊥ FT [φ3ρ(yi)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (10)  �  ρ−(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' e)  �� ¯d(z1)/nγ5g ˜Gµn(λz)u(−z) |0⟩ = −ifρmρζ3(pz)2e∗µ  ⊥ FT  �  ˜φ3ρ(yi)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (11)  where Gµn ≡ Gµνnν and  FT [f(ui)] =  �  Dui eiu1(p′¯n)(z1n)/2+iu2(p′¯n)(z2n)/2+iu3(p′¯n)(z3n)/2φ3π(u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' u3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (12)  with  Dui = du1du2du3δ(1 − u1 − u2 − u3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (13)  The FT[φ3ρ(yi)] is defined analogously but with yi(pn)(zi¯n) in the Fourier transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The  normalisation constants fπ,ρ, ζ3, f3π and models for various LCDAs will be discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  The expression for the amplitude can be written as  Aρπ = ⟨0| χ†σ · ϵψ |ψ(n, ϵ)⟩  �2Mψ  2E  1  4π  �  dΩ Tr  �  Π1 ˆAQ  �  ,  (14)  where ˆAQ describes subprocess Q ¯Q → ρπ with the quark-antiquark pair in the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The  heavy quark projector on the triplet spin state Π1 reads [19]  Π1 =  −1  2  √  2E(E + m)  �1  2 /P + m + q/  � /P + 2E  4E  ϵ/  �1  2 /P − m − q/  �  ⊗  1  √Nc  ,  (15)  and is normalised as  Tr  �  Π1Π†  1  �  = 4E2,  (16)  1For simplicity, we do not explicitly show the gauge links in the light-cone operators assuming the appropriate  light-cone gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  4  where E is the heavy quark energy pQ = (E, q), p ¯Q = (E, −q) and E =  �  m2c + q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  The  integration dΩ over the angles of the relative momentum q in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' (14) is used to get the state  with L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Therefore the relevant amplitude ˆAQ is the function of relative momentum square  q2 only, which is substituted q2 → m2  c  �  v2�  in the final expression (14), various technical details  concerning NRQCD matching can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' [17,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  Calculation of the diagrams as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='1 gives  Aρ−π+ = − ⟨0| χ†σ · ϵψ  ��ψ(3S1)  � �  2Mψ (παs)2 10  27  �  1 + mc  E  � f⊥  ρ f3π  [4E2]2  �  Jπ + fρmρζ3 fπ  fρ⊥f3π  Jρ  �  ,  (17)  where the dimensionless collinear convolution integrals Jπ and Jρ describe contributions with  twist-3 π- and ρ-LCDAs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' These integrals also depend on the NRQCD parameter  �  v2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' In the leading-order limit  �  v2�  → 0, E → m2  c Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' (17) reproduces the result from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' [4]  Alo  ρ−π+ = − ⟨0| χ†σ · ϵψ |ψ(n, ϵ)⟩  �  2Mψ (παs)2 20  27  f⊥  ρ f3π  [4m2c]2  �  Jlo  π + fρmρζ3fπ  fρ⊥f3π  Jlo  ρ  �  ,  (18)  where  Jlo  π =  �  Dui  φ3π(ui)  u1u2u3  � 1  0  dy φ⊥  2ρ(y)  1 − y  2u1  (y¯u2 + u2¯y) (yu1 + ¯y¯u1),  (19)  Jlo  ρ =  � 1  0  du φ2π(u)  1 − u  �  Dyi  �  φ3ρ + ˜φ3ρ  �  (yi)  y1y2y3  1  y2¯u + u¯y2  ,  (20)  with ¯x = 1 − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  The analytical expressions for the integrals Jπ,ρ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' (17) are somewhat  lengthy and presented in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  In order to estimate these integrals we use the following models of LCDAs  φ⊥  2ρ(y) = 6y(1 − y)  �  1 + a2ρ C3/2  2  (2y − 1)  �  ,  (21)  φ2π(u) = 6u(1 − u)  �  1 + aπ  2C3/2  2  (2u − 1)  �  ,  (22)  φ3ρ(yi) = 360y1y2y2  3(y1 − y2)ω3ρ,  (23)  ˜φ3ρ(yi) = 360y1y2y2  3  �  1 + ˜ω3ρ  ζ3  1  2 (7y3 − 3)  �  ,  (24)  φ3π(ui) = 360u1u2u2  3  �  1 + ω3π  1  2(7α3 − 3)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (25)  The different nonperturbative moments, which enter in the definitions (4)-(11) and (21)-(25),  were estimated in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' [4, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Their values are summarised in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' In the numerical  estimates we fix for the factorisation scale the value µ = 2 GeV and use αs ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  All the convolution integrals calculated with the models (21)-(25) are well defined, which  confirms that collinear factorisation is also valid beyond the leading-order approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  As a first step of the numerical analysis let us consider the the leading-order estimate for the  branching ratio of J/ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' For that purpose we use the estimates for the NRQCD matrix element  obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' [18]  ���⟨0| χ†σ · ϵψ |J/ψ⟩  ���  2  ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (26)  5  Table 1:  The values of the moments, which parametrise the hadronic LCDAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' All values are  given at the scale µ = 2 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' For the pion moments, the values are taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' [7], for the  ρ-meson from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  fπ,MeV  fρ, MeV  f⊥  ρ , MeV  a2π  a2ρ  f3π, GeV2  ζ3ρ  ω3ρ  ˜ω3ρ  ω3π  131  216  143  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='31 × 10−2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='09  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='04  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='1  For the various masses in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' (18) we use Mψ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='1 GeV, mρ = 775 MeV, for the pole c-quark  mass mc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='4 GeV and for the total width ΓJ/ψ = 93 KeV [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Then for the sum of all final  states ρ±π∓ and ρ0π0 we obtain  Br[J/ψ → ρπ]lo ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='0%,  (27)  which is somewhat smaller then the corresponding experimental value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='69(15)%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' This updated  result confirm the conclusion of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' [4], that the LO NRQCD approximation works sufficiently  well for the J/ψ decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='2 On the other hand this approximation can not describe branching ratio  ψ′ → ρπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  Consider now the effect provided by the relativistic corrections in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='(17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The one part  is provided by the resummation of relativistic corrections in the factor E = m2  c  �  1 + ⟨v2⟩ in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='(17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' This effect can be understood as transition from the scale 4m2  c to the scale M2  ψ ≃  4m2  c(1 +  �  v2�  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' These corrections reduce the ratio Qρπ due to the factor (1 +  �  v2�  J/ψ)2/(1 +  �  v2�  ψ′)2 ∼ M4  ψ/M 4  ψ′ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' However, this can not explain the very small value Qρπ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  The second effect of the relativistic corrections is associated with the modification of the  hard kernels in the convolution integrals Jρ,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Because these integrals depend on meson LCDAs,  the resulting effect of the relativistic corrections is also sensitive to hadronic nonperturbative  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  For the numerical calculation we use for J/ψ the estimate from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' [18]  �  v2�  J/ψ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='225,  (28)  and for the excite state ψ′ we apply the following estimate  �  v2�  ψ′ = Mψ′ − MJ/ψ + E1  mc  ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='64,  (29)  where E1 =  �  v2�  J/ψ mc ≃ 315 MeV is the binding energy for J/ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The resulting value of  �  v2�  ψ′  is much larger than  �  v2�  J/ψ, which can have a significant numerical effect and, therefore, affect  the value of Qρπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  The given calculation of the relativistic corrections is complete at the relative order v2 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  The resummation of higher orders  �  v2�n with n > 2 describes the part of the relativistic correc-  tions associated with the quark-antiquark wave function only [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' We use this approximation in  order to study a possible effect from higher-order contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Therefore, for the comparison,  we present the values of the integrals in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' (17) obtained in the leading-order approximation  Jlo (  �  v2�  → 0), in the next-to-leading approximation Jnlo, which takes into accont the next-to-  leading correction Jnlo = Jlo +  �  v2�  J(1), and the integral J, which includes all powers  �  v2�n :  J = Jlo + � �  v2�n J(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  2We assume that the difference about factor two is not a large discrepancy taking into account various uncer-  tainties from scale setting, pole mass mc, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=', which we do not consider now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  6  The total integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' (17) is described by the sum of two contributions with the different  LCDAs  Jρπ = Jπ + fρmρζ3 fπ  fρ⊥f3π  Jρ,  (30)  where, schematically, Jπ = φ3π ∗ Tπ ∗ φ⊥  2ρ and Jρ = φ3ρ ∗ Tρ ∗ φ2π + ˜φ3ρ ∗ ˜Tρ ∗ φ2π (the asterisk  denotes the convolution integrals, Tπ,ρ are the hard kernels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Using parameters from Table 1  one finds  fρmρζ3 fπ  fρ⊥f3π  ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (31)  Therefore the normalisation couplings in the definitions (4)-(11) do not provide any numerical  difference between the two terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  The results for convolution integrals (30) are  presented in Table 2  Table 2:  Numerical result for the convolution integrals Jρπ  Jlo  ρπ  Jnlo  ρπ /Jlo  ρπ  Jρπ/Jlo  ρπ  J/ψ  630  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='45  ψ′  630  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='46  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='65  The effect of the relativistic corrections is negative and the values of the LO integrals are  substantially reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Notice that neglecting the higher-order corrections in v2 in the square of  the integral, one gets in case of J/ψ the strong cancellation  |Jnlo  ρπ |2 = (Jlo  ρπ)2(2Jnlo  ρπ /Jlo  ρπ − 1) + O(v4) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='06(Jlo  ρπ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (32)  Therefore we assume that it is better to take the large NLO correction exactly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' do not  expanding the square of the integral in powers of v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' At the same time the numerical effect from  other higher order corrections is already much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  For ψ′ → ρπ the numerical effect is bigger because  �  v2�  ψ′ is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' One can also see that  the dominant part of the correction is also provided by the contribution of relative order v2,  which is obtained exactly in this calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The numerical dominance of this correction can  be explained by the numerical enhancement of the corresponding convolution integrals in the  same way as for the baryon decays [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  Let us assume that the relativistic correction of order v2 provides the dominant numerical  effect for J/ψ and ψ′ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Then, this allows one to suggest a possible explanation of the small  ψ′ → ρπ width, which explains the ”ρπ-puzzle”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The NRQCD description of decay amplitudes  also involves the O(αs) NLO QCD radiative correction, which can also provide substantial  numerical effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Usually this contribution is considered to be of the same order as relativistic  corrections of relative order v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The value of radiative corrections for J/ψ and ψ′ states is the  same except the NRQCD matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Therefore, if the radiative O(αs) correction is positive  and large enough in order to compensate the negative contribution Jnlo for ψ′ then this naturally  explains the small width for ψ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' On the other hand, such positive contribution will improve the  description of J/ψ → ρπ increasing the value of the convolution integral, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' in this case the  O(αs) also compensates the negative effect of the relativistic correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' This is in agreement  with the observation that the leading-order description J/ψ → ρπ provides qualitatively good  estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  7  Potentially this analysis can also be applied for other meson decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The good feature of  the collinear factorisation is that the hadronic nonperturbative content is described in terms  of universal process independent LCDAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Many of these functions were already studied in the  literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Even if the hard kernels are the same the differences in the models for LCDAs can  affect the numerical balance and change the value Qhh′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Consider, for example, the decay of  S-wave charmonia into γπ0 final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' In this case the decay amplitude is described by the same  diagrams as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='1(a) but with the photon LCDAs instead of ρ-meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' These diagrams describe  the photon as a hadron, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' such contributions are sensitive to the nonperturbative components  of the photon wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The contribution with the perturbative photon coupling appear  from the diagrams Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='1(b) only and therefore they are suppressed by electromagnetic coupling  α or by additional QCD coupling αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' In such situation the contributions with nonperturbative  photon can provide a sizeable impact, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' discussion in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  The data for the branching fractions ψ(n) → γπ are known [1]  Br[J/ψ → γπ0] = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='56(17) × 10−5,  Br[ψ′ → γπ0] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='04(22) × 10−6,  (33)  which yields  Qγπ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (34)  The width Γ[J/ψ → γπ0] can be well estimated using data for Γ[J/ψ → ρπ0] and VDM  model [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' This indirectly support the picture with the dominant contribution from the non-  perturbative photon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' However, the ratio Qγπ is about an order of magnitude larger  than Qρπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  We will use the models for the photon LCDAs from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The twist-2 light-cone matrix  element reads  ⟨γ(q, e)| ¯q(z1−)/¯nγµ  ⊥q(z2−) |0⟩ = ieqe fγe∗µ  ⊥ (qn)  � 1  0  dy eiy(pn)(z1¯n)/2+i(1−y)(pn)(z2¯n)/2 φ⊥  2γ(y), (35)  where eu = 2/3, ed = −1/3, electric charge e =  √  4πα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The model for φ⊥  2γ reads [21]  φ⊥  2γ(y) ≃ 6y(1 − y),  fγ(2GeV) ≃ −47 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (36)  Twist-3 DAs matrix elements are defined as  ⟨γ(p)| ¯q(z1−)/ngGµn(z3−)q(z2−) |0⟩ = − eqe f3γ(qn)2ε∗µ  ⊥ FT [φ3γ(yi)] ,  (37)  ⟨γ(p)| q(z1−)/nγ5g ˜Gµn(z2−)q(z3−) |0⟩ = −ieqe f3γ(qn)2ε∗µ  ⊥ FT  �  ˜φ3γ(yi)  �  ,  (38)  where Fourier transformation is the same as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='(12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The models for twist-3 LCDAs read [21]  φ3γ(yi) = 360y1y2y2  3 (y1 − y2) ω3γ,  (39)  ˜φ3γ(yi) = 360y1y2y2  3  �  1 + ˜ω3γ  1  2 (7y3 − 3)  �  ,  ˜ω3γ ≈ ˜ω3ρ/ζ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (40)  where  f3γ(2GeV) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='32 × 10−2GeV2,  ω3γ ≈ ω3ρ,  ˜ω3γ ≈ ˜ω3ρ/ζ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (41)  3We assume that the contribution of J/ψ → γ∗ → γπ is subleading, in contrast to the analysis in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' We  guess that this contribution is overestimated in [4] due to the specific model of the pion LCDA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  8  The γπ-decay amplitude can be obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' (17) substituting photon LCDAs instead of  ρ-meson ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The LO amplitude reads  Alo  γπ0 = − ⟨0| χ†σ · ϵψ |ψ(n, ϵ)⟩  �  2Mψ (παs)2 √  2πα20  27  fγ f3π  [4m2c]2  �  Jlo  γπ + f3γ fπ  f3πfγ  Jlo  γπ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (42)  The ratio of the normalisation couplings in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' (42) yields (µ = 2GeV)  f3γ fπ  f3πfγ  ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='92,  (43)  which is different from the analogous ratio for the ρ-meson (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  The leading-order numerical  estimates gives  Br[J/ψ → γπ0]lo ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='82 × 10−5,  (44)  which very well agrees with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The results for the total convolution integral Jγπ are  presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Comparing with the analogous results for the ρπ-channel one finds that  Table 3: Numerical result for the convolution integrals Jγπ  Jlo  γπ  Jnlo  γπ /Jlo  γπ  Jγπ/Jlo  γπ  J/ψ  932  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='45  ψ′  932  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='49  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='74  the both descriptions are qualitatively similar despite the different ratio (43) and the differences  between the LCDAs φ2γ and φ⊥  2ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Therefore one can again assume that O(αs) radiative correc-  tions also play a crucial role in the understanding of the value of the decay width γπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Moreover,  the contributions with a perturbative photon, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' 1(b) can probably explain the larger  value of Qγπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  3  Conclusions  In conclusion, we calculated and investigated relativistic corrections to the decay amplitudes  ψ(n) → ρπ and ψ(n) → γπ within the pQCD (NRQCD and collinear factorisation) framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  This calculation includes the exact correction of relative order v2 and subset of the higher order  corrections associated with the quark-antiquarks wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Numerical estimates show that  an order v2 correction is large and give the dominant numerical effect, which can be related  to the structure of the collinear integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' If this observation is not affected by other higher  order relativistic corrections, then one has to consider the relative v2-contribution as a special  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The obtained relativistic corrections are negative and large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' In case ψ′ → ρπ the relative  v2 contribution is much larger than the leading-order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The different effects of relativistic  corrections for J/ψ → ρπ and ψ′ → ρπ suggest a possible explanation for the ρπ-puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' If the  QCD radiative correction is positive and large enough then it interferes destructively with the  relativistic correction for ψ′ → ρπ giving the small branching fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' At the same time such  radiative correction will improve the description of J/ψ → ρπ reducing the negative effect of  the relativistic correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Therefore, we believe that further investigation of relative order v4  corrections and QCD radiative corrections can help to clarify this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  We also expect that the same approach can be be used for an analysis other hadronic decay  channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' As a simplest example, the decay ψ(n) → γπ was also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' We studied the  9  contribution, which is given by similar diagrams but with nonperturbative photon instead of ρ-  meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Despite the difference between twist-2 LCDAs for photon and ρ−meson, the qualitative  effect from the relativistic corrections is quite similar to ψ → ρπ, they are also large and negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  Therefore one can guess that the similar scenario with radiative corrections is also applicable  here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' The only difference with ψ(n) → ρπ is provided by the contributions with perturbative  photon coupling, which are suppressed by O(αs) or O(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Therefore, it can be that these expects  are responsible for the larger value of Qγπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  4  Appendix  Here we provide the analytical expressions for the intergrals Jπ and Jρ introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  In order to simplify notations we use  �  v2�  ≡ v2,  δ = 1 − 1/  �  1 + v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (45)  The first integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' (17) reads  Jπ  �  v2�  =  �  Dui  φ3π(ui)  u1u2u3  � 1  0  dy φ⊥  2 (y)  y¯y  � 2Aπ  D1D3  +  Bπ  D1D2  �  ,  ¯y = 1 − y,  (46)  where  Di = δi1 (y1¯u2 + ¯y1u2) + δi2 (y2¯u1 + ¯y2u1) + δi3 u3,  (47)  with  y1 = y, y2 = ¯y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (48)  The symbol δik denotes the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  The numerators Aπ and Bπ are given by the sums  Aπ =  4  �  k=0  fA  k Ik[13],  Bπ =  4  �  k=0  fB  k Ik[12],  (49)  where  Ik[ij] = 1  2  � 1  −1  dη  vkηk  (1 + vη ai) (1 − vη aj) =  vk  ai + aj  ∞  �  n=0  vn an+1  j  + (−1)nan+1  i  n + 1 + k  1  2  �  1 + (−1)n+k�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (50)  with  aj = δ1j (1 − δ)  y1 − u2  y1¯u2 + ¯y1u2  + δ2j (1 − δ)  y2 − u1  y2¯u1 + ¯y2u1  − δ3j (1 − δ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (51)  The coefficients fA,B  k  ≡ fA,B  k  (ui, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' δ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' (49) read  fA  0 = δ  2 (3u3 − 2 − δ) ,  fA  1 = δ  2  (1 − δ)  (2 − δ)u3,  (52)  fA  2 = 1  2  (1 − δ)2  (2 − δ)2 (4 − 3(2 − δ)u3 + 2δ(1 − δ)) ,  (53)  fA  3 = −1  2  (1 − δ)3  (2 − δ)2 u3,  fA  4 = −1  2  (1 − δ)4  (2 − δ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (54)  10  fB  0 = u1y1 + u1y2 − δ  2(u1 + u2 + y1 + y2 − δ)  (55)  +  δ  2 − δ  �  u1y1 + u1y2 − (2 − δ) (u1 + u2 + y1 + y2) + (2 − δ)2�  ,  (56)  fB  1 = 1  2  (1 − δ)  (2 − δ) {4(u1y1 − u2y2) + δ(u1 + y1 − u2 − y2)} ,  (57)  fB  2 = 1  2  (1 − δ)2  (2 − δ)2 {6 (u1 + u2 + y1 + y2) − 2 (u1y1 + u1y2) − 8  (58)  +δ (4 − 3 (u1 + u2 + y1 + y2)) + 2δ(2 − δ)} ,  (59)  fB  3 = −1  2  (1 − δ)3  (2 − δ)2 (u1 + y1 − u2 − y2) ,  fB  4 = −1  2  (1 − δ)4  (2 − δ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (60)  The ρ-meson integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' (17) reads  Jρ =  � 1  0  du φ2π(u)  u¯u  �  Dyi  1  y1y2y3  � 2Aρ  y3D2  +  Bρ  D1D2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (61)  The numerators Aρ and Bρ can be written as  Aρ = φ3ρ(yi)  4  �  k=0  �  fA  k  �  Ik[23] + ˜φ3ρ(yi)  4  �  k=0  �  ˜fA  k  �  Ik[23],  (62)  Bρ = φ3ρ(yi)  4  �  k=0  �  fB  k  �  Ik[12] + ˜φ3ρ(yi)  4  �  k=0  �  ˜fB  k  �  Ik[12],  (63)  where the integrals Ik are defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' (50) with a bit different combination aj  aj = δ1j (1 − δ)  y1 − u2  y1¯u2 + ¯y1u2  + δ2j (1 − δ)  y2 − u1  y2¯u1 + ¯y2u1  + δ3j (1 − δ) ,  (64)  and we again use for the two-particle LCDA u1 = u, u2 = 1 − u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  The coefficients fA,B  k  and ˜fA,B  k  defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' (62) and (63) read  fA  0 = 1  4 (u1(2y3 − δ) + δ (δ − y2 − y3)) + δ2  2  (65)  + 1  4  δ  (2 − δ) (u1 (6 + 4y3 − 3δ) + (2 − δ) (3y2 + 2y3 − 2 − 3δ)) ,  (66)  fA  1 = −1  4  1 − δ  2 − δ (4u1 + δ) y3,  (67)  fA  2 = −1  4  (1 − δ)2  (2 − δ)2 (2u1y3 + (2 − δ) (2u1 + 2y2 + y3) − 2(1 − δ)(2 − δ)) ,  (68)  fA  3 = 1  4  (1 − δ)3  (2 − δ)2 y3,  fA  4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (69)  fB  0 = −fB  2 = δ  4 (u1 − u2 − y1 + y2) ,  (70)  fB  1 = −fB  3 = −δ  4(1 − δ) (u1 + u2 − y1 − y2) ,  fB  4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (71)  11  ˜fA  0 = 1  4 (u1(2y3 − δ) + δ (δ − y2 − y3))  (72)  + 1  4  δ  (2 − δ) (u1 (2 + 4y3 − δ) + (2 − δ) (2 + y2 − 2y3 − 3δ)) ,  (73)  ˜fA  1 = −1  4  1 − δ  2 − δ (4u1y3 + δ (2u1 − 2y2 + y3)) ,  (74)  ˜fA  2 = −1  4  (1 − δ)2  (2 − δ)2 (2u1y3 − 3y3(2 − δ) + δ(2 − δ)) ,  (75)  ˜fA  3 = 1  4  (1 − δ)3  (2 − δ)2 (2u1 − 2y2 + y3) ,  ˜fA  4 = −1  2  (1 − δ)4  (2 − δ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content='  (76)  ˜fB  0 = δ  4 (3 (u1 + u2 + y1 + y2) − 4 − 2δ) ,  ˜fB  1 = −δ  4  1 − δ  2 − δ (u1 − u2 + y1 − y2) ,  (77)  ˜fB  2 = 1  4  (1 − δ)2  (2 − δ)2 (8 − 3 (2 − δ) (u1 + u2 + y1 + y2) + 4δ(1 − δ)) ,  (78)  ˜fB  3 = 1  4  (1 − δ)3  (2 − δ)2 (u1 − u2 + y1 − y2) ,  ˜fB  4 = −1  2  (1 − δ)4  (2 − δ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
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+page_content=' Lee and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
+page_content=' Yu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
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+page_content='  Lee,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfbQfw/content/2301.03884v1.pdf'}
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+arXiv:2301.04759v1  [math.CV]  11 Jan 2023
+SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS
+RICARDO P´EREZ-MARCO
+Abstract. We study shift functional equations that generalize the functional equa-
+tion satisfied by Euler Gamma function. Under a natural growth condition, the
+vector space of meromorphic solutions is finite dimensional. We construct a basis of
+the space of solutions composed by Omega functions that generalize Euler Gamma
+function. Omega functions are defined as exponential periods. They have a mero-
+morphic extension to the complex plane with simple poles at negative integers.
+They are characterized by their growth property on vertical strips and their shift
+functional equation. This generalizes Wielandt’s characterization of Euler Gamma
+function.
+1. Introduction
+1.1. Shift functional equations. The goal of this article is to study and solve shift
+functional equations of the form
+(1)
+sf(s) =
+d
+�
+k=1
+αkf(s + k)
+where α1, . . . , αd ∈ C and αd ̸= 0. The simplest case of this functional equation is
+the functional equation satisfied by Euler Gamma function
+sΓ(s) = Γ(s + 1)
+These equations are linear and we have a vector space of meromorphic solutions.
+A natural motivation for studying these functional equations comes from the study
+of subspaces generated by natural linear operators. For instance, we can consider, in
+the space of meromorphic functions, the shift (or integer translation) linear operator
+T(f(s)) = f(s + 1)
+and the multiplication by s linear operator
+S(f(s)) = sf(s)
+2010 Mathematics Subject Classification. Primary: 30D10. Secondary: 30D15, 30B50.
+Key words and phrases. Euler Gamma function, exponential periods, Omega functions, shift
+functional equation.
+1
+
+2
+R. P´EREZ-MARCO
+Observe that S has no eigenvectors and the minimal invariant subspace invariant
+by S containing the constant functions is the space of polynomials C[s]. The space
+generated by S and the function f(s) is the vector space C[s]f(s)
+⟨f, S(f), S2(f), . . .⟩ = C[s]f(s)
+It is natural to ask for which functions f the space C[s]f(s) is also generated by
+f(s) and the shift operator T. This happens if and only if f is solution of the shift
+functional equation (1).
+Already, in the simplest case of the functional equation of Euler Gamma function,
+the space of solutions is infinite dimensional since any function of the form e2πinsΓ(s)
+for an integer n ∈ Z is also a solution. It is classical to add conditions to characterize
+Euler Gamma function as the only normalized solution to this functional equation.
+One can mention Weierstrass characterization imposing some asymptotic behavior
+when s → +∞ (1856, [14]), or Wielandt’s characterization (1939, [15], see also [12],
+[13]) requiring boundedness on vertical strips of width larger than 1, or, more recently,
+requiring finite order of the solutions and a right half plane free of zeros nor poles
+(2022, [11]). Wielandt’s boundedness condition has been weakened by Fuglede to a
+moderate growth in the vertical strip (2008, [7]).
+In the spirit of Wielandt, we search for solutions with some growth control on
+vertical strips. Under a suitable growth condition, we can prove that the space of
+solutions is finite dimensional:
+Theorem 1.1. The space of meromorphic solutions f of the functional equation
+sf(s) =
+d
+�
+k=1
+αkf(s + k)
+where α1, . . . , αn ∈ C, αd ̸= 0, and f satisfies a growth condition, for 1 ≤ Re s ≤ d,
+|f(s)| ≤ Ce−c Im s
+for some constant C > 0 and 0 ≤ c < 2π, is finite dimensional of dimension d.
+Moreover, we build an explicit basis of the vector space of solutions with a new1
+kind of Special Functions, that we call Omega functions, that generalize Euler Gamma
+function.
+1The author couldn’t find any reference to Omega functions in the literature.
+
+SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS
+3
+1.2. Omega functions. Historically, Euler Gamma function appears for the first
+time in a letter from Euler to Goldbach, dated January 8th 1730 ([6]). Euler defines
+the Gamma function for real values s > 0, by the integral formula
+Γ(s) =
+� +∞
+0
+ts−1e−tdt .
+which is also convergent for complex values of s with Re s > 0.
+In this integral
+formula, the value Γ(s) appears as an exponential period.
+Algebraic periods are integrals of algebraic differential forms over cycles of an alge-
+braic variety. In the special case of an algebraic curve, when we represent the curve as
+a Riemann domain over the complex plane or the Riemann sphere, algebraic periods
+are also the integrals of algebraic differential forms on paths joining two ramification
+points where we have singularities of the differential form. From the transalgebraic
+point of view, it is natural to consider exponential periods, where integrals involve
+exponential expressions, and the singularities can be exponential singularities. More
+general periods can be envisioned where the differential form has transcendental sin-
+gularities with monodromy as ts in a local variable (geometrically these corespond
+to differential forms living in a branched Riemann domain with an infinite ramifica-
+tion). There is a vast literature on classical algebraic periods, but almost none on the
+transalgebraic periods. We refer to [10] for definitions and a review about classical
+periods, and to [3] and [4] for exponential periods and their relation with log-Riemann
+surfaces. Also we refer the reader to [11] for an historical survey of different defini-
+tions of Euler Gamma function and their generalizations, and to [16] for its classical
+properties.
+The second goal of this article is to introduce Omega functions (or Ω-functions)
+which are a natural generalization of the Euler Gamma function. They are defined
+as exponential periods of the form
+Ωk(s) =
+� +∞.ωk
+0
+ts−1eP0(t) dt
+where P0(t) ∈ C[t] and ωk is a root of unity pointing to a direction where the poly-
+nomial P0 diverges to −∞.
+Some critical computations in the proof of the main Theorem are generalizations
+of computations carried out for exponential periods appearing in [4]. In particular,
+the generalization of the Ramificant Determinant considered in that article is the key
+result for the proof of the linear independence of the Omega functions (Ωk). In its
+magical computation, we calculate explicitly a determinant of a matrix of exponential
+periods which are individually not computable.
+
+4
+R. P´EREZ-MARCO
+2. Definition.
+Let P0(t) ∈ C[t] be a degree d ≥ 1 polynomial such that P0(0) = 0 and limt→+∞ Re P0(t) =
+−∞, normalized by
+P0(t) = −1
+dtd +
+d−1
+�
+k=1
+aktk
+We also denote ad = −1/d and a0 = 0. Let ω be the primitive d-th root of unity
+given by
+ω = e
+2πi
+d
+and write ωk = ωk.
+Definition 2.1. Let d ≥ 1. For k = 0, 1, . . . , d − 1, the Omega functions, or Ω-
+functions, associated to P0, are defined for Re s > 0, by
+Ωk(s) =
+� +∞.ωk
+0
+ts−1eP0(t) dt
+The roots ωk point to the directions where the polynomial P0 diverges exponentially
+to −∞,
+lim
+t→+∞.ωk Re P0(t) = −∞
+so the integral is converging and we have a sound definition. Usually, we spare the
+reference to P0, but for some results it will be crucial to keep track of the dependence
+on parameters and we will write
+Ω(s) = Ω(s|P0) = Ω(s|a1, . . . , ad−1) .
+For d = 1, we have P0(t) = −t and Ω1 = Γ is Euler Gamma function.
+If P0 ∈ R[t], then Ω0 is real analytic, and Ωk(¯s|P0) = Ωd−k(s| ¯P0), where ¯P0 is the
+conjugate polynomial
+¯P0(t) = −1
+dtd +
+d−1
+�
+k=1
+¯aktk .
+Sometimes we will be interested in the case where αk ∈ K where K ⊂ C is a number
+field. In that case we say that the Omega functions are defined over K.
+
+SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS
+5
+3. Meromorphic extension, poles and residues.
+Theorem 3.1. The Omega functions (Ωk)0≤k≤d−1 extend to the complex plane into
+meromorphic functions of order 1 satisfying the fundamental functional equation
+(2)
+Ωk(s + d) + αd−1Ωk(s + d − 1) + . . . + α1Ωk(s + 1) = s Ωk(s)
+where αl = −lal.
+Moreover, the function Ωk is holomorphic in C − N, and has simple poles at the
+negative integers. The residue at s = 0 is
+Ress=0 Ωk = 1 .
+Observe that for d = 1, Ω0 = Γ and the functional equation (2) is the classical
+functional equation Γ(s + 1) = sΓ(s).
+Proof. We have for Re s > 0 and by integration by parts,
+Ωk(s + d) +
+d−1
+�
+l=1
+αlΩk(s + l) =
+� +∞.ωk
+0
+ts(−P ′
+0(t)).eP0(t) dt
+=
+�
+−tseP0(t)�+∞.ωk
+0
++ s
+� +∞.ωk
+0
+ts−1eP0(t) dt
+= s Ωk(s)
+and we get the functional equation (2). Now, using once the functional equation we
+extend meromorphically Ωk to {Re s > −1}, and by induction to {Re s > −n}, for
+n = 1, 2, . . ., hence to all of C. The only poles that can be introduced by this extension
+procedure using the functional equation are those created from the pole at s = 0 at
+the negative integers. The functional equation shows that sΩk(s) is holomorphic at
+s = 0, hence the pole at s = 0 is simple. It follows from the functional equation and
+the extension procedure that the other poles are also simple. We compute the residue
+at s = 0 using the functional equation,
+Ress=0 Ωk = lim
+s→0 sΩk(s) =
+d
+�
+l=1
+αlΩk(l) =
+� +∞.ωk
+0
+(−P ′
+0(t)).eP0(t) dt =
+�
+−eP0(t)�+∞.ωk
+0
+= 1
+□
+More generally, we can compute the residues at the negative integers.
+Theorem 3.2. Let (λn)n≥0 be the coefficients of the power series expansion of eP0(t),
+eP0(t) =
++∞
+�
+n=0
+λntn .
+
+6
+R. P´EREZ-MARCO
+Then the residue of Ωk at s = −n is λn,
+Ress=−n Ωk = λn .
+Proof. For n ≥ 0 let rn ∈ C be the residue of Ωk at s = −n, with rn = 0 if there is
+no pole, and rn = 0 for n < 0. The functional equation (2) gives
+rn = lim
+h→0 hΩk(h − n) = lim
+h→0
+1
+h − n
+d
+�
+l=1
+αlhΩk(h − n + l) = −1
+n
+d
+�
+l=1
+αl rn−l
+hence the recurrence relation
+(3)
+nrn = −
+d
+�
+l=1
+αl rn−l
+Now, consider the generating power series
+F(t) =
++∞
+�
+n=0
+rntn
+The recurrence relation (3) gives
+F ′(t) =
++∞
+�
+n=0
+nrntn−1 = −
+d
+�
+l=1
+αl
++∞
+�
+n=0
+rn−ltn−1
+= −
+d
+�
+l=1
+αltl−1
++∞
+�
+n=l
+rn−ltn−l
+= −
+�
+d
+�
+l=1
+αltl−1
+�
+F(t)
+= P ′
+0(t)F(t)
+Since we have F(0) = r0 = 1 from Theorem 3.1, we get
+F(t) = eP0(t)
+thus rn = λn as claimed.
+□
+Example.
+For d = 1, the generating power series is
+F(t) = e−t =
++∞
+�
+n=0
+(−1)n
+n!
+tn
+
+SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS
+7
+and we recover the classical result that
+Ress=−n Γ = (−1)n
+n!
+.
+Note that the residues at the simple poles at the negative integers are the same for
+all the functions Ω0, . . . , Ωd−1. Indeed, for k ̸= l, we can check directly that Ωk − Ωl
+is an entire function because of the convergence for all s ∈ C of the integral
+Ωk(s) − Ωl(s) =
+� +∞.ωk
++∞.ωl
+ts−1eP0(t) dt =
+�
+γlk
+ts−1eP0(t) dt
+where the integral can be taken over any path γlk assymptotic to +∞.ωl and +∞.ωk
+in the proper direction and with 0 winding number around 0. This integral depends
+holomorphically on the parameter s ∈ C.
+Observe also that if the coefficients of P0 belong to a number field K, P0(t) ∈ K[t],
+then the residues of Ωk belong also to K. Another arithmetical observation is the
+following:
+Corollary 3.3. We assume that the only non-zero coefficients of P0 are for powers
+divisible by an integer n0 ≥ 2, that is, if ak ̸= 0 then n0|k.
+Then, if n0 does not divide n, we have rn = 0.
+Proof. From the previous Theorem we have
+eP0(t) =
+d
+�
+k=1
+eaktk =
+d
+�
+k=1
+��
+m≥0
+am
+k
+m!tmk
+�
+and when we expand the last product we get the result.
+□
+We have a more precise result than just the computation of the residues.
+We
+can determine the Mittag-Leffler decomposition of Ωk.This is an analytic result that
+requires some estimates.
+Theorem 3.4. The Omega function Ωk has the Mittag-Leffler decomposition:
+Ωk(s) =
++∞
+�
+n=0
+λn
+z + n +
+� +∞.ωk
+1
+ts−1eP0(t) dt
+where the integral is an entire function of order 1.
+Observe that this Theorem shows that the Omega function Ωk is a meromorphic
+function of order 1.
+Corollary 3.5. The Omega functions Ωk are meromorphic functions of order 1.
+
+8
+R. P´EREZ-MARCO
+Proof. We write
+Ωk(s) =
+� 1
+0
+ts−1eP0(t) dt +
+� +∞.ωk
+1
+ts−1eP0(t) dt
+and we compute the first integral expanding the exponential in power series (uniformly
+convergent in [0, 1]),
+� 1
+0
+ts−1eP0(t) dt =
++∞
+�
+n=0
+λn
+� 1
+0
+ts+n−1 dt =
++∞
+�
+n=0
+λn
+� ts+n
+s + n
+�1
+0
+=
++∞
+�
+n=0
+λn
+s + n
+The second integral can be bounded by the next Lemma that shows that it is an
+entire function of order 1 (using that Euler Gamma function is of order 1).
+□
+Lemma 3.6. We have the estimate
+����
+� +∞.ωk
+1
+ts−1eP0(t) dt
+���� ≤ e−2π k
+d Im s
+�
+C0 + C1dRe s/dΓ
+�Re s
+d
+��
+Proof. We make the change of variables t = ωku
+� +∞.ωk
+1
+ts−1eP0(t) dt = ωs
+k
+� +∞
+ω−1
+k
+us−1e− 1
+d ud(1+O(u−1)) du
+= e2πi k
+d s
+� +∞
+ω−1
+k
+us−1e− 1
+d ud(1+O(u−1)) du
+This gives the bound
+����
+� +∞.ωk
+1
+ts−1eP0(t) dt
+���� ≤ e−2π k
+d Im s
+�����
+� +∞
+ω−1
+k
+us−1e− 1
+d ud(1+O(u−1)) du
+�����
+Now, taking an integration path of finite length from 1 to ωk and bounded away from
+0, we get (using the same letter C to denote several universal constants C > 0)
+����
+� +∞.ωk
+1
+ts−1eP0(t) dt
+���� ≤ e−2π k
+d Im s
+�
+C +
+����
+� +∞
+1
+us−1e− 1
+d ud(1+O(u−1)) du
+����
+�
+The last integral can be estimated by
+����
+� +∞
+1
+us−1e− 1
+d ud(1+O(u−1)) du
+���� ≤ (1 + C)
+� +∞
+1
+us−1e− 1
+d ud du
+≤ (1 + C)
+� +∞
+0
+us−1e− 1
+d ud du
+≤ (1 + C)dRe s/dΓ
+�Re s
+d
+�
+(for the computation of the last integral we use the change of variable v = ud/d).
+□
+
+SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS
+9
+4. Incomplete Omega functions.
+We define the Incomplete Omega functions that generalize the Incomplete Gamma
+function.
+Definition 4.1. For z, s ∈ C, Re s > 0, the Incomplete Omega function Ω(s, z) is
+defined by
+Ω(s, z) =
+� z
+0
+ts−1eP0(t) dt
+For P0(t) = −t this is the classical Incomplete Gamma function. Observe that we
+recover all the (Ωk) functions by taking the appropriate limit of Ω(s, z) as z → ∞,
+Ωk(s) =
+lim
+z→+∞.ωk Ω(s, z)
+For the particular values of s = 1, 2, . . . , d − 1 these are the transcendental entire
+functions of the variable z ∈ C studied in [4] which form a basis of the fundamental
+for the transcendental vector space of functions on a simply connected log-Riemann
+surface with exactly d infinite ramification points. Some of the results proved here
+generalize some results from [4]. Following the same Abel’s philosophy that inspires
+[4], we prove that we only need to use a finite number of transcendentals (Ω(s +
+k, z))0≤k≤d−1 to compute integrals of the form
+� z
+0
+Q(t, ts)eP0(t) dt
+where Q(x, y) ∈ C[x, y] is a polynomial. We start by studying the simpler case when
+Q(t) ∈ C[t].
+Proposition 4.2. Let Q(t) ∈ C[t]. For d ≥ 2, the integral
+� z
+0
+tsQ(t)eP0(t) dt
+is of the form
+� z
+0
+tsQ(t)eP0(t) dt = zsA(s, z) eP0(z) +
+d−1
+�
+k=0
+ck(s) Ω(s + k, z)
+where A ∈ C[s, z], and the polynomial coefficients ck(s) ∈ C[s] have coefficients de-
+pending polynomially on the coefficients of P0, (a1, . . . , ad−1).
+
+10
+R. P´EREZ-MARCO
+Proof. First we consider the case d = 1. We prove the result for Q(t) = tn integrating
+by parts n + 1 times
+� z
+0
+ts+ne−t dt =
+�
+−ts+ne−t�z
+0 + (s + n − 1)
+� z
+0
+ts+n−1e−t dt
+= −zs+ne−z + (s + n)
+� z
+0
+ts+n−1e−t dt
+= −(zs+n + (s + n)zs+n−1)e−z + (s + n)(s + n − 1)
+� z
+0
+ts+n−2e−t dt
+...
+= −zs(zn + (s + n)zn−1 + . . .)e−z + (s + n)(s + n − 1) . . . s Ω(s, z)
+For a general polynomial Q(t) we have the the result by linear decomposition of the
+integral.
+In the rest of the proof we assume d ≥ 2. If q = deg Q ≤ d − 2, by linearity, the
+integral is a linear combination (with the coefficients of tQ(t)) of (Ω(s + k, z))0≤k≤d−1
+and the result follows.
+If deg Q ≥ d − 1, then we consider the Euclidean division of Q(t) by P ′
+0(t),
+Q(t) = A1(t)P ′
+0(t) + B1(t)
+with A1, B1 ∈ C[t], deg B1 ≤ d−2 and deg A1 = deg Q−(d−1) = q −(d−1) ≤ q −1.
+We proceed splitting the integral:
+� z
+0
+tsQ(t)eP0(t) dt =
+� z
+0
+tsA1(t)P ′
+0(t)eP0(t) dt +
+� z
+0
+tsB1(t)eP0(t) dt
+Since deg B1 ≤ d − 2, the second integral is a linear combination of Ω(s, z), Ω(s +
+1, z), . . . , Ω(s + d − 1, z), thus of the desired form, and we can forget about it. We
+work on the first integral integrating by parts,
+� z
+0
+tsA1(t)P ′
+0(t)eP0(t) dt =
+�
+tsA1(t)eP0(t)�z
+0 −
+� z
+0
+(tsA1(t))′ eP0(t) dt
+Then we get:
+� z
+0
+tsA1(t)P ′
+0(t)eP0(t) dt = zsA1(z)eP0(z) −
+� z
+0
+tsA′
+1(t)eP0(t) dt − s
+� z
+0
+ts−1A1(t)eP0(t) dt
+The first integral in the right hand side is of the same form as the initial one with
+Q(t) but with deg A′
+1 = deg A1 − 1 ≤ deg Q − (d − 1) − 1 = q − d ≤ q − 2 (using
+here d ≥ 2), hence by descending induction we can forget about it. For the second
+integral, we can write
+s
+� z
+0
+ts−1A1(t)eP0(t) dt = A1(0)s Ω(s, z) + s
+� z
+0
+ts
+�A1(t) − A1(0)
+t
+�
+eP0(t) dt
+
+SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS
+11
+and t−1(A1(t) − A1(0)) is a polynomial of degree deg A1 − 1 ≤ q − 2.
+Then the
+descending induction gives the expression for the integral as announced.
+The coefficients of P0 appear first linearly in the Euclidean divisions by P ′
+0 then,
+by repeated Euclidean divisions the dependence of the ck(s) is polynomial on the
+coefficients of P0.
+□
+Now, we can get easily the general result for Q(t, ts).
+Corollary 4.3. Let Q(t, ts) ∈ C[t, ts]. For d ≥ 2, the integral
+� z
+0
+Q(t, ts)eP0(t) dt
+is of the form
+� z
+0
+ts−1Q(t, ts)eP0(t) dt = A(s, z, zs)eP0(z) +
+d−1
+�
+k=0
+ck(s)Ω(s + k, z)
+where A ∈ C[s, z, zz] and the coefficients ck(s) ∈ C[s] have coefficients that are poly-
+nomials on the coefficients of P0 (a1, . . . , ad−1).
+Proof. We write Q(t, ts) = Q(t)(ts) as a polynomial on the variable ts with coeffi-
+cients polynomials in t, and we split linearly the integral, observing that the part
+corresponding to each monomial is like the integral when Q is a polynomial in t as
+before, but with s shifted into s + l by some positive integer l. The result follows
+from the previous Proposition.
+□
+Now, we can prove that the Omega functions (Ωk(s))0≤k≤d−1 generate a large class
+of exponential periods:
+Corollary 4.4. Let Q(t, ts) ∈ C[t, ts] and 0 ≤ n ≤ d − 1. The exponential period
+� +∞.ωn
+0
+Q(t, ts) eP0(t) dt
+is a linear combination of the exponential periods (Ωk(s))0≤k≤d−1
+� +∞.ωn
+0
+Q(t, ts) eP0(t) dt =
+d−1
+�
+k=0
+ck(s) Ωk(s)
+where the coefficients ck(s) are polynomials on s and on the coefficients (a1, . . . , ad−1).
+Proof. For d ≥ 2, we have from the previous Proposition that
+� z
+0
+Q(t, ts)eP0(t) dt = A(s, z, zs)eP0(z) +
+d−1
+�
+k=0
+ck(s)Ω(s + k, z)
+
+12
+R. P´EREZ-MARCO
+When z → +∞.ωn the terms in the first sum vanish, since A(s, z, zs)eP0(z) → 0
+for a polynomial A(s, z, zs) ∈ C[z] (the exponential decay of eP0(z) takes over the
+polynomial divergence of A(z)), and we get
+� +∞.ωn
+0
+Q(t, ts)eP0(t) dt =
+d−1
+�
+k=0
+ck(s)Ωn(s + k)
+□
+5. Linear independence.
+The row vector build with Omega functions Ω(s) = (Ωk(s))0≤k≤d−1 has the follow-
+ing important linear independence property:
+Theorem 5.1. For any s ∈ C − N∗
+−, the vectors Ω(s + 1), Ω(s + 2), . . . , Ω(s + d) are
+linearly independent,
+∆(s) ̸= 0
+where
+∆(s|a1, . . . , ad−1) = det
+
+
+Ω11
+Ω12
+. . .
+Ω1d
+Ω21
+Ω22
+. . .
+Ω2d
+...
+...
+...
+...
+Ωd1
+Ωd2
+. . .
+Ωdd
+
+
+.
+where Ωkl = Ωk−1(s + l).
+More precisely, we can compute
+∆(s|a1, . . . , ad−1) = ∆(s|0, . . . , 0) exp (Πd(s, a1, . . . , ad−1))
+where Πd(s, a1, . . . , ad−1) is a universal polynomial with rational coefficients.
+In view of the last formula, the result follows from ∆(s|0, . . . , 0) ̸= 0. We will
+prove the last formula and compute explicitly the determinant ∆(s|0, . . . , 0). These
+computations are similar to the ones for the Ramificant Determinant (see [4]) that
+corresponds to the special case s = 0.
+We can compute Ωk(s + l|0, . . . , 0) using Euler Gamma function.
+Lemma 5.2. We have
+Ωk(s + l|0, . . . , 0) = ωk(s+l)d
+s+l
+d −1Γ
+�s + l
+d
+�
+
+SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS
+13
+Proof. We first make the change of variables t = ωku, and then v = ud/d,
+Ωk(s + l|0, . . . , 0) =
+� +∞.ωk
+0
+ts+l−1e− 1
+d td dt
+= ωk(s+l)
+� +∞
+0
+us+l−1e− 1
+d ud du
+= ωk(s+l)d
+s+l
+d −1
+� +∞
+0
+v
+s+l
+d −1e−v dv
+= ωk(s+l)d
+s+l
+d −1Γ
+�s + l
+d
+�
+□
+Now we recall the following well known elementary Vandermonde Lemma:
+Lemma 5.3. If ξ1, . . . , ξd are the d roots of a monic polynomial Q(X), then we can
+compute the Vandermonde determinant V (ξ1, . . . , ξd) of the (ξ1, . . . , ξd) as
+V (ξ1, . . . , ξd) =
+���������
+1
+ξ1
+ξ2
+1
+. . .
+ξd−1
+1
+1
+ξ2
+ξ2
+2
+. . .
+ξd−1
+2
+...
+...
+...
+...
+...
+1
+ξd
+ξ2
+d
+. . .
+ξd−1
+d
+���������
+=
+�
+i̸=j
+(ξi − ξj) =
+d
+�
+i=1
+Q′(ξi) .
+Using this Lemma with Q(X) = Xd−1 we compute the Vandermonde determinant:
+Vd =
+���������
+1
+ω1
+ω2
+1
+. . .
+ωd−1
+1
+1
+ω2
+ω2
+2
+. . .
+ωd−1
+2
+...
+...
+...
+...
+...
+1
+ωd
+ω2
+d
+. . .
+ωd−1
+d
+���������
+=
+�
+i̸=j
+(ωi−ωj) =
+�
+i
+(dωd−1
+i
+) = dd
+��
+i
+ωi
+�d−1
+= (−1)d−1dd .
+We use this result to compute ∆(s|0, . . . , 0).
+Lemma 5.4. We have
+∆(s|0, . . . , 0) = (2πd)
+d
+2
+√
+2π ω
+d(d−1)
+2
+ss Γ(s)
+and in particular ∆(s|0, . . . , 0) ̸= 0 for s ̸= −1, −2, . . ..
+Taking the limit s → 0 we recover the formula from Lemma 3.5 from [4].
+
+14
+R. P´EREZ-MARCO
+Proof. Using Lemma 5.2 we have
+∆(s|0, . . . , 0) =
+=
+���������
+ω0.(s+1)d
+s+1
+d −1Γ
+�s+1
+d
+�
+ω0.(s+2)d
+s+2
+d −1Γ
+� s+2
+d
+�
+. . .
+ω0.(s+d)d
+s+d
+d −1Γ
+� s+d
+d
+�
+ω1.(s+1)d
+s+1
+d −1Γ
+�s+1
+d
+�
+ω1.(s+2)d
+s+2
+d −1Γ
+�s+d
+d
+�
+. . .
+ω1.(s+d)d
+s+d
+d −1Γ
+� s+d
+d
+�
+...
+...
+...
+...
+ω(d−1).(s+1)d
+s+1
+d −1Γ
+�s+1
+d
+�
+ω(d−1).(s+2)d
+s+2
+d −1Γ
+� s+2
+d
+�
+. . .
+ω(d−1).(s+d)d
+s+d
+d −1Γ
+� s+d
+d
+�
+���������
+= ω
+d(d−1)
+2
+sdsd
+d−1
+2 −dΓ
+�s
+d + 1
+d
+�
+Γ
+�s
+d + 2
+d
+�
+. . . Γ
+�s
+d + d
+d
+�
+��������
+ω1
+0
+ω2
+0
+. . .
+ωd
+0
+ω1
+1
+ω2
+1
+. . .
+ωd
+1
+...
+...
+...
+...
+ω1
+d−1
+ω2
+d−1
+. . .
+ωd
+d−1
+��������
+= ω
+d(d−1)
+2
+sd− d
+2s(2π)
+d−1
+2 Γ(s)dd
+= d
+d
+2 (2π)
+d−1
+2 sω
+d(d−1)
+2
+s Γ(s)
+= (2πd)
+d
+2
+√
+2π ω
+d(d−1)
+2
+ss Γ(s)
+where we have used Gauss multiplication formula in the second line (that is in fact
+due to Euler and not to Gauss, see [1]) with z = s/d,
+Γ(z).Γ
+�
+z + 1
+d
+�
+. . . Γ
+�
+z + d − 1
+d
+�
+= (2π)
+d−1
+2 d
+1
+2−dzΓ(dz) .
+and that the determinant in the fourth line is equal to (−1)d−1Vd where Vd is the
+Vandermonde determinant computed previously.
+□
+Proof of Theorem 5.1. Consider the entire function of several complex variables
+∆(s|a1, a2, . . . , ad−1) on the variables (a1, a2, . . . , ad−1). Observe that Corollary 4.4
+proves that each integral
+� +∞.ωk
+0
+ts+n−1eP0(t) dt ,
+is a linear combination with coefficients that are polynomial on s and the (ak) of the
+integrals Ωk(s) for k = 0, 1, . . . , d − 1, Therefore, differentiating column by column,
+we observe that for each k = 0, 1, . . . , d − 1, we have
+∂ak∆ = Qk ∆
+
+SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS
+15
+where Qk is a polynomial on s and the (ak).
+We conclude that the logarithmic
+derivative of ∆ with respect to each variable ak is a universal polynomial on the
+variables s and (ak). This gives the existence of the universal polynomial Υd such
+that
+∆(s|a1, a2, . . . , ad−1) = c(s).eΥd(s;a1,a2,...,ad−1) ,
+with c(s).eΥd(s;0,0,...,0) = ∆(s|0, . . . , 0) ∈ C. Then if we define Πd(s, a1, . . . , ad−1) =
+Υd(s; a1, a2, . . . , ad−1) − Υd(s; 0, 0, . . . , 0) we get the result
+∆(s|a1, a2, . . . , ad−1) = ∆(s|0, . . . , 0)eΠd(s,a1,...,ad−1)
+□
+Corollary 5.5. The functions Ω0, . . . , Ωd−1 do not have a common zero in C − N−.
+Proof. Otherwise, if s0 ∈ C − N− is a common zero, then s0 + 1 ∈ C − N∗
+− and the
+functional equation shows that the non-zero vector (1, αd−1, . . . , α1) is in the kernel
+of the matrix [Ωkl(s0 + 1)], which contradicts that it has non-zero determinant by
+Theorem 5.1.
+□
+Observe that this simultaneously non-vanishing result relies on the fact that Euler
+Gamma function has no zeros. This is something that was explained to be a “mini-
+Riemann hypothesis” in [11], and was the subject of correspondence between Hermite
+and Stieltjes [9]. Although used in the proof, the non-vanishing of Euler Gamma
+function is a particular case of this general result for Omega functions.
+Corollary 5.6. The functions Ω0, . . . , Ωd−1 are C-linearly independent.
+Proof. Otherwise there will be a non-trivial null linear combination of the rows of the
+matrix [Ωkl] and the determinant will be zero.
+□
+6. Solutions of the functional equation.
+Observe that the functional equation (1) reduces to the functional equation (2) by
+dividing the equation by αd that is assumed to be non-zero. We can make a first
+observation that the space of solutions of the functional equation (2) is an infinite
+dimensional vector space.
+Proposition 6.1. The space of meromorphic solutions f of the functional equation
+(4)
+f(s + d) + αd−1f(s + d − 1) + . . . + α1f(s + 1) = s f(s)
+is an infinite dimensional vector space.
+
+16
+R. P´EREZ-MARCO
+Proof. The functional equation is linear and there are non-zero solutions (the Ωk
+functions). Given a non-zero meromorphic solution f(s), we can construct an infinite
+number of linear independent solutions
+g(s) = e2πinsf(s)
+where n ∈ Z is any integer.
+□
+If we restrict to solutions with a controlled growth, the situation the space of
+solutions is finite dimensional.
+Definition 6.2. We consider the C-vector space V of meromorphic functions Ω sat-
+isfying the functional equation (2) and the estimate in the vertical strip S1,d = {1 ≤
+Re s ≤ d}, for s ∈ S1,d,
+(5)
+|Ω(s)| ≤ Ce−c Im s
+for some constant 0 ≤ c < 2π.
+It is clear that the space V is a subspace of the vector space of general solutions
+(without a prescribed growth condition).
+We prove first that V is non-empty by
+proving the estimates for the functions Ωk for k = 0, 1, . . . d − 1.
+Proposition 6.3. For k = 0, 1, . . . , d − 1, for any strip Sa,b = {a ≤ Re s ≤ b} with
+0 < a < b, there exists a constant C = C(a, b, P0) > 0, depending only on a, b > 0
+and the polynomial P0, such that for s ∈ Sa,b, we have
+|Ωk(s)| ≤ Ce− 2πk
+d
+Im s
+Obviously we can take a = 1 and b = d and since 0 ≤ c = 2πk
+d < 2π we get that Ωk
+satisfies the estimate (5).
+Proof. We make the change of variables t = ωku
+Ωk(s) =
+� +∞.ωk
+0
+ts−1eP0(t) dt
+= ωs
+k
+� +∞
+0
+us−1e− 1
+d ud(1+O(u−1)) du
+= e2πi k
+d s
+� +∞
+0
+us−1e− 1
+d ud(1+O(u−1)) du
+
+SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS
+17
+so, we get for 0 < a ≤ Re s ≤ b
+|Ωk(s)| ≤ e−2π k
+d Im s(1 + C1)
+� +∞
+0
+uRe s−1e− 1
+d ud du
+≤ e−2π k
+d Im s(1 + C1)dRe s/dΓ
+�Re s
+d
+�
+≤ Ce− 2πk
+d
+Im s
+where C, C1 > 0 are constants depending only on a, b > 0 and P0.
+□
+The growth condition on the strip S(1, d) implies an exponential type estimate on
+the halph plane {Re s ≥ 1}. More precisely, we have
+Proposition 6.4. Let f ∈ V. Then f is of exponential type, i.e. there exists constants
+C, τ > 0 such that for Re s ≥ 1
+|f(s)| ≤ Ceτ|s|
+Proof. The estimate holds in the strip S(1, b) because of estimate (5) and Im s ≤ |s|.
+In fact we have in the strip S(1, b)
+|f(s)| ≤ Cec|s|
+If we define
+M(f, y, x) =
+max
+1≤Re s≤x
+Im s=y
+|f(s)|
+then for s ∈ S(1, b) we have M(f, y, x) ≤ Cec|s|.
+Let A = maxl |αl|. Let s ∈ C with Re s ≥ d + 1 and Im s = y. The functional
+equation shows that
+|f(s)| ≤
+A
+|s − d| max
+1≤k≤d |f(s − k)| ≤ AM(f, y, x − 1)
+Consider now τ0 > max(log A, c) and suppose that M(f, y, x − 1) ≤ Ceτ0(x−1). Then
+we get
+M(f, y, x) ≤ AM(f, y, x − 1) ≤ eτ0Ceτ0(x−1) = Ceτ0x
+For any s ∈ C with Re s ≥ 1 we have some integer 0 ≤ n ≤ [Re s] ≤ Re s such that
+x − n ∈ S(1, d), and by repeating n times the previous estimate we get
+|f(s)| ≤ Ceτ0(|s−n|+Re s) ≤ Ceτ0(|s|+2 Re s) ≤ Ceτ|s|
+with τ = 3τ0 and this proves the estimate.
+□
+Now we prove the main Theorem:
+
+18
+R. P´EREZ-MARCO
+Theorem 6.5. The space of solutions V is a finite dimensional vector space generated
+by the basis (Ωk)0≤k≤d−1.
+We recall Carlson’s Theorem [5]:
+Theorem 6.6 (Carlson, 1914). Let C+ = {s ∈ C; Re s > 0} and f : C+ → C
+be a holomorphic function extending continuously to C+. We assume that f is of
+exponential type, that is, there is C, τ > 0 such that for all s ∈ C+,
+|f(s)| ≤ Ceτ|s|
+We assume that on the imaginary axes we have a more precise control, for y ∈ R,
+|f(iy)| ≤ Cec|y|
+for some constant c < π.
+If f(n) = 0 for all n ∈ N, then f is identically 0.
+We use Carlson’s Theorem in the half plane {Re s > 1} to prove the main Theorem.
+Proof. We consider a meromorphic solution f(s) of the functional equation and satis-
+fying the estimate (5). The matrix [Ωkl(1)] being invertible, we have a linear combi-
+nation g(s) = c0Ω0(s) + . . . + cd−1Ωd−1(s) with c0, . . . , cd−1 ∈ C such that g(l) = f(l)
+for l = 1, 2, . . . , d. Since g satisfies also the functional equation, we get by induction
+using the functional equation that f and g take the same values at all the positive
+integers s ∈ N∗. So the function f − g vanish at all integers and satisfies the estimate
+(5). Therefore, the function h(s) = e−iπs(f(s) − g(s)) satisfies on Re s = 1,
+|h(s)| ≤ Ce−(c−π) Im s
+with 0 ≤ c < 2π. Therefore we have on Re s = 1
+|h(s)| ≤ Cec′| Im s|
+with 0 ≤ c′ < π.
+Also by Proposition 6.4 the function f has exponential growth in the right half
+plane {Re s ≥ 1}, as well as the function h. Therefore using Carlson’s Theorem we
+conclude that h is identically 0, thus f(s) = g(s) for all values s in this half plane,
+hence in C.
+□
+We have proved that the vector space generated by Omega functions can be charac-
+terized by the functional equation (2) and the growth property (5). This generalizes
+to Omega functions Wielandt’s characterization for Euler Gamma function (1939,
+[15], [12], [13]). We can characterize individually each Omega function Ωk by their
+asymptotic growth when s → +∞ analogue to Stirling asymptotic of Euler Gamma
+function.
+
+SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS
+19
+We also observe that Omega functions provide the general solutions of the func-
+tional equation (2) with estimates (5) since given such a functional equations with
+coefficients (αl) we can build the coefficients al = −l−1αl, then the polynomial P0
+and the Omega functions (Ωk)0≤k≤d−1 that form a basis for the space of solutions.
+It is also easy to see that we can replace the (5) by an estimate of the form, for
+s ∈ S(1, b),
+|f(s)| ≤ Ce−c Im s
+with 2πn ≤ c < 2π(n + 1) for an integer n ∈ Z. Then the space of solutions is also
+finite dimensional as the map f(s) �→ e−2πinsf(s) provides an isomorphism of the
+space of solutions with V.
+The structure of the space of solutions is interesting. The space of holomrphic
+solutions is a subspace of dimension d − 1.
+Proposition 6.7. The subspace of holomorphic solutions in V is a subspace of di-
+mension d − 1 generated by the entire functions
+Ωl(s) − Ω0(s) =
+�
+γ0l
+ts−1eP0(t) dt
+Proof. As observed before, the functions Ωl(s) − Ω0(s) are entire functions and are
+linearly independent.
+□
+Some of the results in [4] can be generalized. In particular the Integrability criterion
+and Abel-like Theorem (Theorems 4.2 and 4.3). This will be studied in a separate
+article. K. Biswas has extended those results from [4] to curves of higher genus [2].
+It is interesting to speculate on the extension of the results for Omega functions in
+higher genus.
+References
+[1] AYCOCK, A.; Euler and the multiplication formula for the Γ-function, arXiv:1901.03400, 2019.
+[2] BISWAS,
+K.;
+Algebraic
+de
+Rham
+cohomology
+of
+log-Riemann
+surfaces
+of
+finite
+type,
+arXiv:1602.08219, 2015.
+[3] BISWAS, K.; P´EREZ-MARCO, R.; Log-Riemann surfaces, arXiv:1512.03776, 2015.
+[4] BISWAS, K.; P´EREZ-MARCO, R.; The Ramificant Determinant, Symmetry, Integrability and
+Geometry: Methods and Applications (SIGMA), 15, 086, arXiv:1903.0677, 2019.
+[5] CARLSON, F.; Sur une classe de s´eries de Taylor, Thesis, Uppsala, 1914.
+[6] EULER, L.; Letter to Goldbach, 8 January 1730, Euler Archive [E00717], eulerarchive.maa.org,
+1730.
+[7] FUGLEDE, B.; A sharpening of Wielandt’s characterization of the Gamma function, The Amer-
+ican Math. Monthly, 115, 9, p.845-850, 2008.
+
+20
+R. P´EREZ-MARCO
+[8] HANKEL, H. ; Die Euler’schen Integrale bei unbeschr¨ankter Variabilit¨at des Argumentes,
+Zeitschr. Math. Phys., 9, p.1-21, 1864.
+[9] HERMITE, Ch.; STIELTJES, T.J. ; Correspondance d’Hermite et Stieltjes, publi´ee par Baillaud
+et Bouguet, Gauthier-Villars, Paris, 1905.
+[10] KONSEVICH, M.; ZAGIER, D. ; Periods, Mathematics unlimited-2001 and beyond, Springer,
+p.771-808,, 2001.
+[11] P´EREZ-MARCO,
+R.
+On
+the
+definition
+of
+Euler
+Gamma
+function,
+L’Enseignement
+Math´ematique, 68, 1/2, p.135-160, 2022.
+[12] REMMERT, R. Wielandt’s theorem about the Γ-function, The Amer. Math. Monthly, 103,3,
+1996.
+[13] REMMERT, R. Classical topics in complex function theory, Graduate Texts in Mathematics,
+172, Springer, 1998.
+[14] WEIERSTRASS, K.; ¨Uber die Theorie der analytischen Fakult¨aten, Journal f¨ur Mathematik,
+51, p.1-60, 1856.
+[15] WIELANDT, H.; , Mathematische Werke, 2, 2, De Gruyter, New York, published 1996. See
+also [12].
+[16] WHITTAKER, E.T.; WATSON, G.N.; A course in modern analysis, Cambridge Univ. Press,
+4th edition, 1927.
+CNRS, IMJ-PRG, Universit´e Paris Cit´e, Paris, France
+Email address: ricardo.perez.marco@gmail.com
+
diff --git a/etE3T4oBgHgl3EQf3Asg/content/tmp_files/load_file.txt b/etE3T4oBgHgl3EQf3Asg/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..aef7330929347d956df4abd07f944b041c6bcddf
--- /dev/null
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@@ -0,0 +1,688 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf,len=687
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='04759v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='CV]  11 Jan 2023  SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS  RICARDO P´EREZ-MARCO  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We study shift functional equations that generalize the functional equa-  tion satisfied by Euler Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Under a natural growth condition, the  vector space of meromorphic solutions is finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We construct a basis of  the space of solutions composed by Omega functions that generalize Euler Gamma  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Omega functions are defined as exponential periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' They have a mero-  morphic extension to the complex plane with simple poles at negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  They are characterized by their growth property on vertical strips and their shift  functional equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' This generalizes Wielandt’s characterization of Euler Gamma  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Introduction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Shift functional equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The goal of this article is to study and solve shift  functional equations of the form  (1)  sf(s) =  d  �  k=1  αkf(s + k)  where α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , αd ∈ C and αd ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The simplest case of this functional equation is  the functional equation satisfied by Euler Gamma function  sΓ(s) = Γ(s + 1)  These equations are linear and we have a vector space of meromorphic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  A natural motivation for studying these functional equations comes from the study  of subspaces generated by natural linear operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' For instance, we can consider, in  the space of meromorphic functions, the shift (or integer translation) linear operator  T(f(s)) = f(s + 1)  and the multiplication by s linear operator  S(f(s)) = sf(s)  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Primary: 30D10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Secondary: 30D15, 30B50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Euler Gamma function, exponential periods, Omega functions, shift  functional equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  1  2  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' P´EREZ-MARCO  Observe that S has no eigenvectors and the minimal invariant subspace invariant  by S containing the constant functions is the space of polynomials C[s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The space  generated by S and the function f(s) is the vector space C[s]f(s)  ⟨f, S(f), S2(f), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='⟩ = C[s]f(s)  It is natural to ask for which functions f the space C[s]f(s) is also generated by  f(s) and the shift operator T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' This happens if and only if f is solution of the shift  functional equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Already, in the simplest case of the functional equation of Euler Gamma function,  the space of solutions is infinite dimensional since any function of the form e2πinsΓ(s)  for an integer n ∈ Z is also a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' It is classical to add conditions to characterize  Euler Gamma function as the only normalized solution to this functional equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  One can mention Weierstrass characterization imposing some asymptotic behavior  when s → +∞ (1856, [14]), or Wielandt’s characterization (1939, [15], see also [12],  [13]) requiring boundedness on vertical strips of width larger than 1, or, more recently,  requiring finite order of the solutions and a right half plane free of zeros nor poles  (2022, [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Wielandt’s boundedness condition has been weakened by Fuglede to a  moderate growth in the vertical strip (2008, [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  In the spirit of Wielandt, we search for solutions with some growth control on  vertical strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Under a suitable growth condition, we can prove that the space of  solutions is finite dimensional:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The space of meromorphic solutions f of the functional equation  sf(s) =  d  �  k=1  αkf(s + k)  where α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , αn ∈ C, αd ̸= 0, and f satisfies a growth condition, for 1 ≤ Re s ≤ d,  |f(s)| ≤ Ce−c Im s  for some constant C > 0 and 0 ≤ c < 2π, is finite dimensional of dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Moreover, we build an explicit basis of the vector space of solutions with a new1  kind of Special Functions, that we call Omega functions, that generalize Euler Gamma  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  1The author couldn’t find any reference to Omega functions in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Omega functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Historically, Euler Gamma function appears for the first  time in a letter from Euler to Goldbach, dated January 8th 1730 ([6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Euler defines  the Gamma function for real values s > 0, by the integral formula  Γ(s) =  � +∞  0  ts−1e−tdt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  which is also convergent for complex values of s with Re s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  In this integral  formula, the value Γ(s) appears as an exponential period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Algebraic periods are integrals of algebraic differential forms over cycles of an alge-  braic variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' In the special case of an algebraic curve, when we represent the curve as  a Riemann domain over the complex plane or the Riemann sphere, algebraic periods  are also the integrals of algebraic differential forms on paths joining two ramification  points where we have singularities of the differential form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' From the transalgebraic  point of view, it is natural to consider exponential periods, where integrals involve  exponential expressions, and the singularities can be exponential singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' More  general periods can be envisioned where the differential form has transcendental sin-  gularities with monodromy as ts in a local variable (geometrically these corespond  to differential forms living in a branched Riemann domain with an infinite ramifica-  tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' There is a vast literature on classical algebraic periods, but almost none on the  transalgebraic periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We refer to [10] for definitions and a review about classical  periods, and to [3] and [4] for exponential periods and their relation with log-Riemann  surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Also we refer the reader to [11] for an historical survey of different defini-  tions of Euler Gamma function and their generalizations, and to [16] for its classical  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  The second goal of this article is to introduce Omega functions (or Ω-functions)  which are a natural generalization of the Euler Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' They are defined  as exponential periods of the form  Ωk(s) =  � +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk  0  ts−1eP0(t) dt  where P0(t) ∈ C[t] and ωk is a root of unity pointing to a direction where the poly-  nomial P0 diverges to −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Some critical computations in the proof of the main Theorem are generalizations  of computations carried out for exponential periods appearing in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' In particular,  the generalization of the Ramificant Determinant considered in that article is the key  result for the proof of the linear independence of the Omega functions (Ωk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' In its  magical computation, we calculate explicitly a determinant of a matrix of exponential  periods which are individually not computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  4  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' P´EREZ-MARCO  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Let P0(t) ∈ C[t] be a degree d ≥ 1 polynomial such that P0(0) = 0 and limt→+∞ Re P0(t) =  −∞, normalized by  P0(t) = −1  dtd +  d−1  �  k=1  aktk  We also denote ad = −1/d and a0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Let ω be the primitive d-th root of unity  given by  ω = e  2πi  d  and write ωk = ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Let d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' For k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , d − 1, the Omega functions, or Ω-  functions, associated to P0, are defined for Re s > 0, by  Ωk(s) =  � +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk  0  ts−1eP0(t) dt  The roots ωk point to the directions where the polynomial P0 diverges exponentially  to −∞,  lim  t→+∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk Re P0(t) = −∞  so the integral is converging and we have a sound definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Usually, we spare the  reference to P0, but for some results it will be crucial to keep track of the dependence  on parameters and we will write  Ω(s) = Ω(s|P0) = Ω(s|a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , ad−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  For d = 1, we have P0(t) = −t and Ω1 = Γ is Euler Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  If P0 ∈ R[t], then Ω0 is real analytic, and Ωk(¯s|P0) = Ωd−k(s| ¯P0), where ¯P0 is the  conjugate polynomial  ¯P0(t) = −1  dtd +  d−1  �  k=1  ¯aktk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Sometimes we will be interested in the case where αk ∈ K where K ⊂ C is a number  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' In that case we say that the Omega functions are defined over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Meromorphic extension, poles and residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The Omega functions (Ωk)0≤k≤d−1 extend to the complex plane into  meromorphic functions of order 1 satisfying the fundamental functional equation  (2)  Ωk(s + d) + αd−1Ωk(s + d − 1) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' + α1Ωk(s + 1) = s Ωk(s)  where αl = −lal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Moreover, the function Ωk is holomorphic in C − N, and has simple poles at the  negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The residue at s = 0 is  Ress=0 Ωk = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Observe that for d = 1, Ω0 = Γ and the functional equation (2) is the classical  functional equation Γ(s + 1) = sΓ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We have for Re s > 0 and by integration by parts,  Ωk(s + d) +  d−1  �  l=1  αlΩk(s + l) =  � +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk  0  ts(−P ′  0(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='eP0(t) dt  =  �  −tseP0(t)�+∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk  0  + s  � +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk  0  ts−1eP0(t) dt  = s Ωk(s)  and we get the functional equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Now, using once the functional equation we  extend meromorphically Ωk to {Re s > −1}, and by induction to {Re s > −n}, for  n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=', hence to all of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The only poles that can be introduced by this extension  procedure using the functional equation are those created from the pole at s = 0 at  the negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The functional equation shows that sΩk(s) is holomorphic at  s = 0, hence the pole at s = 0 is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' It follows from the functional equation and  the extension procedure that the other poles are also simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We compute the residue  at s = 0 using the functional equation,  Ress=0 Ωk = lim  s→0 sΩk(s) =  d  �  l=1  αlΩk(l) =  � +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk  0  (−P ′  0(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='eP0(t) dt =  �  −eP0(t)�+∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk  0  = 1  □  More generally, we can compute the residues at the negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Let (λn)n≥0 be the coefficients of the power series expansion of eP0(t),  eP0(t) =  +∞  �  n=0  λntn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  6  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' P´EREZ-MARCO  Then the residue of Ωk at s = −n is λn,  Ress=−n Ωk = λn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' For n ≥ 0 let rn ∈ C be the residue of Ωk at s = −n, with rn = 0 if there is  no pole, and rn = 0 for n < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The functional equation (2) gives  rn = lim  h→0 hΩk(h − n) = lim  h→0  1  h − n  d  �  l=1  αlhΩk(h − n + l) = −1  n  d  �  l=1  αl rn−l  hence the recurrence relation  (3)  nrn = −  d  �  l=1  αl rn−l  Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' consider the generating power series  F(t) =  +∞  �  n=0  rntn  The recurrence relation (3) gives  F ′(t) =  +∞  �  n=0  nrntn−1 = −  d  �  l=1  αl  +∞  �  n=0  rn−ltn−1  = −  d  �  l=1  αltl−1  +∞  �  n=l  rn−ltn−l  = −  �  d  �  l=1  αltl−1  �  F(t)  = P ′  0(t)F(t)  Since we have F(0) = r0 = 1 from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='1, we get  F(t) = eP0(t)  thus rn = λn as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  □  Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  For d = 1, the generating power series is  F(t) = e−t =  +∞  �  n=0  (−1)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  tn  SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS  7  and we recover the classical result that  Ress=−n Γ = (−1)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Note that the residues at the simple poles at the negative integers are the same for  all the functions Ω0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , Ωd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Indeed, for k ̸= l, we can check directly that Ωk − Ωl  is an entire function because of the convergence for all s ∈ C of the integral  Ωk(s) − Ωl(s) =  � +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk  +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωl  ts−1eP0(t) dt =  �  γlk  ts−1eP0(t) dt  where the integral can be taken over any path γlk assymptotic to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωl and +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk  in the proper direction and with 0 winding number around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' This integral depends  holomorphically on the parameter s ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Observe also that if the coefficients of P0 belong to a number field K, P0(t) ∈ K[t],  then the residues of Ωk belong also to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Another arithmetical observation is the  following:  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We assume that the only non-zero coefficients of P0 are for powers  divisible by an integer n0 ≥ 2, that is, if ak ̸= 0 then n0|k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Then, if n0 does not divide n, we have rn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' From the previous Theorem we have  eP0(t) =  d  �  k=1  eaktk =  d  �  k=1  ��  m≥0  am  k  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='tmk  �  and when we expand the last product we get the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  □  We have a more precise result than just the computation of the residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  We  can determine the Mittag-Leffler decomposition of Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='This is an analytic result that  requires some estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The Omega function Ωk has the Mittag-Leffler decomposition:  Ωk(s) =  +∞  �  n=0  λn  z + n +  � +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk  1  ts−1eP0(t) dt  where the integral is an entire function of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Observe that this Theorem shows that the Omega function Ωk is a meromorphic  function of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The Omega functions Ωk are meromorphic functions of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  8  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' P´EREZ-MARCO  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We write  Ωk(s) =  � 1  0  ts−1eP0(t) dt +  � +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk  1  ts−1eP0(t) dt  and we compute the first integral expanding the exponential in power series (uniformly  convergent in [0, 1]),  � 1  0  ts−1eP0(t) dt =  +∞  �  n=0  λn  � 1  0  ts+n−1 dt =  +∞  �  n=0  λn  � ts+n  s + n  �1  0  =  +∞  �  n=0  λn  s + n  The second integral can be bounded by the next Lemma that shows that it is an  entire function of order 1 (using that Euler Gamma function is of order 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We have the estimate  ����  � +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk  1  ts−1eP0(t) dt  ���� ≤ e−2π k  d Im s  �  C0 + C1dRe s/dΓ  �Re s  d  ��  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We make the change of variables t = ωku  � +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk  1  ts−1eP0(t) dt = ωs  k  � +∞  ω−1  k  us−1e− 1  d ud(1+O(u−1)) du  = e2πi k  d s  � +∞  ω−1  k  us−1e− 1  d ud(1+O(u−1)) du  This gives the bound  ����  � +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk  1  ts−1eP0(t) dt  ���� ≤ e−2π k  d Im s  �����  � +∞  ω−1  k  us−1e− 1  d ud(1+O(u−1)) du  �����  Now, taking an integration path of finite length from 1 to ωk and bounded away from  0, we get (using the same letter C to denote several universal constants C > 0)  ����  � +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk  1  ts−1eP0(t) dt  ���� ≤ e−2π k  d Im s  �  C +  ����  � +∞  1  us−1e− 1  d ud(1+O(u−1)) du  ����  �  The last integral can be estimated by  ����  � +∞  1  us−1e− 1  d ud(1+O(u−1)) du  ���� ≤ (1 + C)  � +∞  1  us−1e− 1  d ud du  ≤ (1 + C)  � +∞  0  us−1e− 1  d ud du  ≤ (1 + C)dRe s/dΓ  �Re s  d  �  (for the computation of the last integral we use the change of variable v = ud/d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  □  SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS  9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Incomplete Omega functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  We define the Incomplete Omega functions that generalize the Incomplete Gamma  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' For z, s ∈ C, Re s > 0, the Incomplete Omega function Ω(s, z) is  defined by  Ω(s, z) =  � z  0  ts−1eP0(t) dt  For P0(t) = −t this is the classical Incomplete Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Observe that we  recover all the (Ωk) functions by taking the appropriate limit of Ω(s, z) as z → ∞,  Ωk(s) =  lim  z→+∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk Ω(s, z)  For the particular values of s = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , d − 1 these are the transcendental entire  functions of the variable z ∈ C studied in [4] which form a basis of the fundamental  for the transcendental vector space of functions on a simply connected log-Riemann  surface with exactly d infinite ramification points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Some of the results proved here  generalize some results from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Following the same Abel’s philosophy that inspires  [4], we prove that we only need to use a finite number of transcendentals (Ω(s +  k, z))0≤k≤d−1 to compute integrals of the form  � z  0  Q(t, ts)eP0(t) dt  where Q(x, y) ∈ C[x, y] is a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We start by studying the simpler case when  Q(t) ∈ C[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Let Q(t) ∈ C[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' For d ≥ 2, the integral  � z  0  tsQ(t)eP0(t) dt  is of the form  � z  0  tsQ(t)eP0(t) dt = zsA(s, z) eP0(z) +  d−1  �  k=0  ck(s) Ω(s + k, z)  where A ∈ C[s, z], and the polynomial coefficients ck(s) ∈ C[s] have coefficients de-  pending polynomially on the coefficients of P0, (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , ad−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  10  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' P´EREZ-MARCO  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' First we consider the case d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We prove the result for Q(t) = tn integrating  by parts n + 1 times  � z  0  ts+ne−t dt =  �  −ts+ne−t�z  0 + (s + n − 1)  � z  0  ts+n−1e−t dt  = −zs+ne−z + (s + n)  � z  0  ts+n−1e−t dt  = −(zs+n + (s + n)zs+n−1)e−z + (s + n)(s + n − 1)  � z  0  ts+n−2e−t dt  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  = −zs(zn + (s + n)zn−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' )e−z + (s + n)(s + n − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' s Ω(s, z)  For a general polynomial Q(t) we have the the result by linear decomposition of the  integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  In the rest of the proof we assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' If q = deg Q ≤ d − 2, by linearity, the  integral is a linear combination (with the coefficients of tQ(t)) of (Ω(s + k, z))0≤k≤d−1  and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  If deg Q ≥ d − 1, then we consider the Euclidean division of Q(t) by P ′  0(t),  Q(t) = A1(t)P ′  0(t) + B1(t)  with A1, B1 ∈ C[t], deg B1 ≤ d−2 and deg A1 = deg Q−(d−1) = q −(d−1) ≤ q −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  We proceed splitting the integral:  � z  0  tsQ(t)eP0(t) dt =  � z  0  tsA1(t)P ′  0(t)eP0(t) dt +  � z  0  tsB1(t)eP0(t) dt  Since deg B1 ≤ d − 2, the second integral is a linear combination of Ω(s, z), Ω(s +  1, z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , Ω(s + d − 1, z), thus of the desired form, and we can forget about it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We  work on the first integral integrating by parts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  � z  0  tsA1(t)P ′  0(t)eP0(t) dt =  �  tsA1(t)eP0(t)�z  0 −  � z  0  (tsA1(t))′ eP0(t) dt  Then we get:  � z  0  tsA1(t)P ′  0(t)eP0(t) dt = zsA1(z)eP0(z) −  � z  0  tsA′  1(t)eP0(t) dt − s  � z  0  ts−1A1(t)eP0(t) dt  The first integral in the right hand side is of the same form as the initial one with  Q(t) but with deg A′  1 = deg A1 − 1 ≤ deg Q − (d − 1) − 1 = q − d ≤ q − 2 (using  here d ≥ 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' hence by descending induction we can forget about it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' For the second  integral, we can write  s  � z  0  ts−1A1(t)eP0(t) dt = A1(0)s Ω(s, z) + s  � z  0  ts  �A1(t) − A1(0)  t  �  eP0(t) dt  SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS  11  and t−1(A1(t) − A1(0)) is a polynomial of degree deg A1 − 1 ≤ q − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Then the  descending induction gives the expression for the integral as announced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  The coefficients of P0 appear first linearly in the Euclidean divisions by P ′  0 then,  by repeated Euclidean divisions the dependence of the ck(s) is polynomial on the  coefficients of P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  □  Now, we can get easily the general result for Q(t, ts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Let Q(t, ts) ∈ C[t, ts].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' For d ≥ 2, the integral  � z  0  Q(t, ts)eP0(t) dt  is of the form  � z  0  ts−1Q(t, ts)eP0(t) dt = A(s, z, zs)eP0(z) +  d−1  �  k=0  ck(s)Ω(s + k, z)  where A ∈ C[s, z, zz] and the coefficients ck(s) ∈ C[s] have coefficients that are poly-  nomials on the coefficients of P0 (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , ad−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We write Q(t, ts) = Q(t)(ts) as a polynomial on the variable ts with coeffi-  cients polynomials in t, and we split linearly the integral, observing that the part  corresponding to each monomial is like the integral when Q is a polynomial in t as  before, but with s shifted into s + l by some positive integer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The result follows  from the previous Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  □  Now, we can prove that the Omega functions (Ωk(s))0≤k≤d−1 generate a large class  of exponential periods:  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Let Q(t, ts) ∈ C[t, ts] and 0 ≤ n ≤ d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The exponential period  � +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωn  0  Q(t, ts) eP0(t) dt  is a linear combination of the exponential periods (Ωk(s))0≤k≤d−1  � +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωn  0  Q(t, ts) eP0(t) dt =  d−1  �  k=0  ck(s) Ωk(s)  where the coefficients ck(s) are polynomials on s and on the coefficients (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , ad−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' For d ≥ 2, we have from the previous Proposition that  � z  0  Q(t, ts)eP0(t) dt = A(s, z, zs)eP0(z) +  d−1  �  k=0  ck(s)Ω(s + k, z)  12  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' P´EREZ-MARCO  When z → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωn the terms in the first sum vanish, since A(s, z, zs)eP0(z) → 0  for a polynomial A(s, z, zs) ∈ C[z] (the exponential decay of eP0(z) takes over the  polynomial divergence of A(z)), and we get  � +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωn  0  Q(t, ts)eP0(t) dt =  d−1  �  k=0  ck(s)Ωn(s + k)  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Linear independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  The row vector build with Omega functions Ω(s) = (Ωk(s))0≤k≤d−1 has the follow-  ing important linear independence property:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' For any s ∈ C − N∗  −, the vectors Ω(s + 1), Ω(s + 2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , Ω(s + d) are  linearly independent,  ∆(s) ̸= 0  where  ∆(s|a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , ad−1) = det  \uf8ee  \uf8ef\uf8ef\uf8f0  Ω11  Ω12  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Ω1d  Ω21  Ω22  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Ω2d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Ωd1  Ωd2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Ωdd  \uf8f9  \uf8fa\uf8fa\uf8fb  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  where Ωkl = Ωk−1(s + l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  More precisely, we can compute  ∆(s|a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , ad−1) = ∆(s|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , 0) exp (Πd(s, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , ad−1))  where Πd(s, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , ad−1) is a universal polynomial with rational coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  In view of the last formula, the result follows from ∆(s|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , 0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We will  prove the last formula and compute explicitly the determinant ∆(s|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' These  computations are similar to the ones for the Ramificant Determinant (see [4]) that  corresponds to the special case s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  We can compute Ωk(s + l|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , 0) using Euler Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We have  Ωk(s + l|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , 0) = ωk(s+l)d  s+l  d −1Γ  �s + l  d  �  SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS  13  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We first make the change of variables t = ωku, and then v = ud/d,  Ωk(s + l|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , 0) =  � +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk  0  ts+l−1e− 1  d td dt  = ωk(s+l)  � +∞  0  us+l−1e− 1  d ud du  = ωk(s+l)d  s+l  d −1  � +∞  0  v  s+l  d −1e−v dv  = ωk(s+l)d  s+l  d −1Γ  �s + l  d  �  □  Now we recall the following well known elementary Vandermonde Lemma:  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' If ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , ξd are the d roots of a monic polynomial Q(X), then we can  compute the Vandermonde determinant V (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , ξd) of the (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , ξd) as  V (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , ξd) =  ���������  1  ξ1  ξ2  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  ξd−1  1  1  ξ2  ξ2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  ξd−1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  1  ξd  ξ2  d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  ξd−1  d  ���������  =  �  i̸=j  (ξi − ξj) =  d  �  i=1  Q′(ξi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Using this Lemma with Q(X) = Xd−1 we compute the Vandermonde determinant:  Vd =  ���������  1  ω1  ω2  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  ωd−1  1  1  ω2  ω2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  ωd−1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  1  ωd  ω2  d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  ωd−1  d  ���������  =  �  i̸=j  (ωi−ωj) =  �  i  (dωd−1  i  ) = dd  ��  i  ωi  �d−1  = (−1)d−1dd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  We use this result to compute ∆(s|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We have  ∆(s|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , 0) = (2πd)  d  2  √  2π ω  d(d−1)  2  ss Γ(s)  and in particular ∆(s|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , 0) ̸= 0 for s ̸= −1, −2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='.  Taking the limit s → 0 we recover the formula from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='5 from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  14  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' P´EREZ-MARCO  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='2 we have  ∆(s|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , 0) =  =  ���������  ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' (s+1)d  s+1  d −1Γ  �s+1  d  �  ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' (s+2)d  s+2  d −1Γ  � s+2  d  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' (s+d)d  s+d  d −1Γ  � s+d  d  �  ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' (s+1)d  s+1  d −1Γ  �s+1  d  �  ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' (s+2)d  s+2  d −1Γ  �s+d  d  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' (s+d)d  s+d  d −1Γ  � s+d  d  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  ω(d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' (s+1)d  s+1  d −1Γ  �s+1  d  �  ω(d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' (s+2)d  s+2  d −1Γ  � s+2  d  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  ω(d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' (s+d)d  s+d  d −1Γ  � s+d  d  �  ���������  = ω  d(d−1)  2  sdsd  d−1  2 −dΓ  �s  d + 1  d  �  Γ  �s  d + 2  d  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Γ  �s  d + d  d  �  ��������  ω1  0  ω2  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  ωd  0  ω1  1  ω2  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  ωd  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  ω1  d−1  ω2  d−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  ωd  d−1  ��������  = ω  d(d−1)  2  sd− d  2s(2π)  d−1  2 Γ(s)dd  = d  d  2 (2π)  d−1  2 sω  d(d−1)  2  s Γ(s)  = (2πd)  d  2  √  2π ω  d(d−1)  2  ss Γ(s)  where we have used Gauss multiplication formula in the second line (that is in fact  due to Euler and not to Gauss, see [1]) with z = s/d,  Γ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='Γ  �  z + 1  d  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Γ  �  z + d − 1  d  �  = (2π)  d−1  2 d  1  2−dzΓ(dz) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  and that the determinant in the fourth line is equal to (−1)d−1Vd where Vd is the  Vandermonde determinant computed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  □  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Consider the entire function of several complex variables  ∆(s|a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , ad−1) on the variables (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , ad−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Observe that Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='4  proves that each integral  � +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk  0  ts+n−1eP0(t) dt ,  is a linear combination with coefficients that are polynomial on s and the (ak) of the  integrals Ωk(s) for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , d − 1, Therefore, differentiating column by column,  we observe that for each k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , d − 1, we have  ∂ak∆ = Qk ∆  SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS  15  where Qk is a polynomial on s and the (ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  We conclude that the logarithmic  derivative of ∆ with respect to each variable ak is a universal polynomial on the  variables s and (ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' This gives the existence of the universal polynomial Υd such  that  ∆(s|a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , ad−1) = c(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='eΥd(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='a1,a2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=',ad−1) ,  with c(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='eΥd(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='0,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=',0) = ∆(s|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , 0) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Then if we define Πd(s, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , ad−1) =  Υd(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , ad−1) − Υd(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , 0) we get the result  ∆(s|a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , ad−1) = ∆(s|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , 0)eΠd(s,a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=',ad−1)  □  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The functions Ω0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , Ωd−1 do not have a common zero in C − N−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Otherwise, if s0 ∈ C − N− is a common zero, then s0 + 1 ∈ C − N∗  − and the  functional equation shows that the non-zero vector (1, αd−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , α1) is in the kernel  of the matrix [Ωkl(s0 + 1)], which contradicts that it has non-zero determinant by  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  □  Observe that this simultaneously non-vanishing result relies on the fact that Euler  Gamma function has no zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' This is something that was explained to be a “mini-  Riemann hypothesis” in [11], and was the subject of correspondence between Hermite  and Stieltjes [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Although used in the proof, the non-vanishing of Euler Gamma  function is a particular case of this general result for Omega functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The functions Ω0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , Ωd−1 are C-linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Otherwise there will be a non-trivial null linear combination of the rows of the  matrix [Ωkl] and the determinant will be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Solutions of the functional equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Observe that the functional equation (1) reduces to the functional equation (2) by  dividing the equation by αd that is assumed to be non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We can make a first  observation that the space of solutions of the functional equation (2) is an infinite  dimensional vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The space of meromorphic solutions f of the functional equation  (4)  f(s + d) + αd−1f(s + d − 1) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' + α1f(s + 1) = s f(s)  is an infinite dimensional vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  16  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' P´EREZ-MARCO  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The functional equation is linear and there are non-zero solutions (the Ωk  functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Given a non-zero meromorphic solution f(s), we can construct an infinite  number of linear independent solutions  g(s) = e2πinsf(s)  where n ∈ Z is any integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  □  If we restrict to solutions with a controlled growth, the situation the space of  solutions is finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We consider the C-vector space V of meromorphic functions Ω sat-  isfying the functional equation (2) and the estimate in the vertical strip S1,d = {1 ≤  Re s ≤ d}, for s ∈ S1,d,  (5)  |Ω(s)| ≤ Ce−c Im s  for some constant 0 ≤ c < 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  It is clear that the space V is a subspace of the vector space of general solutions  (without a prescribed growth condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  We prove first that V is non-empty by  proving the estimates for the functions Ωk for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' For k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , d − 1, for any strip Sa,b = {a ≤ Re s ≤ b} with  0 < a < b, there exists a constant C = C(a, b, P0) > 0, depending only on a, b > 0  and the polynomial P0, such that for s ∈ Sa,b, we have  |Ωk(s)| ≤ Ce− 2πk  d  Im s  Obviously we can take a = 1 and b = d and since 0 ≤ c = 2πk  d < 2π we get that Ωk  satisfies the estimate (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We make the change of variables t = ωku  Ωk(s) =  � +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='ωk  0  ts−1eP0(t) dt  = ωs  k  � +∞  0  us−1e− 1  d ud(1+O(u−1)) du  = e2πi k  d s  � +∞  0  us−1e− 1  d ud(1+O(u−1)) du  SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS  17  so, we get for 0 < a ≤ Re s ≤ b  |Ωk(s)| ≤ e−2π k  d Im s(1 + C1)  � +∞  0  uRe s−1e− 1  d ud du  ≤ e−2π k  d Im s(1 + C1)dRe s/dΓ  �Re s  d  �  ≤ Ce− 2πk  d  Im s  where C, C1 > 0 are constants depending only on a, b > 0 and P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  □  The growth condition on the strip S(1, d) implies an exponential type estimate on  the halph plane {Re s ≥ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' More precisely, we have  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Let f ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Then f is of exponential type, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' there exists constants  C, τ > 0 such that for Re s ≥ 1  |f(s)| ≤ Ceτ|s|  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The estimate holds in the strip S(1, b) because of estimate (5) and Im s ≤ |s|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  In fact we have in the strip S(1, b)  |f(s)| ≤ Cec|s|  If we define  M(f, y, x) =  max  1≤Re s≤x  Im s=y  |f(s)|  then for s ∈ S(1, b) we have M(f, y, x) ≤ Cec|s|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Let A = maxl |αl|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Let s ∈ C with Re s ≥ d + 1 and Im s = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The functional  equation shows that  |f(s)| ≤  A  |s − d| max  1≤k≤d |f(s − k)| ≤ AM(f, y, x − 1)  Consider now τ0 > max(log A, c) and suppose that M(f, y, x − 1) ≤ Ceτ0(x−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Then  we get  M(f, y, x) ≤ AM(f, y, x − 1) ≤ eτ0Ceτ0(x−1) = Ceτ0x  For any s ∈ C with Re s ≥ 1 we have some integer 0 ≤ n ≤ [Re s] ≤ Re s such that  x − n ∈ S(1, d), and by repeating n times the previous estimate we get  |f(s)| ≤ Ceτ0(|s−n|+Re s) ≤ Ceτ0(|s|+2 Re s) ≤ Ceτ|s|  with τ = 3τ0 and this proves the estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  □  Now we prove the main Theorem:  18  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' P´EREZ-MARCO  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The space of solutions V is a finite dimensional vector space generated  by the basis (Ωk)0≤k≤d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  We recall Carlson’s Theorem [5]:  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='6 (Carlson, 1914).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Let C+ = {s ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Re s > 0} and f : C+ → C  be a holomorphic function extending continuously to C+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We assume that f is of  exponential type, that is, there is C, τ > 0 such that for all s ∈ C+,  |f(s)| ≤ Ceτ|s|  We assume that on the imaginary axes we have a more precise control, for y ∈ R,  |f(iy)| ≤ Cec|y|  for some constant c < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  If f(n) = 0 for all n ∈ N, then f is identically 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  We use Carlson’s Theorem in the half plane {Re s > 1} to prove the main Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We consider a meromorphic solution f(s) of the functional equation and satis-  fying the estimate (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The matrix [Ωkl(1)] being invertible, we have a linear combi-  nation g(s) = c0Ω0(s) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' + cd−1Ωd−1(s) with c0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , cd−1 ∈ C such that g(l) = f(l)  for l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Since g satisfies also the functional equation, we get by induction  using the functional equation that f and g take the same values at all the positive  integers s ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' So the function f − g vanish at all integers and satisfies the estimate  (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Therefore, the function h(s) = e−iπs(f(s) − g(s)) satisfies on Re s = 1,  |h(s)| ≤ Ce−(c−π) Im s  with 0 ≤ c < 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Therefore we have on Re s = 1  |h(s)| ≤ Cec′| Im s|  with 0 ≤ c′ < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Also by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='4 the function f has exponential growth in the right half  plane {Re s ≥ 1}, as well as the function h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Therefore using Carlson’s Theorem we  conclude that h is identically 0, thus f(s) = g(s) for all values s in this half plane,  hence in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  □  We have proved that the vector space generated by Omega functions can be charac-  terized by the functional equation (2) and the growth property (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' This generalizes  to Omega functions Wielandt’s characterization for Euler Gamma function (1939,  [15], [12], [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' We can characterize individually each Omega function Ωk by their  asymptotic growth when s → +∞ analogue to Stirling asymptotic of Euler Gamma  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  SHIFT FUNCTIONAL EQUATIONS AND OMEGA FUNCTIONS  19  We also observe that Omega functions provide the general solutions of the func-  tional equation (2) with estimates (5) since given such a functional equations with  coefficients (αl) we can build the coefficients al = −l−1αl, then the polynomial P0  and the Omega functions (Ωk)0≤k≤d−1 that form a basis for the space of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  It is also easy to see that we can replace the (5) by an estimate of the form, for  s ∈ S(1, b),  |f(s)| ≤ Ce−c Im s  with 2πn ≤ c < 2π(n + 1) for an integer n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Then the space of solutions is also  finite dimensional as the map f(s) �→ e−2πinsf(s) provides an isomorphism of the  space of solutions with V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  The structure of the space of solutions is interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The space of holomrphic  solutions is a subspace of dimension d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' The subspace of holomorphic solutions in V is a subspace of di-  mension d − 1 generated by the entire functions  Ωl(s) − Ω0(s) =  �  γ0l  ts−1eP0(t) dt  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' As observed before, the functions Ωl(s) − Ω0(s) are entire functions and are  linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  □  Some of the results in [4] can be generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' In particular the Integrability criterion  and Abel-like Theorem (Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' This will be studied in a separate  article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Biswas has extended those results from [4] to curves of higher genus [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  It is interesting to speculate on the extension of the results for Omega functions in  higher genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  References  [1] AYCOCK, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
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+page_content=' Sur une classe de s´eries de Taylor, Thesis, Uppsala, 1914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
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+page_content=' Letter to Goldbach, 8 January 1730, Euler Archive [E00717], eulerarchive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='maa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
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+page_content=' A sharpening of Wielandt’s characterization of the Gamma function, The Amer-  ican Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
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+page_content=' P´EREZ-MARCO  [8] HANKEL, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Die Euler’schen Integrale bei unbeschr¨ankter Variabilit¨at des Argumentes,  Zeitschr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
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+page_content=' Correspondance d’Hermite et Stieltjes, publi´ee par Baillaud  et Bouguet, Gauthier-Villars, Paris, 1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
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+page_content=' ZAGIER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Periods, Mathematics unlimited-2001 and beyond, Springer,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
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+page_content='  On  the  definition  of  Euler  Gamma  function,  L’Enseignement  Math´ematique, 68, 1/2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
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+page_content='  [12] REMMERT, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Wielandt’s theorem about the Γ-function, The Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
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+page_content=' Monthly, 103,3,  1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  [13] REMMERT, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Classical topics in complex function theory, Graduate Texts in Mathematics,  172, Springer, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  [14] WEIERSTRASS, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' ¨Uber die Theorie der analytischen Fakult¨aten, Journal f¨ur Mathematik,  51, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='1-60, 1856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  [15] WIELANDT, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' , Mathematische Werke, 2, 2, De Gruyter, New York, published 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' See  also [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  [16] WHITTAKER, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' WATSON, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' A course in modern analysis, Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content=' Press,  4th edition, 1927.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='  CNRS, IMJ-PRG, Universit´e Paris Cit´e, Paris, France  Email address: ricardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='perez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='marco@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
+page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQf3Asg/content/2301.04759v1.pdf'}
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+Long Rayleigh length confocal microscope: A 
+fast evaluation tool for obtaining quantum 
+propensities of color centers 
+Yuta Masuyama,1,* Chikara Shinei,2 Shuya Ishii,1 Hiroshi Abe,1 
+Takashi Taniguchi, 2  Tokuyuki Teraji, 2 & Takeshi Ohshima 1 
+1 National Institutes for Quantum Science and Technology, Takasaki, Gunma,  
+370 – 1292, Japan 
+2 National Institute for Materials Science, Tsukuba, Ibaraki 305 – 0044, Japan 
+* masuyama.yuta@qst.go.jp 
+ 
+Abstract 
+Color centers in wide band-gap semiconductors, which have superior quantum properties even 
+at room temperature and atmospheric pressure, have been actively applied to quantum sensing 
+devices. Characterization of the quantum properties of the color centers in the semiconductor 
+materials and ensuring that these properties are uniform over a wide area are key issues for 
+developing quantum sensing devices based on color center. In this article, we will describe the 
+principle and performance of a newly developed confocal microscope system with a long 
+Rayleigh length (LRCFM). This system can characterize a wider area faster than the confocal 
+microscope systems commonly used for color center evaluation.  
+ 
+Introduction 
+Color centers such as NV centers in diamonds are expected to be an important tool for 
+quantum sensing because they are qubits that can be operated at room temperature and 
+atmospheric pressure [1 - 4]. In addition to NV centers in diamond, various other color centers, 
+such as silicon vacancies in SiC [5, 6], are beginning to be used depending on the sensing target 
+application [7]. Magnetic sensing using quantum properties of NV centers has achieved the 
+sensitivity of less than 1 pT/sqrt (Hz) for both AC magnetic field sensing [8] and DC magnetic 
+field sensing [9] by applying a large sensor volume. Refining quantum sensor materials will 
+make further sensitivity improvements possible [10]. In addition to magnetic field detection, 
+the quantum sensor using color centers can measure temperature [11, 12], electric field [13], 
+and pressure [14]. Furthermore, measurement techniques using multiple quantum sensors are 
+also beginning to develop and improve sensor performance [15, 16]. 
+One effective way to increase sensitivity is to improve the spatial uniformity of quantum 
+properties within the sensor material. Confocal microscopy (CFM) and electron spin resonance 
+(ESR) has been commonly used to evaluate quantum properties of color centers. Although 
+CFM is a measurement technique with excellent spatial resolution, it is not suitable for 
+materials with large sensor sizes because of small detection volume. Thus, the CFM is best 
+suited for evaluating a small number of color centers, such as a single color center. However, 
+measuring the spatial distribution of the entire sample by the CFM is impractical because of the 
+long measurement time required due to the small detection volume. ESR is a method that 
+evaluates the quantum properties of the entire sample as an average value and is not suitable 
+for evaluating the spatial distribution of quantum properties. In order to obtain a guideline for 
+improving sensitivity, a method that can evaluate the spatial distribution of quantum properties 
+in the sensor materials within a millimeter-order spatial range is desired. 
+
+In this study, we describe a design protocol for an optical system that realizes a 
+measurement system capable of evaluating a wide spatial distribution of quantum properties of 
+color centers and their concentration while minimizing the effects of surface and background 
+light. Here, we introduce a newly developed confocal microscope system that is featured by a 
+long Rayleigh length (LRCFM) in an instrument configuration similar to a CFM system. By 
+injecting a laser beam that is uniform in the depth direction and has a relatively larger spot size, 
+the detection volume becomes significantly large. This effect makes it possible to evaluate the 
+spatial distribution of quantum properties of color centers over the entire sample within a 
+reasonable time. By optimizing the optical excitation technique, optical detection system, and 
+microwave circuit, our proposed LRCFM method can evaluate color centers in a wide range of 
+concentration ensembles, from about one ppb and above, up to high concentration color centers 
+such as ten ppm, for example. 
+ 
+Results and discussion 
+Long Rayleigh length confocal microscope (LRCFM) 
+The LRCFM is characterized by increasing the excitation volume of the sample, thereby 
+reducing the amount of unpolarized fluorescence that becomes noise. This increase in excitation 
+volume can be achieved by exciting the entire thickness direction of the sample. Figure 1 shows 
+a schematic diagram for comparison with existing color center evaluation methods. The spatial 
+resolution, which correspond to the sample volume to be measured, of LRCFM is inferior than 
+that of CFM while better than that of ESR. In the practical use, LRCFM is suitable for obtaining 
+spatial distribution mapping of quantum properties across the entire mm-size sample in a 
+realistic time. The proposed method polarizes spin states as efficiently as CFM. In addition, 
+unlike CFM, this method is not affected so much by stray light of the material surface or 
+background light because of the large excitation volume to be measured, resulting in high 
+detection signal. Eventually, the measurement time becomes very short when using LRCFM. 
+Details of features of LRCFM are described below. 
+ 
+
+ 
+Fig. 1. Comparison of this study with confocal microscopy (CFM) and electron spin resonance 
+(ESR). 
+ 
+Rayleigh length and excitation volume 
+First, the optical features of this study are described. Figure 2 (a) shows a schematic 
+comparison of the beam diameter of LRCFM and CFM. CFM commonly uses lens system with 
+high magnification objective lens. On the other hand, LRCFM with low magnification objective 
+lens has a relatively larger laser beam diameter of about 20 μm and enables uniform color center 
+excitation in the depth direction. The Rayleigh length is used as a measure of laser depth. The 
+Rayleigh length zR is the length from the focal point to the point where the beam radius is √2 
+times the waist radius  𝑤0  which is beam radius at the focal point of the laser. And the Rayleigh 
+length is defined as 
+zR =
+π w02
+λ ,                                                                                                                          (1)  
+where 𝜆 is the wavelength of the laser [Fig. 2 (b)] [17]. Increasing the beam waist radius allows 
+laser irradiation with a long Rayleigh length. 
+In this study, the calculations were made by approximating the laser excitation region as a 
+cylindrical shape [the shaded area in Fig. 2 (c)]. The average power density of the laser light 
+passing through this cylinder is 
+𝑃0
+𝜋𝑤02, where 𝑃0 is the total power of the laser. A 50x objective 
+lens, which has an effective focal length of 3.6 mm, is commonly used in the case of CFM 
+systems, as shown in Fig. 2 (c). When the twice Rayleigh length 2 ZR is shorter than the sample 
+thickness, a cylinder of length (indicated by “Rayleigh length” in the Fig.2) 2 𝑧𝑅 and waist 
+radius 𝑤0 is taken as the excited volume in the calculation. On the other hand, when twice the 
+Rayleigh length is longer than the thickness of the sample, the calculation was done using the 
+thickness of the sample as the length of the cylinder. As the laser beam waist radius increases, 
+
+Focal length
+Waist radius
+Sample
+Sample
+Laser
+Laser
+magnetic
+field
+Sample
+Sample
+o Small
+Q Very small
+X Large
+X Large
+APartially
+o Possible
+X Impossible
+X Impossible
+possible
+o Negligible
+X Critical
+o Negligible
+o Negligible
+o Short
+X Long
+X Long
+X Longthe laser excitation volume rapidly increases. In the calculations below, the sample thickness 
+of 500 μm and the incident 532 nm laser beam radius of 450 μm were applied. 
+ 
+ 
+Fig. 2. (a) Schematic comparison of this study (LRCFM) and a confocal microscope (CFM) 
+system. (b) Dependence of Rayleigh length on waist radius for laser wavelength 532 nm. (c) 
+Excitation volume along the optical axis direction for focal lengths of 3.6 mm (red) and 30 mm 
+(blue). The effective focal length of 3.6 mm corresponds to a 50x objective lens, and the focal 
+length of 30 mm is used in this study. The blue and red colored squares with twice the Rayleigh 
+length and twice the waist radius on each side represent the effective beam areas with focal 
+lengths of 30 mm and 3.6 mm, respectively. 
+ 
+The amount of detected light was evaluated by the calculation steps shown in figure 3(a) to 
+find the optimal laser beam condition for the LRCFM. Specifically, the excitation volume was 
+calculated by changing the Rayleigh length of the excitation laser, and then the amount of 
+fluorescence emitted by polarized color centers within this excitation volume and the amount 
+of collected light were evaluated. Here, these calculations were performed using the energy 
+levels of the negatively charged NV centers in the diamond [Figure 3(b)] (see supplemental 
+material for the details). The values of the transition rates of the NV center required for this 
+numerical calculation were taken from the reference by [18].  
+
+CFM
+LRCFM
+CFM
+LRCFM
+(This study)
+(This study)
+0.5 μm
+x 5
+X 5
+20 μm
+20 μm
+Rayleigh length ZR
+Waist radisu Wo 
+Fig. 3 (a) Calculation step to determine the focal length of the objective lens for LRCFM. (b) 
+Energy level diagram for the NV center in diamond. The green arrow indicates optical excitation 
+from the ground state of the NV center. The red arrow indicates fluorescence from the excited 
+state of the NV center. ms is magnetic sublevel of the NV center. 𝑘𝑖𝑗 indicates the transition rate 
+from level i to level j. 
+ 
+Excitation signal amount 
+When the waist radius is very large, the Rayleigh length increases but the photon power 
+density decreases greatly, resulting in a significant decrease in the polarization ratio of the color 
+center. This phenomenon is confirmed from the numerically calculated amount of fluorescence 
+emitted from the region excited by the laser beam. Figure 4 shows the calculated results for the 
+NV center. 
+The amount of fluorescence emitted from the NV center is calculated using the steady-state 
+population in states |3> and |4> as 
+𝐼𝐶𝑊 =
+𝑘31+𝑘32 
+𝑘31+𝑘32+𝑘35 𝜌33
+𝑠𝑠 +
+𝑘41+𝑘42 
+𝑘41+𝑘42+𝑘45 𝜌44
+𝑠𝑠,                                                                            (2) 
+where 𝜌33
+𝑠𝑠 and  𝜌44
+𝑠𝑠  are the elements of the steady state density matrix corresponding to the 
+states |3> and |4> and 𝑘𝑖𝑗 is the transition rate from level i to level j (see Appendix) [19]. Figure 
+4 (a) shows the numerical results of the Rayleigh length dependence of the excitation volume 
+at a laser power of 10 mW. The solid blue line corresponds to a sample with a thickness of 500 
+μm, and the dashed red line corresponds to a sample with infinite thickness. As shown in Fig. 
+4 (b), the fluorescence per unit volume decreases due to the lowering of laser beam power 
+density. 
+ 
+
+K32
+KA7
+K31 
+Fig.4 Numerical calculations for a 532nm laser with a beam diameter of 0.9 mm at a laser power 
+of 10 mW. (a) Dependence of excitation volume on the Rayleigh length for sample thicknesses 
+of 500 μm (solid blue line) and infinity (dashed red line). (b) Rayleigh length dependence of 
+fluorescence intensity. (c) Rayleigh length dependence of polarization rate. (inset) A wider view 
+of the Rayleigh length region in Fig. 4 (c). (d) Rayleigh length dependence of the product of 
+excitation volume, fluorescence per unit volume, and polarization ratio. Blue and red circles 
+correspond to the LRCFM and CFM cases, respectively. (inset) A wider view of the Rayleigh 
+length region in Fig. 4 (d). 
+ 
+The polarization rate of a color center is an important indicator because a quantum state 
+cannot be manipulated or read out unless it is polarized. The polarization rate of a quantum 
+state is defined as 
+𝑃 =
+𝜌11
+𝑠𝑠 − 𝜌22
+𝑠𝑠
+𝜌11
+𝑠𝑠 + 𝜌22
+𝑠𝑠,                                                                                                                          (3) 
+where 𝜌11
+𝑠𝑠 and 𝜌22
+𝑠𝑠 are the elements of the steady state density matrix corresponding to the 
+states |1> and |2>. In Fig. 4 (c), The polarization rate at a laser power of 10 mW is calculated. 
+According to the Eq. (1), there is a positive correlation between the laser beam diameter and 
+the Rayleigh length. The polarization ratio decreases with increasing beam diameter that can 
+be written with the Rayleigh length because the laser beam power density becomes lower [Fig. 
+4 (c, inset)].  
+Furthermore, we study the total signal volume effective for quantum state readout. We 
+calculated the dependence of the product of the detection volume (= π ×w02× sample thickness), 
+the fluorescence per unit volume 𝐼𝑐𝑤, and the polarization rate 𝑃 on the Rayleigh length at a 
+laser power of 10 mW [Fig. 4 (d)]. The polarization ratio is good for the short Rayleigh length 
+
+0.6
+X1016
+0.4
+0.2
+0
+500
+500corresponding to the narrow waist radius, but the detection volume is so small that these product 
+value becomes very small. The fluorescence intensity increases rapidly until twice the Rayleigh 
+length reaches near the thickness of the sample, which corresponds to 500 μm. Then, as the 
+twice the Rayleigh length increases further, the product's value decreases, mainly due to the 
+decrease in laser power density [Fig. 4 (d, inset)]. Note that this calculation is for overall 
+fluorescence emission. The focus calculation is given in the next subsection. 
+ 
+Detection signal amount 
+In the previous subsection, the total amount of emitted fluorescence was calculated which 
+is represented by diagrams in Fig. 4. This subsection deals with the photon correction efficiency 
+of the LRCFM system. Practically, we calculate the amount of detected fluorescence, taking 
+into account the focusing performance of the objective lens [Fig. 5 (a)]. In this calculation, the 
+beam radius of the incident laser is fixed at 450 μm, so the lens's focal length can be determined 
+from the waist radius w0. The numerical aperture (NA) was calculated from the focal length 
+and the objective lens radius. The detection ratio was then calculated by comparing the ratio 
+with NA = 1 [Fig. 5 (b)]. The amount of detected fluorescence for a laser power of 10 mW is 
+obtained by multiplying the product of the detection volume (= π × w02 × sample thickness), 
+the amount of fluorescence per unit volume 𝐼𝑐𝑤, and the polarization ratio 𝑃. Fig. 5 (c) shows 
+the detected fluorescence as a function of the Rayleigh length. This simulation shows that the 
+maximum amount of the detected fluorescence occurs when twice the Rayleigh length matches 
+the sample thickness. Our designed system can be used without concern for laser power because 
+the Rayleigh length at maximum detected fluorescence is constant in the practical range of laser 
+power (see supplemental material for the details). 
+Based on the results obtained so far, using the relation between the focal length of the 
+objective lens and the waist radius w0 = 2λF/πD, where F is the lens's focal length and D is 
+the diameter of the incident beam, the required focal length of the lens is obtained by 
+F =
+𝐷
+2 √
+𝑧𝑅π
+λ .                                                                                                                            (4) 
+This laser beam, focused by an objective lens with a focal length of F, excites the sample 
+uniformly in-depth and produces a large amount of fluorescence. Because of the strong 
+collected signal and larger beam spot size, typically about 20 μm, the LRCFM can measure the 
+spatial distribution of properties throughout the entire mm-scale size sample in a realistic time.    
+ 
+ 
+
+ 
+Fig. 5 (a) Schematic of a sample being excited by a laser and its fluorescence being collected. 
+(b) Dependence of detection rate on excitation Rayleigh length for a 1-inch lens (blue) and a 0.5-
+inch lens (magenta). (c) Dependence of the product of the polarized fluorescence [Fig. 4 (d)] and 
+the detection rate [Fig. 5 (b)] for a 1-inch lens (blue) and a 0.5-inch lens (magenta) on the 
+excitation Rayleigh length at 10 mW laser power. The dashed line represents the location where 
+the Rayleigh length is 0.25 mm, and twice this Rayleigh length equals the thickness of the sample. 
+ 
+Choosing the objective lens and the optical fiber 
+In our experimental setup using LRCFM, the incident beam diameter of a 532 nm laser is 
+0.9 mm, and an objective lens with a focal length of 30 mm collects the most polarized signal 
+when selected from a commercially available 1-inch achromatic lens. The 1-inch achromatic 
+lens with a focal length of 30 mm was used in the following experiment. 
+
+Fluorescence
+Waist radius Wo
+Diameter of
+the incident
+beam D
+Laser
+Sample
+Collected
+fluorescenceIn conventional CFM, the light collection efficiency increases due to the higher NA of the 
+objective lens compared to LRCFM, at the same time, the spatial resolution improves. On the 
+other hand, the amount of collected fluorescence decreases due to the effect of pinholes in the 
+CFM, as indicated by red rectangle in Fig. 6 (a). The advantage of the LRCFM is that the 
+measurement time is shortened by making identical size between the detection and the 
+excitation regions, which corresponds to a detection proportion equal to one. The drawback is 
+poor spatial resolution. A large pinhole, which corresponds to optical fiber core in the case of 
+LRCFM, makes this large detection proportion possible, as indicated by red and green 
+rectangles in Fig. 6 (a). The multimode fiber with a core diameter of 200 μm was used in the 
+following experiment. By replacing the objective lens, optical fiber, and photodetector, it is 
+possible to turn a conventional CFM into a LRCFM. 
+Compared to CFM with infinitely small pinhole diameters, the detected fluorescence of 
+LRCFM is more than 104 times higher [Fig. 6 (b)]. Thus, assuming that the noise levels of both 
+systems are the same, CFM requires 108 times the measurement time to achieve the same signal-
+to-noise ratio as LRCFM. In addition, the larger detection volume means that the surface 
+occupies a tiny percentage of the total volume, making it less susceptible to the emission light 
+from impurities and other substances on the surface. Due to its high photon focusing efficiency, 
+LRCFM has a wide concentration range of detectable color centers. 
+ 
+ 
+Fig. 6 (a) Schematic comparison of excitation and focusing volumes for confocal microscopy 
+(CFM) and this study. (b) Product of detection rate [ Fig .4 (d)] and polarized fluorescence at 10 
+
+Excitation
+Detection
+(Proportion = 3.5x1o-4
+Excitatior
+Detection
+(Proportion = 1)
+LRCFMmW laser power [Fig. 5 (d)] as a function of detection proportion and excitation beam radius. 
+Blue and red circles correspond to the LRCFM and CFM cases, respectively. 
+ 
+Measurement of the spatial distribution of the quantum properties 
+Figure 7 shows the spatial distribution of the quantum properties of NV centers in the 
+diamond. One pixel size is 50 μm × 50 μm, and the spatial measurement area was up to 350 μm 
+× 1050 μm. Here, the x and y coordinates are those shown in Fig. S1(b). Figure 7 (a) shows the 
+results of Rabi oscillation measurements at the point (x, y) = (-200um, 100um) of Fig. 7(b), 
+fitted by the function 𝑎1 𝑒𝑥𝑝(−τ/𝑎2) 𝑐𝑜𝑠(2π𝑎3τ + 𝑎4) + 𝑎5 , where τ is the microwave 
+irradiation time, 𝑎𝑖 (i = 1 ~ 5) is a fitting parameter. Figure 7(b) shows the spatial distribution 
+of the π pulse duration that inverts the quantum state. This data was used to correct for 
+differences in π pulse duration due to different positions on the microwave circuit, thereby 
+improving the accuracy of the T1 and T2 measurements. The π/2 pulse was set to half of the π 
+pulse time. Figure 7 (c) shows the results of the energy relaxation time T1 measurement at the 
+point (x, y) = (-200um, 100um), fitted by the function 𝑎1 𝑒𝑥𝑝(−τ/𝑎2) + 𝑎3, where τ is the time 
+duration between π pulse and readout, 𝑎𝑖 (i = 1 ~ 3) is a fitting parameter. In T1 measurement, 
+π pulses of microwaves were irradiated, and the quantum state was read out as a 
+photoluminescence intensity by a 532 nm laser excitation after a time τ. Figure 7 (d) shows the 
+spatial distribution of the energy relaxation time T1. The average value over the entire 
+measurement region of the sample was 11.0 ms with a standard deviation of 4.0 ms. Figure 7 
+(e) is the result of the phase relaxation time T2 measurement at the point (x, y) = (-200um, 
+100um), fitted by the function 𝑎1 𝑒𝑥𝑝(−(τ/𝑎2)𝑎3), where τ is the time between the π pulse and 
+the readout, 𝑎𝑖 (i = 1 ~ 3) is a fitting parameter. The microwave pulse sequence is π/2 pulse- π 
+pulse - π/2, with π/2 pulse and π pulse separated by time τ. After an exposure of the microwave 
+pulse sequence, the quantum states were read out with a 532 nm laser. Figure 7(f) shows the 
+spatial distribution of the phase relaxation time T2. The average value over the entire 
+measurement region of the sample was 21.5 us and the standard deviation was 1.9 us. The value 
+of T2. is gradually decreasing from the left-side area to the right-side area. This trend of T2 
+spatial distribution corresponds to the trend of the spatial distribution of fluorescence from the 
+NV− center, as shown in Fig. S1(b). Fig. 7(f) and Fig. S1 (b) indicate that T2 is decreasing in 
+increasing density of nitrogen and NV− center as a decoherence source. 
+
+ 
+Fig. 7 (a) Rabi oscillation measurement at the point (x, y) = (-200um, 100um). (b) Spatial 
+distribution of π pulse duration. (c) Energy relaxation time T1 measurement at the point (x, y) = 
+(-200um, 100um). (d) Spatial distribution of NV center energy relaxation time T1. (e) Phase 
+relaxation time T2 measured at the point (x, y) = (-200um, 100um). (f) Spatial distribution of 
+phase relaxation time T2 at the NV center. 
+ 
+Conclusion  
+In summary, we have developed a method to rapidly measure the sample information such as 
+quantum properties of an entire mm-scale sample by constructing a confocal microscope with 
+a long Rayleigh length. By considering the excitation volume, the polarization ratio of the 
+quantum state, and the detection efficiency, we found that matching twice the Rayleigh length 
+to the thickness of the sample produces the highest amount of detected signal. The detected 
+signal is about 104 times larger than the conventional ideal confocal microscopy. LRCFM uses 
+an identical microwave setup with conventional confocal microscopy to evaluate color centers. 
+
+T pulse time = 105.0 ns
+107
+1.00
+1000
+106
+I signal
+0.98
+800
+105
+Normalized
+(μm)
+0.96
+600
+104
+>
+0.94
+103
+400
+0.92
+102
+200
+101
+0
+100
+200
+300
+400
+500
+-200
+0
+Time (ns)
+x (um)
+T1 = 10.5 ms
+10.5
+0.004
+1000
+10.0
+0.003
+800
+0.002
+(un)
+9.5
+(sw)
+600
+0.001
+>
+9.0
+0.000
+400
+8.5
+-0.001
+200
+-0.002
+8.0
+0
+5
+10
+15
+-200
+0
+Time (ms)
+X (μm)
+T2 = 25.4 μs, n = 1.22
+26
+0.0025
+1000
+0.0020
+800
+24
+0.0015
+(wr)
+(srl)
+600
+0.0010
+>
+22 /2
+400
+0.0005
+200
+20
+0.0000
+50
+0
+100
+-200
+0
+Time (μs)
+X (μm)Thus, it is possible to use these microwave pulse sequences for LRCFM, such as evaluating 
+NV center density using instantaneous diffusion [22, 23]. Density evaluation of spin defects in 
+the quantum material, usually performed on the entire sample by ESR [24], can be 
+accomplished using LRCFM, with additional information on spatial distribution. In addition, 
+this method can be used not only for the evaluation of existing color centers, but also for the 
+discovery of new color centers, as it can take advantage of the high signal-to-noise ratio to 
+increase the speed for measuring photoluminescence spectra of unknown color centers. [25, 
+26]. 
+ 
+Method 
+A diamond sample containing NV centers was placed on the microwave resonator [20]. The 
+resonator is fixed to a motorized stage with a movable range of 13 mm. A permanent magnet 
+to fix the quantization axis was placed behind the resonator. The strength of the magnetic field 
+was approximately 2.5 mT. The fluorescence from the NV centers was collected through the 
+objective lens (AC254-030-AB-ML; Thorlabs) and detected using an avalanche photodiode 
+(APD410A/M; Thorlabs). An oscilloscope (MDO34; Tektronix) recorded the signal from the 
+avalanche photodiode and analyzed it to distinguish the spin state of the NV centers. Using an 
+arbitrary waveform generator (M3202A; Keysight) to generate the laser and microwave pulse 
+sequence, we controlled an acousto-optic modulator (#35 250-0.2-0.53-XQ; Gooch & 
+Housego) and a transistor-transistor logic (TTL) switch. After the microwave pulse sequence, 
+a 180 µs measurement laser pulse of 532 nm laser (gem 532; Laser Quantum) for measurement 
+with 12 mW was irradiated to the NV centers in the diamond. The diamond {111} single crystal 
+grown by high-temperature/high-pressure (HPHT) synthetic method was used in this study [21]. 
+The HPHT sample was irradiated with a 2.0 MeV electron beam with a total fluence of 5.0 × 
+1017 cm−2 for creating vacancies in the crystal and then annealed at 1000 ◦C for 2 hours in 
+vacuum to create NV centers (see supplemental material for the details).  
+ 
+Data availability  
+The data that support the findings of this study are available from the corresponding author 
+upon reasonable request. 
+ 
+References 
+1. 
+Taylor, J. M. et al. High-sensitivity diamond magnetometer with nanoscale resolution. Nat. Phys. 4, 810 (2008). 
+2. 
+Doherty, M. W. et al. The nitrogen-vacancy colour centre in diamond. Physics Reports 528, 1 (2013) 
+3. 
+Rondin, L.  et al. Magnetometry with nitrogen-vacancy defects in diamond. Rep. Prog. Phys. 77, 056503 (2014)  
+4. 
+Degen, C. L., Reinhard, F., & Cappellaro, P., Quantum sensing. Rev. Mod. Phys. 89, 035002 (2017) 
+5. 
+Tarasenko, S. A. et al. Spin and Optical Properties of Silicon Vacancies in Silicon Carbide− A Review. physica 
+status solidi (b) 255, 1700258 (2018) 
+6. 
+Zhang, G., Cheng, Y., Chou, J-P., & Gali, A., Material platforms for defect qubits and single-photon emitters. 
+Applied Physics Reviews 7, 031308 (2020) 
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+thermometry enhanced by the quantum coherence of single spins in diamond. Proceedings of the National 
+Academy of Sciences 110, 8417 (2013) 
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+waveguide. Nature Physics 11, 393 (2015) 
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+14. Doherty, M. W. et al. Electronic Properties and Metrology Applications of the Diamond NV− Center under 
+Pressure. Phys. Rev. Lett. 112, 047601 (2014) 
+
+15. Zhang, C. Diamond Magnetometry and Gradiometry Towards Subpicotesla dc Field Measurement. Phys. Rev. 
+Applied 15 064075 (2021) 
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+17. Meschede, D. Optics, Light and Lasers: The Practical Approach to Modern Aspects of Photonics and Laser 
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+19. El-Ella, H. A. R., Ahmadi, S., Wojciechowski, A. M., Huck, A., & Andersen, U. L., Optimised frequency 
+modulation for continuous-wave optical magnetic resonance sensing using nitrogen-vacancy ensembles. Optics 
+express 25.13 (2017): 14809-14821. 
+20. Masuyama, Y. et al. Extending coherence time of macro-scale diamond magnetometer by dynamical 
+decoupling with coplanar waveguide resonator. Rev. Sci. Instrum. 89, 125007 (2018) 
+21. Miyakawa, M., Shinei, C., & Taniguchi, T. Nitrogen concentration control in diamonds grown in Co–(Fe)–
+Ti/Al solvents under high-pressure and high-temperature. Jpn. J. Appl.  61, 045507 (2022) 
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+Press, New York, (2001). 
+24. Ishii, S. et al. Ensemble Negatively-Charged Nitrogen-Vacancy Centers in Type-Ib Diamond Created by High 
+Fluence Electron Beam Irradiation. Quantum Beam Sci. 6, 2 (2022) 
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+26. Kanai, S. et al.  Generalized scaling of spin qubit coherence in over 12,000 host materials. Proceedings of the 
+National Academy of Sciences 119.15 (2022): e2121808119. 
+ 
+Acknowledgements 
+This work was supported by MEXT Q-LEAP (JPMXS0118067395 and JPMXS0118068379). 
+YM acknowledges the support of JSPS KAKENHI (20K14392). T.T. acknowledges the 
+support of JST Moonshot R&D (JPMJMS2062), MIC R&D for construction of a global 
+quantum cryptography network (JPMI00316) and JSPS KAKENHI (20H02187, 20H05661). 
+ 
+Author information 
+Authors and Affiliations 
+National Institutes for Quantum Science and Technology, Takasaki, Gunma,  
+370 – 1292, Japan 
+Y. Masuyama, S. Ishii, H. Abe & T. Ohshima  
+ 
+National Institute for Materials Science, Tsukuba, Ibaraki 305 – 0044, Japan 
+C. Shinei, T. Taniguchi & T. Teraji 
+Contributions 
+The measurement system was designed, constructed, and tested by Y.M. Y.M. and C.S. 
+performed the measurements and the data analysis. T.T. synthesized the diamond. The diamond 
+was electron-irradiated by S. I. and H.A. The overall supervision was performed by T.T. and 
+T.O. 
+ 
+Corresponding author 
+Correspondence to Yuta Masuyama 
+ 
+Ethics declarations 
+Competing Interests 
+The authors declare no conflicts of interest. 
+
diff --git a/fdFMT4oBgHgl3EQf1zGx/content/tmp_files/load_file.txt b/fdFMT4oBgHgl3EQf1zGx/content/tmp_files/load_file.txt
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@@ -0,0 +1,471 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf,len=470
+page_content='Long Rayleigh length confocal microscope: A  fast evaluation tool for obtaining quantum  propensities of color centers  Yuta Masuyama,1,* Chikara Shinei,2 Shuya Ishii,1 Hiroshi Abe,1  Takashi Taniguchi, 2  Tokuyuki Teraji, 2 & Takeshi Ohshima 1  1 National Institutes for Quantum Science and Technology, Takasaki, Gunma,  370 – 1292, Japan  2 National Institute for Materials Science, Tsukuba, Ibaraki 305 – 0044, Japan  masuyama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='yuta@qst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='jp  Abstract  Color centers in wide band-gap semiconductors, which have superior quantum properties even  at room temperature and atmospheric pressure, have been actively applied to quantum sensing  devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Characterization of the quantum properties of the color centers in the semiconductor  materials and ensuring that these properties are uniform over a wide area are key issues for  developing quantum sensing devices based on color center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' In this article, we will describe the  principle and performance of a newly developed confocal microscope system with a long  Rayleigh length (LRCFM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' This system can characterize a wider area faster than the confocal  microscope systems commonly used for color center evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Introduction  Color centers such as NV centers in diamonds are expected to be an important tool for  quantum sensing because they are qubits that can be operated at room temperature and  atmospheric pressure [1 - 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' In addition to NV centers in diamond, various other color centers,  such as silicon vacancies in SiC [5, 6], are beginning to be used depending on the sensing target  application [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Magnetic sensing using quantum properties of NV centers has achieved the  sensitivity of less than 1 pT/sqrt (Hz) for both AC magnetic field sensing [8] and DC magnetic  field sensing [9] by applying a large sensor volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Refining quantum sensor materials will  make further sensitivity improvements possible [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' In addition to magnetic field detection,  the quantum sensor using color centers can measure temperature [11, 12], electric field [13],  and pressure [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Furthermore, measurement techniques using multiple quantum sensors are  also beginning to develop and improve sensor performance [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  One effective way to increase sensitivity is to improve the spatial uniformity of quantum  properties within the sensor material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Confocal microscopy (CFM) and electron spin resonance  (ESR) has been commonly used to evaluate quantum properties of color centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Although  CFM is a measurement technique with excellent spatial resolution, it is not suitable for  materials with large sensor sizes because of small detection volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Thus, the CFM is best  suited for evaluating a small number of color centers, such as a single color center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' However,  measuring the spatial distribution of the entire sample by the CFM is impractical because of the  long measurement time required due to the small detection volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' ESR is a method that  evaluates the quantum properties of the entire sample as an average value and is not suitable  for evaluating the spatial distribution of quantum properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' In order to obtain a guideline for  improving sensitivity, a method that can evaluate the spatial distribution of quantum properties  in the sensor materials within a millimeter-order spatial range is desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  In this study, we describe a design protocol for an optical system that realizes a  measurement system capable of evaluating a wide spatial distribution of quantum properties of  color centers and their concentration while minimizing the effects of surface and background  light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Here, we introduce a newly developed confocal microscope system that is featured by a  long Rayleigh length (LRCFM) in an instrument configuration similar to a CFM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' By  injecting a laser beam that is uniform in the depth direction and has a relatively larger spot size,  the detection volume becomes significantly large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' This effect makes it possible to evaluate the  spatial distribution of quantum properties of color centers over the entire sample within a  reasonable time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' By optimizing the optical excitation technique, optical detection system, and  microwave circuit, our proposed LRCFM method can evaluate color centers in a wide range of  concentration ensembles, from about one ppb and above, up to high concentration color centers  such as ten ppm, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Results and discussion  Long Rayleigh length confocal microscope (LRCFM)  The LRCFM is characterized by increasing the excitation volume of the sample, thereby  reducing the amount of unpolarized fluorescence that becomes noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' This increase in excitation  volume can be achieved by exciting the entire thickness direction of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Figure 1 shows  a schematic diagram for comparison with existing color center evaluation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The spatial  resolution, which correspond to the sample volume to be measured, of LRCFM is inferior than  that of CFM while better than that of ESR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' In the practical use, LRCFM is suitable for obtaining  spatial distribution mapping of quantum properties across the entire mm-size sample in a  realistic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The proposed method polarizes spin states as efficiently as CFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' In addition,  unlike CFM, this method is not affected so much by stray light of the material surface or  background light because of the large excitation volume to be measured, resulting in high  detection signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Eventually, the measurement time becomes very short when using LRCFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Details of features of LRCFM are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Comparison of this study with confocal microscopy (CFM) and electron spin resonance  (ESR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Rayleigh length and excitation volume  First, the optical features of this study are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Figure 2 (a) shows a schematic  comparison of the beam diameter of LRCFM and CFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' CFM commonly uses lens system with  high magnification objective lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' On the other hand, LRCFM with low magnification objective  lens has a relatively larger laser beam diameter of about 20 μm and enables uniform color center  excitation in the depth direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The Rayleigh length is used as a measure of laser depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The  Rayleigh length zR is the length from the focal point to the point where the beam radius is √2  times the waist radius  𝑤0  which is beam radius at the focal point of the laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' And the Rayleigh  length is defined as  zR =  π w02  λ ,                                                                                                                          (1)  where 𝜆 is the wavelength of the laser [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 2 (b)] [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Increasing the beam waist radius allows  laser irradiation with a long Rayleigh length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  In this study, the calculations were made by approximating the laser excitation region as a  cylindrical shape [the shaded area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 2 (c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The average power density of the laser light  passing through this cylinder is  𝑃0  𝜋𝑤02, where 𝑃0 is the total power of the laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' A 50x objective  lens, which has an effective focal length of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='6 mm, is commonly used in the case of CFM  systems, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 2 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' When the twice Rayleigh length 2 ZR is shorter than the sample  thickness, a cylinder of length (indicated by “Rayleigh length” in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='2) 2 𝑧𝑅 and waist  radius 𝑤0 is taken as the excited volume in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' On the other hand, when twice the  Rayleigh length is longer than the thickness of the sample, the calculation was done using the  thickness of the sample as the length of the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' As the laser beam waist radius increases,  Focal length  Waist radius  Sample  Sample  Laser  Laser  magnetic  field  Sample  Sample  o Small  Q Very small  X Large  X Large  APartially  o Possible  X Impossible  X Impossible  possible  o Negligible  X Critical  o Negligible  o Negligible  o Short  X Long  X Long  X Longthe laser excitation volume rapidly increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' In the calculations below, the sample thickness  of 500 μm and the incident 532 nm laser beam radius of 450 μm were applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (a) Schematic comparison of this study (LRCFM) and a confocal microscope (CFM)  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (b) Dependence of Rayleigh length on waist radius for laser wavelength 532 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (c)  Excitation volume along the optical axis direction for focal lengths of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='6 mm (red) and 30 mm  (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The effective focal length of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='6 mm corresponds to a 50x objective lens, and the focal  length of 30 mm is used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The blue and red colored squares with twice the Rayleigh  length and twice the waist radius on each side represent the effective beam areas with focal  lengths of 30 mm and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='6 mm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  The amount of detected light was evaluated by the calculation steps shown in figure 3(a) to  find the optimal laser beam condition for the LRCFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Specifically, the excitation volume was  calculated by changing the Rayleigh length of the excitation laser, and then the amount of  fluorescence emitted by polarized color centers within this excitation volume and the amount  of collected light were evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Here, these calculations were performed using the energy  levels of the negatively charged NV centers in the diamond [Figure 3(b)] (see supplemental  material for the details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The values of the transition rates of the NV center required for this  numerical calculation were taken from the reference by [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  CFM  LRCFM  CFM  LRCFM  (This study)  (This study)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='5 μm  x 5  X 5  20 μm  20 μm  Rayleigh length ZR  Waist radisu Wo  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 3 (a) Calculation step to determine the focal length of the objective lens for LRCFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (b)  Energy level diagram for the NV center in diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The green arrow indicates optical excitation  from the ground state of the NV center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The red arrow indicates fluorescence from the excited  state of the NV center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' ms is magnetic sublevel of the NV center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 𝑘𝑖𝑗 indicates the transition rate  from level i to level j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Excitation signal amount  When the waist radius is very large, the Rayleigh length increases but the photon power  density decreases greatly, resulting in a significant decrease in the polarization ratio of the color  center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' This phenomenon is confirmed from the numerically calculated amount of fluorescence  emitted from the region excited by the laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Figure 4 shows the calculated results for the  NV center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  The amount of fluorescence emitted from the NV center is calculated using the steady-state  population in states |3> and |4> as  𝐼𝐶𝑊 =  𝑘31+𝑘32  𝑘31+𝑘32+𝑘35 𝜌33  𝑠𝑠 +  𝑘41+𝑘42  𝑘41+𝑘42+𝑘45 𝜌44  𝑠𝑠,                                                                            (2)  where 𝜌33  𝑠𝑠 and  𝜌44  𝑠𝑠  are the elements of the steady state density matrix corresponding to the  states |3> and |4> and 𝑘𝑖𝑗 is the transition rate from level i to level j (see Appendix) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Figure  4 (a) shows the numerical results of the Rayleigh length dependence of the excitation volume  at a laser power of 10 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The solid blue line corresponds to a sample with a thickness of 500  μm, and the dashed red line corresponds to a sample with infinite thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  4 (b), the fluorescence per unit volume decreases due to the lowering of laser beam power  density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  K32  KA7  K31  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='4 Numerical calculations for a 532nm laser with a beam diameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='9 mm at a laser power  of 10 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (a) Dependence of excitation volume on the Rayleigh length for sample thicknesses  of 500 μm (solid blue line) and infinity (dashed red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (b) Rayleigh length dependence of  fluorescence intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (c) Rayleigh length dependence of polarization rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (inset) A wider view  of the Rayleigh length region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 4 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (d) Rayleigh length dependence of the product of  excitation volume, fluorescence per unit volume, and polarization ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Blue and red circles  correspond to the LRCFM and CFM cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (inset) A wider view of the Rayleigh  length region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 4 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  The polarization rate of a color center is an important indicator because a quantum state  cannot be manipulated or read out unless it is polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The polarization rate of a quantum  state is defined as  𝑃 =  𝜌11  𝑠𝑠 − 𝜌22  𝑠𝑠  𝜌11  𝑠𝑠 + 𝜌22  𝑠𝑠,                                                                                                                          (3)  where 𝜌11  𝑠𝑠 and 𝜌22  𝑠𝑠 are the elements of the steady state density matrix corresponding to the  states |1> and |2>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 4 (c), The polarization rate at a laser power of 10 mW is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  According to the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (1), there is a positive correlation between the laser beam diameter and  the Rayleigh length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The polarization ratio decreases with increasing beam diameter that can  be written with the Rayleigh length because the laser beam power density becomes lower [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  4 (c, inset)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Furthermore, we study the total signal volume effective for quantum state readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' We  calculated the dependence of the product of the detection volume (= π ×w02× sample thickness),  the fluorescence per unit volume 𝐼𝑐𝑤, and the polarization rate 𝑃 on the Rayleigh length at a  laser power of 10 mW [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 4 (d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The polarization ratio is good for the short Rayleigh length  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='6  X1016  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='2  0  500  500corresponding to the narrow waist radius, but the detection volume is so small that these product  value becomes very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The fluorescence intensity increases rapidly until twice the Rayleigh  length reaches near the thickness of the sample, which corresponds to 500 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=" Then, as the  twice the Rayleigh length increases further, the product's value decreases, mainly due to the  decrease in laser power density [Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 4 (d, inset)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Note that this calculation is for overall  fluorescence emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The focus calculation is given in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Detection signal amount  In the previous subsection, the total amount of emitted fluorescence was calculated which  is represented by diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' This subsection deals with the photon correction efficiency  of the LRCFM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Practically, we calculate the amount of detected fluorescence, taking  into account the focusing performance of the objective lens [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 5 (a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=" In this calculation, the  beam radius of the incident laser is fixed at 450 μm, so the lens's focal length can be determined  from the waist radius w0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The numerical aperture (NA) was calculated from the focal length  and the objective lens radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The detection ratio was then calculated by comparing the ratio  with NA = 1 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 5 (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The amount of detected fluorescence for a laser power of 10 mW is  obtained by multiplying the product of the detection volume (= π × w02 × sample thickness),  the amount of fluorescence per unit volume 𝐼𝑐𝑤, and the polarization ratio 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 5 (c) shows  the detected fluorescence as a function of the Rayleigh length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' This simulation shows that the  maximum amount of the detected fluorescence occurs when twice the Rayleigh length matches  the sample thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Our designed system can be used without concern for laser power because  the Rayleigh length at maximum detected fluorescence is constant in the practical range of laser  power (see supplemental material for the details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content="  Based on the results obtained so far, using the relation between the focal length of the  objective lens and the waist radius w0 = 2λF/πD, where F is the lens's focal length and D is  the diameter of the incident beam, the required focal length of the lens is obtained by  F =  𝐷  2 √  𝑧𝑅π  λ ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (4)  This laser beam, focused by an objective lens with a focal length of F, excites the sample  uniformly in-depth and produces a large amount of fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Because of the strong  collected signal and larger beam spot size, typically about 20 μm, the LRCFM can measure the  spatial distribution of properties throughout the entire mm-scale size sample in a realistic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 5 (a) Schematic of a sample being excited by a laser and its fluorescence being collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  (b) Dependence of detection rate on excitation Rayleigh length for a 1-inch lens (blue) and a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='5-  inch lens (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (c) Dependence of the product of the polarized fluorescence [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 4 (d)] and  the detection rate [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 5 (b)] for a 1-inch lens (blue) and a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='5-inch lens (magenta) on the  excitation Rayleigh length at 10 mW laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The dashed line represents the location where  the Rayleigh length is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='25 mm, and twice this Rayleigh length equals the thickness of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Choosing the objective lens and the optical fiber  In our experimental setup using LRCFM, the incident beam diameter of a 532 nm laser is  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='9 mm, and an objective lens with a focal length of 30 mm collects the most polarized signal  when selected from a commercially available 1-inch achromatic lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The 1-inch achromatic  lens with a focal length of 30 mm was used in the following experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Fluorescence  Waist radius Wo  Diameter of  the incident  beam D  Laser  Sample  Collected  fluorescenceIn conventional CFM, the light collection efficiency increases due to the higher NA of the  objective lens compared to LRCFM, at the same time, the spatial resolution improves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' On the  other hand, the amount of collected fluorescence decreases due to the effect of pinholes in the  CFM, as indicated by red rectangle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 6 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The advantage of the LRCFM is that the  measurement time is shortened by making identical size between the detection and the  excitation regions, which corresponds to a detection proportion equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The drawback is  poor spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' A large pinhole, which corresponds to optical fiber core in the case of  LRCFM, makes this large detection proportion possible, as indicated by red and green  rectangles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 6 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The multimode fiber with a core diameter of 200 μm was used in the  following experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' By replacing the objective lens, optical fiber, and photodetector, it is  possible to turn a conventional CFM into a LRCFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Compared to CFM with infinitely small pinhole diameters, the detected fluorescence of  LRCFM is more than 104 times higher [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 6 (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Thus, assuming that the noise levels of both  systems are the same, CFM requires 108 times the measurement time to achieve the same signal-  to-noise ratio as LRCFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' In addition, the larger detection volume means that the surface  occupies a tiny percentage of the total volume, making it less susceptible to the emission light  from impurities and other substances on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Due to its high photon focusing efficiency,  LRCFM has a wide concentration range of detectable color centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 6 (a) Schematic comparison of excitation and focusing volumes for confocal microscopy  (CFM) and this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (b) Product of detection rate [ Fig .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='4 (d)] and polarized fluorescence at 10  Excitation  Detection  (Proportion = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='5x1o-4  Excitatior  Detection  (Proportion = 1)  LRCFMmW laser power [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 5 (d)] as a function of detection proportion and excitation beam radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Blue and red circles correspond to the LRCFM and CFM cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Measurement of the spatial distribution of the quantum properties  Figure 7 shows the spatial distribution of the quantum properties of NV centers in the  diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' One pixel size is 50 μm × 50 μm, and the spatial measurement area was up to 350 μm  × 1050 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Here, the x and y coordinates are those shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' S1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Figure 7 (a) shows the  results of Rabi oscillation measurements at the point (x, y) = (-200um, 100um) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 7(b),  fitted by the function 𝑎1 𝑒𝑥𝑝(−τ/𝑎2) 𝑐𝑜𝑠(2π𝑎3τ + 𝑎4) + 𝑎5 , where τ is the microwave  irradiation time, 𝑎𝑖 (i = 1 ~ 5) is a fitting parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Figure 7(b) shows the spatial distribution  of the π pulse duration that inverts the quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' This data was used to correct for  differences in π pulse duration due to different positions on the microwave circuit, thereby  improving the accuracy of the T1 and T2 measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The π/2 pulse was set to half of the π  pulse time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Figure 7 (c) shows the results of the energy relaxation time T1 measurement at the  point (x, y) = (-200um, 100um), fitted by the function 𝑎1 𝑒𝑥𝑝(−τ/𝑎2) + 𝑎3, where τ is the time  duration between π pulse and readout, 𝑎𝑖 (i = 1 ~ 3) is a fitting parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' In T1 measurement,  π pulses of microwaves were irradiated, and the quantum state was read out as a  photoluminescence intensity by a 532 nm laser excitation after a time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Figure 7 (d) shows the  spatial distribution of the energy relaxation time T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The average value over the entire  measurement region of the sample was 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='0 ms with a standard deviation of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='0 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Figure 7  (e) is the result of the phase relaxation time T2 measurement at the point (x, y) = (-200um,  100um), fitted by the function 𝑎1 𝑒𝑥𝑝(−(τ/𝑎2)𝑎3), where τ is the time between the π pulse and  the readout, 𝑎𝑖 (i = 1 ~ 3) is a fitting parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The microwave pulse sequence is π/2 pulse- π  pulse - π/2, with π/2 pulse and π pulse separated by time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' After an exposure of the microwave  pulse sequence, the quantum states were read out with a 532 nm laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Figure 7(f) shows the  spatial distribution of the phase relaxation time T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The average value over the entire  measurement region of the sample was 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='5 us and the standard deviation was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='9 us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The value  of T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' is gradually decreasing from the left-side area to the right-side area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' This trend of T2  spatial distribution corresponds to the trend of the spatial distribution of fluorescence from the  NV− center, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' S1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 7(f) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' S1 (b) indicate that T2 is decreasing in  increasing density of nitrogen and NV− center as a decoherence source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' 7 (a) Rabi oscillation measurement at the point (x, y) = (-200um, 100um).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (b) Spatial  distribution of π pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (c) Energy relaxation time T1 measurement at the point (x, y) =  (-200um, 100um).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (d) Spatial distribution of NV center energy relaxation time T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (e) Phase  relaxation time T2 measured at the point (x, y) = (-200um, 100um).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' (f) Spatial distribution of  phase relaxation time T2 at the NV center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Conclusion  In summary, we have developed a method to rapidly measure the sample information such as  quantum properties of an entire mm-scale sample by constructing a confocal microscope with  a long Rayleigh length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' By considering the excitation volume, the polarization ratio of the  quantum state, and the detection efficiency, we found that matching twice the Rayleigh length  to the thickness of the sample produces the highest amount of detected signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The detected  signal is about 104 times larger than the conventional ideal confocal microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' LRCFM uses  an identical microwave setup with conventional confocal microscopy to evaluate color centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  T pulse time = 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='0 ns  107  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='00  1000  106  I signal  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='98  800  105  Normalized  (μm)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='96  600  104  >  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='94  103  400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='92  102  200  101  0  100  200  300  400  500  200  0  Time (ns)  x (um)  T1 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='5 ms  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='004  1000  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='003  800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='002  (un)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='5  (sw)  600  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='001  >  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='000  400  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='001  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='002  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='0  0  5  10  15  200  0  Time (ms)  X (μm)  T2 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='4 μs, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='22  26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='0025  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='0020  800  24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='0015  (wr)  (srl)  600  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='0010  >  22 /2  400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='0005  200  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='0000  50  0  100  200  0  Time (μs)  X (μm)Thus, it is possible to use these microwave pulse sequences for LRCFM, such as evaluating  NV center density using instantaneous diffusion [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Density evaluation of spin defects in  the quantum material, usually performed on the entire sample by ESR [24], can be  accomplished using LRCFM, with additional information on spatial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' In addition,  this method can be used not only for the evaluation of existing color centers, but also for the  discovery of new color centers, as it can take advantage of the high signal-to-noise ratio to  increase the speed for measuring photoluminescence spectra of unknown color centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' [25,  26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Method  A diamond sample containing NV centers was placed on the microwave resonator [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The  resonator is fixed to a motorized stage with a movable range of 13 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' A permanent magnet  to fix the quantization axis was placed behind the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The strength of the magnetic field  was approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='5 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The fluorescence from the NV centers was collected through the  objective lens (AC254-030-AB-ML;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Thorlabs) and detected using an avalanche photodiode  (APD410A/M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Thorlabs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' An oscilloscope (MDO34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Tektronix) recorded the signal from the  avalanche photodiode and analyzed it to distinguish the spin state of the NV centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Using an  arbitrary waveform generator (M3202A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Keysight) to generate the laser and microwave pulse  sequence, we controlled an acousto-optic modulator (#35 250-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='53-XQ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Gooch &  Housego) and a transistor-transistor logic (TTL) switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' After the microwave pulse sequence,  a 180 µs measurement laser pulse of 532 nm laser (gem 532;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Laser Quantum) for measurement  with 12 mW was irradiated to the NV centers in the diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The diamond {111} single crystal  grown by high-temperature/high-pressure (HPHT) synthetic method was used in this study [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  The HPHT sample was irradiated with a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='0 MeV electron beam with a total fluence of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='0 ×  1017 cm−2 for creating vacancies in the crystal and then annealed at 1000 ◦C for 2 hours in  vacuum to create NV centers (see supplemental material for the details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Data availability  The data that support the findings of this study are available from the corresponding author  upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Taylor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' High-sensitivity diamond magnetometer with nanoscale resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
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+page_content='  Acknowledgements  This work was supported by MEXT Q-LEAP (JPMXS0118067395 and JPMXS0118068379).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  YM acknowledges the support of JSPS KAKENHI (20K14392).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' acknowledges the  support of JST Moonshot R&D (JPMJMS2062), MIC R&D for construction of a global  quantum cryptography network (JPMI00316) and JSPS KAKENHI (20H02187, 20H05661).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Author information  Authors and Affiliations  National Institutes for Quantum Science and Technology, Takasaki, Gunma,  370 – 1292, Japan  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Masuyama, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Ishii, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Abe & T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Ohshima  National Institute for Materials Science, Tsukuba, Ibaraki 305 – 0044, Japan  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Shinei, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Taniguchi & T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Teraji  Contributions  The measurement system was designed, constructed, and tested by Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  performed the measurements and the data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' synthesized the diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The diamond  was electron-irradiated by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' The overall supervision was performed by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content=' and  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
+page_content='  Corresponding author  Correspondence to Yuta Masuyama  Ethics declarations  Competing Interests  The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFMT4oBgHgl3EQf1zGx/content/2301.12441v1.pdf'}
diff --git a/gdAyT4oBgHgl3EQfxPne/content/tmp_files/2301.00665v1.pdf.txt b/gdAyT4oBgHgl3EQfxPne/content/tmp_files/2301.00665v1.pdf.txt
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+arXiv:2301.00665v1  [cs.CL]  30 Dec 2022
+Targeted Phishing Campaigns using Large Scale
+Language Models
+Rabimba Karanjai
+Department of Computer Science
+University Of Houston
+Houston, United States
+rkaranjai@uh.edu
+Abstract—Natural language models (NLMs) such as GPT-3,
+GPT-2, and other large language models have achieved impressive
+results in various natural language processing tasks, including
+language translation, summarization, and text generation. In
+recent years, there has been a growing concern about the
+potential use of NLMs to generate phishing emails, which are
+fraudulent emails that trick individuals into revealing sensitive
+information or performing actions that benefit the attackers.
+This research paper aims to investigate the feasibility and
+effectiveness of NLMs in generating phishing emails. To this end,
+we propose a framework for evaluating the performance of NLMs
+in generating phishing emails based on various metrics, including
+the quality of the generated text, the ability to bypass spam filters,
+and the success rate of tricking individuals into falling for the
+phishing attack.
+We evaluate the performance of several NLMs on a dataset
+of phishing emails and compare their results with those of a
+baseline model. Our results show that NLMs can indeed generate
+phishing emails that are difficult to detect and that have a high
+success rate in tricking individuals. However, we also find that
+the performance of NLMs in generating phishing emails depends
+on the specific NLM and the training data used, and that there
+are limitations to their effectiveness.
+Overall, our research suggests that NLMs have the potential
+to significantly impact the landscape of phishing attacks and
+highlights the need for further research on the ethical and
+security implications of using NLMs for malicious purposes.
+I. INTRODUCTION
+Recent advances in natural language generation (NLG) have
+greatly improved the diversity, control, and quality of machine-
+generated text. However, this increased ability to quickly
+and efficiently create unique, manipulable, human-like text
+also presents new challenges for detecting the abuse of NLG
+models in phishing attacks.
+Machine-generated texts can pose various risks depending
+on the context and how they are used. For example, in the case
+of NLG models, the ability to generate legitimate texts atht
+looks like emails can lead to attacks like phishing, where the
+attacker tricks the victim into disclosing sensitive information
+by impersonating someone else.
+Another effect of machine generated text is mass dis-
+information campaigns. With the ability to generate large
+amounts of text automatically and quickly, it is possible for
+malicious actors to create fake news, hoaxes, and other forms
+of false or misleading information that can harm individuals,
+organizations, and even entire societies.
+Moreover, machine-generated texts can also raise ethical
+concerns, such as the impact on employment and the potential
+for bias and discrimination. For example, the use of NLG
+models to automate certain writing tasks may lead to job losses
+for human writers, and the algorithms used in NLG may reflect
+and amplify the biases and stereotypes present in the data they
+are trained on.
+Abuses
+of
+NLG
+models,
+such
+as
+phishing
+[1],
+[2],disinformation[3], [4], [5] has been on the rise.
+Email is a common method used by phishers to deliver
+malicious links and attachments to victims. Anti-Phishing
+Working Group found over 121860 phishing email incidents
+in march 2017 and in 2016, the APWG received more than
+1313771 unique phishing reports. In the first quarter of 2017,
+around 870 organizations were targeted by W2-based phishing
+scams, a significant increase from the 100 organizations in
+2016. These attacks are becoming more sophisticated and
+difficult to detect.
+Phishers often use techniques such as bulk mailing, spam-
+ming, and including action words and links in phishing emails
+to increase their chances of success. However, these techniques
+can be easily detected by improved statistical detection mod-
+els. Another popular method is email masquerading, where the
+attacker gains access to the victim’s email inbox or outbox
+and studies the content and nature of the emails to create a
+synthetic malicious email that resembles a benign one. This
+reduces the chances of detection by automated classifiers and
+increases the likelihood of a successful attack. Modern large
+language models have enabled users to generate text based
+on context. These models can be trained to generate text
+using predefined grammars, such as the Dada Engine[1], or
+by leveraging deep learning neural networks, such as recurrent
+neural networks (RNNs)[6], to learn and emulate the input to
+the system.
+NLG systems that use advanced deep learning neural net-
+works (DNNs) can be used by phishers to generate coherent
+and convincing sequences of text. These systems have been
+shown to be effective for generating text in various genres,
+from tweets[7] to poetry[8]. It is likely that phishers and
+spammers will soon start using email datasets, both legitimate
+and malicious, in conjunction with DNNs to create decep-
+tive malicious emails that mimic the properties of legitimate
+emails. This makes it harder for pre-trained email detectors to
+
+identify and block these attacks.
+In this report, we try to show a class of attacks where
+existing large-scale language models have been trained on both
+legitimate and malicious (phishing and spam) email data. We
+also aim to show how the generated emails can bypass existing
+production-level email protection mechanisms and propose a
+future work to detect such attacks.
+II. RELATED WORK
+Phishing detection is a well-studied area in cybersecurity,
+but many victims still fall for these attacks. In their work,
+Drake et al [9] provide a detailed analysis of the structure
+and tactics used in phishing emails. In this section, we
+review previous research on natural language generation, deep
+learning, and their applications in generating and detecting
+phishing attacks.
+Natural language generation techniques have been widely
+used to synthesize unique pieces of text. Previous work by
+Reiter and Dale et al [10] relied on pre-constructed templates
+for specific purposes, while the fake email generation system
+in Baki et al[1] used manually constructed rules to define the
+structure of fake emails. Recent advances in deep learning
+have enabled the generation of creative and equitable text
+with enough training data. RNN(Recurrent Neural Networks)
+language models are used to generate a range of genres,
+including poetry by Ghazvininejad et al [8], fake reviews by
+Yao et al [6], tweets [7], and geographical information by
+Turner et al [11], among others.
+III. EXPERIMENTAL METHODOLOGY
+The section is divided into four subsections. The first
+subsection (Section 3.1) describes the nature and source of the
+training and evaluation data. The second subsection (Section
+3.2) discusses the pre-processing steps applied to the data. The
+third subsection (Section 3.3) presents the system setup and
+experimental settings used in the study.
+A. Data Description
+To create a legitimate looking phishing email we first need
+to start from actually benign and legitimate emails. The text
+generation algorithms must be trained in legitimate emails.
+Hence it was imperative to have valid benign emails in the
+dataset used for training. However, since the goal here is to
+create emails that even though can serve as a phishing email,
+should still look like legitimate emails, a mix of legitimate and
+bad emails was used as a dataset for training and augmenting
+the models.
+For legitimate datasets, instead of using one dataset on our
+own, we use pre-trained models from Meta and Google to cre-
+ate benign emails. The pre-trained models utilized are Roberta,
+The Pile, and PushShift.io Reddit. Since training these large
+language models is almost impossible in normal infrastructure,
+we utilize [12] to generate the texts. This has been augmented
+with [13] to have email generation capabilities. Python clean
+text [14] has been used to remove email, and phone numbers
+from the dataset.
+For malicious datasets, we primarily use two datasets to
+augment the benign email data. Notably, the Phishing emails
+from Jose Nazario’s Phishing corpus [15] and [16] along with
+the Enron email dataset [17].
+B. Data Processing
+Most of the pre-processing was done by trying to remove
+personal information using Python clean text [14]. As well as
+Removal of special characters like , #, $, % as well as common
+punctuations from the email body.
+However, as we have realized later generating emails was
+not perfect.
+C. Experimental Setup
+The experimental setup has been designed with certain
+different methods in mind. We primarily focused on
+• Using GPT-2 to generate emails. Augmented with email
+dataset [18]
+• GPT-3 to generate emails without any training
+• Contextual support for GPT-3 with da-vinci-beta which
+has been trained in email by openai
+• The DADA engine [1]
+• Word based RNN’s proposed by Xie et al [19], Das et al
+[20]
+• Augmenting Open Pre-trained Transformer Language
+Models[12] on [13]
+While using the general large language models were
+interesting in trying to produce emails.
+The spam and phishing email datasets used for train-
+ing the models to produce malicious looking email
+produced better results. The Jose Nazario dataset has
+32,000 spams and 415 phishing email. These are all
+in Unix mbox formatted dataset which were cleaned
+using clean-text.
+The Enron corpus was email dataset from Enron
+Corporation. It has been used in email research for
+quite some time and was made public during their legal
+investigation. This however gives us a rare glimpse
+in the working of legitimate email data of a big
+corporation. It consists of 619446 emails from 158
+users. It has folders like ”discussion threads, ”notes
+inbox” and more in its outlook inboxes. These were
+created by default for most of the users but were nota
+ctively used for tagging. The ”All Documents” folder
+present also had large number of duplicates present
+from inbox. To further clean our dataset, these all were
+removed. The cleaned corpus has 200399 emails from
+158 users.
+We have tried to validate our ways with existing find-
+ings available here https://github.com/egmp777/basic
+data cleaning enron case study. But our purpose
+was different than exploratory analysis.
+
+D. Experiment
+A PoC was done to evaluate if we can auto-detect any emails
+from the generated text. GPT3 API, GPT2, OPT, Chatbot
+and our trained models were all used to generate the texts.
+Some samples have been provided below. In our test we had
+generated 100 samples using each. However some of these
+gave us inconsistent result for a same prompt. We also had
+sample fo real phishing email (only one for that topic) to
+compare it with.
+A variety fo prompts were used to see what kind of emails
+the AI’s are more comfortable generating.
+We applied the GPT-3 davinci model with default sampling
+parameters (temperature=1, top p=1, presence penalty=0, fre-
+quency penalty=0) to predict the log probabilities for the first
+100 tokens in each sample using all the preceding tokens as
+the prompt. We then calculated the actual token’s probability,
+absolute rank, contextual entropy, and whether it matched
+the predicted token by the GPT-3 API. To account for the
+maximum of 100 log probabilities returned by the API, we
+masked out any outliers with an absolute rank greater than
+100. We also normalized the results if there were less than
+100 tokens in a sample.
+IV. ANALYSIS AND RESULTS
+1) Example Results generated: ChatGPT: The following
+was generated with the prompt Can you generate an email that
+will be from The President of a University asking its students
+to fill up course completion survey form
+Dear Students,
+As the academic year comes to a close, we would
+like to gather your feedback on your experiences and
+courses at the University. To help us improve our
+programs and support your learning, we encourage
+you to take a few minutes to complete the course
+completion survey form.
+Your honest and thoughtful responses will be greatly
+appreciated and will help us better understand your
+needs and expectations. The survey form is available
+online and can be accessed through your student portal.
+We thank you in advance for your participation and
+look forward to hearing from you.
+Best regards,
+[Name], President [University]
+GPT3 da-vinci: The following was generated with the
+prompt Can you generate an email that will be from The
+President of a University asking its students to fill up course
+completion survey form
+Hello,
+As the President of the University, I am writing to
+request your help in completing a course completion
+survey form. This form will help us to understand how
+well our students are doing in their courses and how
+we can improve our offerings.
+I would greatly appreciate it if you could take a few
+minutes to fill out the form. You can access the form
+at the following link:
+[insert form URL here]
+Thank you for your time and assistance.
+Sincerely,
+[Your Name]
+GPT2: The following was generated with the prompt of
+Hello, As the President of the University, I am writing to
+request your help in completing a course completion survey
+form.
+Hello,
+As the President of the University, I am writing to
+request your help in completing a course completion
+survey form. As requested, we would like your answer
+by May NUMBER, NUMBER. The survey should
+only take a few minutes of your time (it takes only
+about NUMBER minutes to complete the survey). All
+survey responses are anonymous and confidential, and
+will not be shared outside of our University commu-
+nity. If you have
+OPT: The following was generated with the prompt of
+Hello, As the President of the University, I am writing to
+request your help in completing a course completion survey
+form.
+(¡PROMPT¿ Hello, As the President of the University,
+I am writing to request your help in completing a
+course completion survey form.¡END-OF-PROMPT¿
+The purpose of this survey is to gather feedback
+from ¡NUMBER¿ current and former students on their
+experiences at UofH. Please complete the survey no
+later than Friday, October 00th by sending it back to
+me via fax no. ¡PHONE¿. Thank you very much for
+your time. Best Regards,
+UofH Office of Admissions ¡EMAIL¿ (See attached
+file: uofh0000 sample.pdf))
+The most interesting part of this prompt was the ending
+where it signed off as UofH Office of admissions and a sample
+pdf. Whereas our prompt never indicates those details[21].
+Further testing with the GPT2 dataset produced the
+following output from the same prompt [21]
+
+Hello,
+As the President of the University, I am writing to
+request your help in completing a course completion
+survey form. By completing this form, you are wel-
+come to access the COMPANY Club website and ask
+additional questions about the UCLA Club and our
+events. To access the website you may click the link
+at the top of this page.If you prefer to not complete
+this form at this time, please let me know and I will
+Notably, UCLA was not present in the prompt. This shows
+us that with enough clever prompt discovery it is probably
+possible to extract meaningful information from the trained
+dataset even with safeguards in place.
+2) Training Parameters: The training parameters used for
+the HF opt model was
+• learning rate: 6e−5
+• train batch size: 8
+• eval batch size: 8
+• seed: 42
+• distributed type: GPU
+• gradient accumulation steps: 16
+• total train batch size: 128
+• optimizer: Adam with betas = (0.9, 0.999) and epsilon =
+1e−8
+• lr scheduler type: cosine
+• lr scheduler warmup ratio: 0.03
+• num epochs: 8
+And the training parameters used for HF postbot GPT2
+• learning rate: 0.001
+• train batch size: 16
+• eval batch size: 16
+• seed: 42
+• distributed type: multi-GPU
+• gradient accumulation steps: 8
+• total train batch size: 128
+• optimizer: Adam with betas = (0.9, 0.999) and epsilon =
+1e−8
+• lr scheduler type: cosine
+• lr scheduler warmup ratio: 0.02
+• num epochs: 3
+V. FUTURE WORK
+Research on the risks of using natural language generation
+(NLG) models suggests that being able to detect machine-
+generated text is useful for reducing the harm caused by abuse
+of these models. When we want to detect machine-generated
+text, it can be treated as a binary classification problem. We
+train a classifier to differentiate between machine-generated
+and human-generated text [22].
+We can use generative models without fine-tuning to detect
+their own outputs or the outputs of other similar models.
+Autoregressive generative models like GPT-2, GPT-3 are uni-
+directional, where each token has an embedding that depends
+on the embeddings of the tokens that come before it. This
+shows us that an embedding can be created if we add a token
+at the end of an input sequence, thus creating a sequence
+of tokens. This now can be used as a new feature vector.
+Now once we have these newly created features, they can be
+utilized along with human data to train a layer of neurons for
+classification.
+Research on how to detect machine-generated text has
+looked at the problem of detecting text when a different
+dataset was used to train RoBERTa than GPT-2. But here,
+it was observed that just tuning the detection model with
+couple of hundred different attack samples provided by domain
+esperts had a significant effect on the detector’s performance
+on different domains[23].
+One another possibility is when an attacker decides to gener-
+ate the attack from an existing hand-written content. Much like
+how we have started in this email generation problem. Using
+human like sample but tweaking the generating parameters to
+closely meet his goals. Analysis showed that making these
+targeted changes to texts reduces the effectiveness of GPT-2
+or RoBerta-based detectors [24].
+A generalized solution to this is trying to differentiate
+between human and machine generated text. Giant Language
+Model Test Room is a software developed to improve the
+detection of machine-generated text by adding human review
+in the pipeline. The tool helps humans classify text by high-
+lighting texts based on how likely of them being chosen by the
+Transformer model. However, this tool was designed to target
+GPT-2, which was found to be easier for untrained human
+evaluators to detect. In addition, GLTR uses ”top-k” sampling
+to determine the likelihood of a word being selected, but this
+method has been largely replaced by nucleus sampling, which
+is used in GPT-3 and other works that build on the GPT-
+2 architecture. While highlighting words based on sampling
+likelihood may improve human classification ability, it is clear
+that it still will pose a problem when they have to detect the
+more advanced models and sampling methods of today.
+In long term, we want to propose a framework that can
+differentiate NLG-generated emails from human-generated
+emails. Prior work has already been done trying to determine
+machine-generated text, however specifically for email and
+malicious emails, there are distinct characteristics we have
+observed that can be exploited to augment prior works to be
+more effective. Few of these are homogeneous to what we
+have seen in language models [25], but some are significantly
+distinct and should be explored more.
+VI. CONCLUSION
+The more we experimented with large language models
+and prior works by Das et al [20], Baki et al [1] it became
+clear that prior RNN-based models and DIDA engines, even
+though show some malicious intent in their generation, don’t
+actually pose threat to be understood as real malicious email.
+All of them went past Gmail and outlook when sent from
+a legitimate email id. The emails generated by GPT3 and
+OPT significantly pose a larger threat to be believed as real
+emails when generated in mass using tools and bulk emailed
+
+with targeted intent. Especially with targeted email dataset
+training and keywords in prompts, the models generated very
+convincing-looking emails. Even with safeguards in place for
+GPT3, we were able to generate these emails and chatGPT
+was a very interesting contender in the tests. Even though
+chatgpt didn’t let us generate the email directly in one go, we
+were able to find creative ways by ’conversing’ with it and
+giving it a plausible context to overcome its barriers. Here we
+identify how these new language models can be weaponized
+to be used as phishing and scamming tools which gets past our
+present email systems like Gmail and Outlook. However, that’s
+hardly surprising considering they look legitimate. We want to
+further this work by integrating it with tools like PhEmail[26]
+which makes sending NLG generated emails to targeted bulk
+userbase a keypress away.
+REFERENCES
+[1] S. Baki, R. Verma, A. Mukherjee, and O. Gnawali, “Scaling and
+effectiveness of email masquerade attacks: Exploiting natural language
+generation,” in Proceedings of the 2017 ACM on Asia Conference on
+Computer and Communications Security, 2017, pp. 469–482.
+[2] A. Giaretta and N. Dragoni, “Community targeted phishing,” in Inter-
+national Conference in Software Engineering for Defence Applications.
+Springer, 2018, pp. 86–93.
+[3] K. Shu, S. Wang, D. Lee, and H. Liu, “Mining disinformation and fake
+news: Concepts, methods, and recent advancements,” in Disinformation,
+misinformation, and fake news in social media.
+Springer, 2020, pp. 1–
+19.
+[4] H. Stiff and F. Johansson, “Detecting computer-generated disinforma-
+tion,” International Journal of Data Science and Analytics, vol. 13, no. 4,
+pp. 363–383, 2022.
+[5] R. Zellers, A. Holtzman, H. Rashkin, Y. Bisk, A. Farhadi, F. Roesner,
+and Y. Choi, “Defending against neural fake news,” Advances in neural
+information processing systems, vol. 32, 2019.
+[6] Y. Yao, B. Viswanath, J. Cryan, H. Zheng, and B. Y. Zhao, “Auto-
+mated crowdturfing attacks and defenses in online review systems,” in
+Proceedings of the 2017 ACM SIGSAC conference on computer and
+communications security, 2017, pp. 1143–1158.
+[7] P. Sidhaye and J. C. K. Cheung, “Indicative tweet generation: An extrac-
+tive summarization problem?” in Proceedings of the 2015 Conference on
+Empirical Methods in Natural Language Processing, 2015, pp. 138–147.
+[8] M. Ghazvininejad, X. Shi, Y. Choi, and K. Knight, “Generating topical
+poetry,” in Proceedings of the 2016 Conference on Empirical Methods
+in Natural Language Processing, 2016, pp. 1183–1191.
+[9] C. E. Drake, J. J. Oliver, and E. J. Koontz, “Anatomy of a phishing
+email.” in CEAS, 2004.
+[10] M. A. Covington, “Building natural language generation systems,”
+Language, vol. 77, no. 3, pp. 611–612, 2001.
+[11] R. Turner, S. Sripada, and E. Reiter, “Generating approximate geo-
+graphic descriptions,” in Empirical methods in natural language gen-
+eration.
+Springer, 2009, pp. 121–140.
+[12] S. Zhang, S. Roller, N. Goyal, M. Artetxe, M. Chen, S. Chen, C. Dewan,
+M. Diab, X. Li, X. V. Lin et al., “Opt: Open pre-trained transformer
+language models,” arXiv preprint arXiv:2205.01068, 2022.
+[13] R. Zhang and J. Tetreault, “This email could save your life: In-
+troducing the task of email subject line generation,” arXiv preprint
+arXiv:1906.03497, 2019.
+[14] (2022)
+clean-text
+·
+pypi.
+[Online].
+Available:
+https://pypi.org/project/clean-text/
+[15] H. Gonzalez, K. Nance, and J. Nazario, “Phishing by form: The abuse
+of form sites,” in 2011 6th International Conference on Malicious and
+Unwanted Software.
+IEEE, 2011, pp. 95–101.
+[16] (2000)
+Jose
+malicious
+email
+dataset:
+https://monkey.org/
+jose/wiki/doku.php
+Link
+Deprecated,
+Uploaded
+to
+my
+own
+github.
+[Online].
+Available:
+https://monkey.org/∼jose/wiki/doku.php
+[17] J. Shetty and J. Adibi, “The enron email dataset database schema and
+brief statistical report,” Information sciences institute technical report,
+University of Southern California, vol. 4, no. 1, pp. 120–128, 2004.
+[18] (2022)
+email-blog
+—
+kaggle:
+https://www.kaggle.com/datasets/mikeschmidtavemac/emailblog.
+[On-
+line]. Available: https://www.kaggle.com/datasets/mikeschmidtavemac/emailblog
+[19] Z. Xie, “Neural text generation: A practical guide,” arXiv preprint
+arXiv:1711.09534, 2017.
+[20] A. Das and R. Verma, “Automated email generation for targeted attacks
+using natural language,” arXiv preprint arXiv:1908.06893, 2019.
+[21] (2022)
+rabimba/email-gen-nlg:
+https://github.com/rabimba/email-gen-
+nlg. [Online]. Available: https://github.com/rabimba/email-gen-nlg
+[22] E. Crothers, N. Japkowicz, H. Viktor, and P. Branco, “Adversarial
+robustness of neural-statistical features in detection of generative trans-
+formers,” arXiv preprint arXiv:2203.07983, 2022.
+[23] J. Rodriguez, T. Hay, D. Gros, Z. Shamsi, and R. Srinivasan, “Cross-
+domain detection of gpt-2-generated technical text,” in Proceedings of
+the 2022 Conference of the North American Chapter of the Association
+for Computational Linguistics: Human Language Technologies, 2022,
+pp. 1213–1233.
+[24] M. M. Bhat and S. Parthasarathy, “How effectively can machines
+defend against machine-generated fake news? an empirical study,” in
+Proceedings of the First Workshop on Insights from Negative Results in
+NLP, 2020, pp. 48–53.
+[25] S. Gehrmann, H. Strobelt, and A. M. Rush, “GLTR: statistical detection
+and visualization of generated text,” CoRR, vol. abs/1906.04043, 2019.
+[Online]. Available: http://arxiv.org/abs/1906.04043
+[26] (2022) dionach/phemail: Phemail is a python open source phishing
+email
+tool
+that
+automates
+the
+process
+of
+sending
+phishing
+emails as part of a social engineering test. [Online]. Available:
+https://github.com/Dionach/PhEmail
+
diff --git a/gdAyT4oBgHgl3EQfxPne/content/tmp_files/load_file.txt b/gdAyT4oBgHgl3EQfxPne/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf,len=333
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='00665v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='CL]  30 Dec 2022  Targeted Phishing Campaigns using Large Scale  Language Models  Rabimba Karanjai  Department of Computer Science  University Of Houston  Houston, United States  rkaranjai@uh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='edu  Abstract—Natural language models (NLMs) such as GPT-3,  GPT-2, and other large language models have achieved impressive  results in various natural language processing tasks, including  language translation, summarization, and text generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' In  recent years, there has been a growing concern about the  potential use of NLMs to generate phishing emails, which are  fraudulent emails that trick individuals into revealing sensitive  information or performing actions that benefit the attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  This research paper aims to investigate the feasibility and  effectiveness of NLMs in generating phishing emails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' To this end,  we propose a framework for evaluating the performance of NLMs  in generating phishing emails based on various metrics, including  the quality of the generated text, the ability to bypass spam filters,  and the success rate of tricking individuals into falling for the  phishing attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  We evaluate the performance of several NLMs on a dataset  of phishing emails and compare their results with those of a  baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Our results show that NLMs can indeed generate  phishing emails that are difficult to detect and that have a high  success rate in tricking individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' However, we also find that  the performance of NLMs in generating phishing emails depends  on the specific NLM and the training data used, and that there  are limitations to their effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  Overall, our research suggests that NLMs have the potential  to significantly impact the landscape of phishing attacks and  highlights the need for further research on the ethical and  security implications of using NLMs for malicious purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' INTRODUCTION  Recent advances in natural language generation (NLG) have  greatly improved the diversity, control, and quality of machine-  generated text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' However, this increased ability to quickly  and efficiently create unique, manipulable, human-like text  also presents new challenges for detecting the abuse of NLG  models in phishing attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  Machine-generated texts can pose various risks depending  on the context and how they are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' For example, in the case  of NLG models, the ability to generate legitimate texts atht  looks like emails can lead to attacks like phishing, where the  attacker tricks the victim into disclosing sensitive information  by impersonating someone else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  Another effect of machine generated text is mass dis-  information campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' With the ability to generate large  amounts of text automatically and quickly, it is possible for  malicious actors to create fake news, hoaxes, and other forms  of false or misleading information that can harm individuals,  organizations, and even entire societies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  Moreover, machine-generated texts can also raise ethical  concerns, such as the impact on employment and the potential  for bias and discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' For example, the use of NLG  models to automate certain writing tasks may lead to job losses  for human writers, and the algorithms used in NLG may reflect  and amplify the biases and stereotypes present in the data they  are trained on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  Abuses  of  NLG  models,  such  as  phishing  [1],  [2],disinformation[3], [4], [5] has been on the rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  Email is a common method used by phishers to deliver  malicious links and attachments to victims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Anti-Phishing  Working Group found over 121860 phishing email incidents  in march 2017 and in 2016, the APWG received more than  1313771 unique phishing reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' In the first quarter of 2017,  around 870 organizations were targeted by W2-based phishing  scams, a significant increase from the 100 organizations in  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' These attacks are becoming more sophisticated and  difficult to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  Phishers often use techniques such as bulk mailing, spam-  ming, and including action words and links in phishing emails  to increase their chances of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' However, these techniques  can be easily detected by improved statistical detection mod-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Another popular method is email masquerading, where the  attacker gains access to the victim’s email inbox or outbox  and studies the content and nature of the emails to create a  synthetic malicious email that resembles a benign one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' This  reduces the chances of detection by automated classifiers and  increases the likelihood of a successful attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Modern large  language models have enabled users to generate text based  on context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' These models can be trained to generate text  using predefined grammars, such as the Dada Engine[1], or  by leveraging deep learning neural networks, such as recurrent  neural networks (RNNs)[6], to learn and emulate the input to  the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  NLG systems that use advanced deep learning neural net-  works (DNNs) can be used by phishers to generate coherent  and convincing sequences of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' These systems have been  shown to be effective for generating text in various genres,  from tweets[7] to poetry[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' It is likely that phishers and  spammers will soon start using email datasets, both legitimate  and malicious, in conjunction with DNNs to create decep-  tive malicious emails that mimic the properties of legitimate  emails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' This makes it harder for pre-trained email detectors to  identify and block these attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  In this report, we try to show a class of attacks where  existing large-scale language models have been trained on both  legitimate and malicious (phishing and spam) email data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' We  also aim to show how the generated emails can bypass existing  production-level email protection mechanisms and propose a  future work to detect such attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' RELATED WORK  Phishing detection is a well-studied area in cybersecurity,  but many victims still fall for these attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' In their work,  Drake et al [9] provide a detailed analysis of the structure  and tactics used in phishing emails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' In this section, we  review previous research on natural language generation, deep  learning, and their applications in generating and detecting  phishing attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  Natural language generation techniques have been widely  used to synthesize unique pieces of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Previous work by  Reiter and Dale et al [10] relied on pre-constructed templates  for specific purposes, while the fake email generation system  in Baki et al[1] used manually constructed rules to define the  structure of fake emails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Recent advances in deep learning  have enabled the generation of creative and equitable text  with enough training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' RNN(Recurrent Neural Networks)  language models are used to generate a range of genres,  including poetry by Ghazvininejad et al [8], fake reviews by  Yao et al [6], tweets [7], and geographical information by  Turner et al [11], among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' EXPERIMENTAL METHODOLOGY  The section is divided into four subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' The first  subsection (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='1) describes the nature and source of the  training and evaluation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' The second subsection (Section  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='2) discusses the pre-processing steps applied to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' The  third subsection (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='3) presents the system setup and  experimental settings used in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Data Description  To create a legitimate looking phishing email we first need  to start from actually benign and legitimate emails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' The text  generation algorithms must be trained in legitimate emails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  Hence it was imperative to have valid benign emails in the  dataset used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' However, since the goal here is to  create emails that even though can serve as a phishing email,  should still look like legitimate emails, a mix of legitimate and  bad emails was used as a dataset for training and augmenting  the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  For legitimate datasets, instead of using one dataset on our  own, we use pre-trained models from Meta and Google to cre-  ate benign emails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' The pre-trained models utilized are Roberta,  The Pile, and PushShift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='io Reddit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Since training these large  language models is almost impossible in normal infrastructure,  we utilize [12] to generate the texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' This has been augmented  with [13] to have email generation capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Python clean  text [14] has been used to remove email, and phone numbers  from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  For malicious datasets, we primarily use two datasets to  augment the benign email data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Notably, the Phishing emails  from Jose Nazario’s Phishing corpus [15] and [16] along with  the Enron email dataset [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Data Processing  Most of the pre-processing was done by trying to remove  personal information using Python clean text [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' As well as  Removal of special characters like , #, $, % as well as common  punctuations from the email body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  However, as we have realized later generating emails was  not perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Experimental Setup  The experimental setup has been designed with certain  different methods in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' We primarily focused on  Using GPT-2 to generate emails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Augmented with email  dataset [18]  GPT-3 to generate emails without any training  Contextual support for GPT-3 with da-vinci-beta which  has been trained in email by openai  The DADA engine [1]  Word based RNN’s proposed by Xie et al [19], Das et al  [20]  Augmenting Open Pre-trained Transformer Language  Models[12] on [13]  While using the general large language models were  interesting in trying to produce emails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  The spam and phishing email datasets used for train-  ing the models to produce malicious looking email  produced better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' The Jose Nazario dataset has  32,000 spams and 415 phishing email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' These are all  in Unix mbox formatted dataset which were cleaned  using clean-text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  The Enron corpus was email dataset from Enron  Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' It has been used in email research for  quite some time and was made public during their legal  investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' This however gives us a rare glimpse  in the working of legitimate email data of a big  corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' It consists of 619446 emails from 158  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' It has folders like ”discussion threads, ”notes  inbox” and more in its outlook inboxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' These were  created by default for most of the users but were nota  ctively used for tagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' The ”All Documents” folder  present also had large number of duplicates present  from inbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' To further clean our dataset, these all were  removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' The cleaned corpus has 200399 emails from  158 users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  We have tried to validate our ways with existing find-  ings available here https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='com/egmp777/basic  data cleaning enron case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' But our purpose  was different than exploratory analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Experiment  A PoC was done to evaluate if we can auto-detect any emails  from the generated text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' GPT3 API, GPT2, OPT, Chatbot  and our trained models were all used to generate the texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  Some samples have been provided below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' In our test we had  generated 100 samples using each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' However some of these  gave us inconsistent result for a same prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' We also had  sample fo real phishing email (only one for that topic) to  compare it with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  A variety fo prompts were used to see what kind of emails  the AI’s are more comfortable generating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  We applied the GPT-3 davinci model with default sampling  parameters (temperature=1, top p=1, presence penalty=0, fre-  quency penalty=0) to predict the log probabilities for the first  100 tokens in each sample using all the preceding tokens as  the prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' We then calculated the actual token’s probability,  absolute rank, contextual entropy, and whether it matched  the predicted token by the GPT-3 API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' To account for the  maximum of 100 log probabilities returned by the API, we  masked out any outliers with an absolute rank greater than  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' We also normalized the results if there were less than  100 tokens in a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' ANALYSIS AND RESULTS  1) Example Results generated: ChatGPT: The following  was generated with the prompt Can you generate an email that  will be from The President of a University asking its students  to fill up course completion survey form  Dear Students,  As the academic year comes to a close, we would  like to gather your feedback on your experiences and  courses at the University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' To help us improve our  programs and support your learning, we encourage  you to take a few minutes to complete the course  completion survey form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  Your honest and thoughtful responses will be greatly  appreciated and will help us better understand your  needs and expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' The survey form is available  online and can be accessed through your student portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  We thank you in advance for your participation and  look forward to hearing from you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  Best regards,  [Name], President [University]  GPT3 da-vinci: The following was generated with the  prompt Can you generate an email that will be from The  President of a University asking its students to fill up course  completion survey form  Hello,  As the President of the University, I am writing to  request your help in completing a course completion  survey form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' This form will help us to understand how  well our students are doing in their courses and how  we can improve our offerings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  I would greatly appreciate it if you could take a few  minutes to fill out the form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' You can access the form  at the following link:  [insert form URL here]  Thank you for your time and assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  Sincerely,  [Your Name]  GPT2: The following was generated with the prompt of  Hello, As the President of the University, I am writing to  request your help in completing a course completion survey  form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  Hello,  As the President of the University, I am writing to  request your help in completing a course completion  survey form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' As requested, we would like your answer  by May NUMBER, NUMBER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' The survey should  only take a few minutes of your time (it takes only  about NUMBER minutes to complete the survey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' All  survey responses are anonymous and confidential, and  will not be shared outside of our University commu-  nity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' If you have  OPT: The following was generated with the prompt of  Hello, As the President of the University, I am writing to  request your help in completing a course completion survey  form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  (¡PROMPT¿ Hello, As the President of the University,  I am writing to request your help in completing a  course completion survey form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='¡END-OF-PROMPT¿  The purpose of this survey is to gather feedback  from ¡NUMBER¿ current and former students on their  experiences at UofH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Please complete the survey no  later than Friday, October 00th by sending it back to  me via fax no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' ¡PHONE¿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Thank you very much for  your time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Best Regards,  UofH Office of Admissions ¡EMAIL¿ (See attached  file: uofh0000 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='pdf))  The most interesting part of this prompt was the ending  where it signed off as UofH Office of admissions and a sample  pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Whereas our prompt never indicates those details[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  Further testing with the GPT2 dataset produced the  following output from the same prompt [21]  Hello,  As the President of the University, I am writing to  request your help in completing a course completion  survey form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' By completing this form, you are wel-  come to access the COMPANY Club website and ask  additional questions about the UCLA Club and our  events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' To access the website you may click the link  at the top of this page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='If you prefer to not complete  this form at this time, please let me know and I will  Notably, UCLA was not present in the prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' This shows  us that with enough clever prompt discovery it is probably  possible to extract meaningful information from the trained  dataset even with safeguards in place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  2) Training Parameters: The training parameters used for  the HF opt model was  learning rate: 6e−5  train batch size: 8  eval batch size: 8  seed: 42  distributed type: GPU  gradient accumulation steps: 16  total train batch size: 128  optimizer: Adam with betas = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='999) and epsilon =  1e−8  lr scheduler type: cosine  lr scheduler warmup ratio: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='03  num epochs: 8  And the training parameters used for HF postbot GPT2  learning rate: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='001  train batch size: 16  eval batch size: 16  seed: 42  distributed type: multi-GPU  gradient accumulation steps: 8  total train batch size: 128  optimizer: Adam with betas = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='999) and epsilon =  1e−8  lr scheduler type: cosine  lr scheduler warmup ratio: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='02  num epochs: 3  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' FUTURE WORK  Research on the risks of using natural language generation  (NLG) models suggests that being able to detect machine-  generated text is useful for reducing the harm caused by abuse  of these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' When we want to detect machine-generated  text, it can be treated as a binary classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' We  train a classifier to differentiate between machine-generated  and human-generated text [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  We can use generative models without fine-tuning to detect  their own outputs or the outputs of other similar models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  Autoregressive generative models like GPT-2, GPT-3 are uni-  directional, where each token has an embedding that depends  on the embeddings of the tokens that come before it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' This  shows us that an embedding can be created if we add a token  at the end of an input sequence, thus creating a sequence  of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' This now can be used as a new feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  Now once we have these newly created features, they can be  utilized along with human data to train a layer of neurons for  classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  Research on how to detect machine-generated text has  looked at the problem of detecting text when a different  dataset was used to train RoBERTa than GPT-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' But here,  it was observed that just tuning the detection model with  couple of hundred different attack samples provided by domain  esperts had a significant effect on the detector’s performance  on different domains[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  One another possibility is when an attacker decides to gener-  ate the attack from an existing hand-written content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Much like  how we have started in this email generation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Using  human like sample but tweaking the generating parameters to  closely meet his goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Analysis showed that making these  targeted changes to texts reduces the effectiveness of GPT-2  or RoBerta-based detectors [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  A generalized solution to this is trying to differentiate  between human and machine generated text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Giant Language  Model Test Room is a software developed to improve the  detection of machine-generated text by adding human review  in the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' The tool helps humans classify text by high-  lighting texts based on how likely of them being chosen by the  Transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' However, this tool was designed to target  GPT-2, which was found to be easier for untrained human  evaluators to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' In addition, GLTR uses ”top-k” sampling  to determine the likelihood of a word being selected, but this  method has been largely replaced by nucleus sampling, which  is used in GPT-3 and other works that build on the GPT-  2 architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' While highlighting words based on sampling  likelihood may improve human classification ability, it is clear  that it still will pose a problem when they have to detect the  more advanced models and sampling methods of today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  In long term, we want to propose a framework that can  differentiate NLG-generated emails from human-generated  emails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Prior work has already been done trying to determine  machine-generated text, however specifically for email and  malicious emails, there are distinct characteristics we have  observed that can be exploited to augment prior works to be  more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Few of these are homogeneous to what we  have seen in language models [25], but some are significantly  distinct and should be explored more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' CONCLUSION  The more we experimented with large language models  and prior works by Das et al [20], Baki et al [1] it became  clear that prior RNN-based models and DIDA engines, even  though show some malicious intent in their generation, don’t  actually pose threat to be understood as real malicious email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content='  All of them went past Gmail and outlook when sent from  a legitimate email id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' The emails generated by GPT3 and  OPT significantly pose a larger threat to be believed as real  emails when generated in mass using tools and bulk emailed  with targeted intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Especially with targeted email dataset  training and keywords in prompts, the models generated very  convincing-looking emails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Even with safeguards in place for  GPT3, we were able to generate these emails and chatGPT  was a very interesting contender in the tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Even though  chatgpt didn’t let us generate the email directly in one go, we  were able to find creative ways by ’conversing’ with it and  giving it a plausible context to overcome its barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' Here we  identify how these new language models can be weaponized  to be used as phishing and scamming tools which gets past our  present email systems like Gmail and Outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' However, that’s  hardly surprising considering they look legitimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
+page_content=' We want to  further this work by integrating it with tools like PhEmail[26]  which makes sending NLG generated emails to targeted bulk  userbase a keypress away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAyT4oBgHgl3EQfxPne/content/2301.00665v1.pdf'}
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diff --git a/gdE2T4oBgHgl3EQfcAds/content/tmp_files/2301.03891v1.pdf.txt b/gdE2T4oBgHgl3EQfcAds/content/tmp_files/2301.03891v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..26183e12c131ce02118ddd875c9b025f5fc10714
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@@ -0,0 +1,1005 @@
+Removal of industrial dye and pharmaceutical
+product using the nano and micron-sized PS
+rough particles studded with Pt nanoparticles
+Faizan Khan, Chandra Shekhar, Tarak Mondal, and Manigandan Sabapathy∗
+Department of Chemical Engineering, Indian Institute of Technology Ropar,
+Rupnagar-140001, Punjab, India
+E-mail: mani@iitrpr.ac.in
+Abstract
+We show that the rough particles studded with platinum nanoparticles can be fabri-
+cated straightforwardly and in a single step at room temperature. These rough particles
+displayed a good catalytic power (100% removal efficiency) against a model industrial
+dye (methylene blue) and pharmaceutical residue (tetracycline) within a reasonable
+time scale. Further, we illustrate the effects of particle size, concentration, and contact
+patterns on the performance of rough catalytic particles. The semi-batch conditions
+favoured the complete decomposition of tetracycline within 40 min, but the batch-wise
+operation offered a good contacting pattern for methylene blue yielding a maximal out-
+put within 10 min. The kinetics of the heterogeneous catalytic process modelled by
+Langmuir-Hinshelwood kinetics predicts that the given methylene blue decomposition
+reaction induced by the rough particles follows the pseudo-first-order kinetics. The rate
+constants for the reaction catalyzed by 0.6 and 1.0 µm-sized rough particles are 0.048
+and 0.032 min−1, respectively. Furthermore, we established the proof-of-concept using
+1
+arXiv:2301.03891v1  [physics.chem-ph]  10 Jan 2023
+
+magnetically-responsive rough particles for real-time applications, including decontam-
+ination and recovery of catalyst particles via an externally applied magnetic field in one
+cycle. Our proposed method helps achieve a near-100% degrading efficiency within 10
+to 40 min at minimal catalytic particle concentration, i.e., 200 ppm. Since we can turn
+the rough particles into super-paramagnetic, we can recover and reuse them for several
+wastewater treatment cycles without incurring any running costs.
+Introduction
+The rapid rise of contamination in the groundwater due to industrial dyes and pharma-
+ceutical products generated from textile and pharma industries has been a most pressing
+issue globally that it has urged the intervention of policymakers, researchers, and engineers
+around the globe.1,2 Although the contamination of several azo-based dyes and pharmaceu-
+tical residues is supposed to be a matter of concern, the primary issue that remains to be
+tackled is establishing efficient removal of the contaminants from the wastewater as it poses
+a severe threat to ecology. In recent years, much attention has been given to the advanced
+oxidation processes (AOPs)3 over traditional water treatment methods such as mechanical
+separation, coagulation, and filtration. The various techniques of AOPs include 1) photoly-
+sis,4,5 2) Fenton reaction,6,7 3) photo-Fenton,7,8 4) photo-catalysis,4,9 5) ozonation,9,10 and
+6) sonochemistry.7,11 Despite the advancement in these techniques being well received under
+AOPs, each method has merits and demerits. For instance, high energy demand, strict pH
+range (Fenton),12 high H2O2 consumption (Fenton/photo-Fenton), and cost of implementa-
+tion.
+Among several AOPs, Fenton reaction-based methods are found to be the most efficient
+one for the removal of organic pollutants.13 Nevertheless, this method is not much attractive
+due to a few shortcomings such as 1) tendency to aggregate and sediment against gravity,
+2) process condition that demands narrow pH range of 2-3 to be maintained, and 3) high
+peroxide consumptions. On the contrary, heterogeneous Fenton-like catalyst-assisted reac-
+2
+
+tion based on iron oxides,13 transition metal oxides,14 and the alloy made up of zero-charge
+metal/metal-iron oxides15 are promising candidates owing to its increased electrocatalytic
+activity even at higher pH. Toda et al. reported the enhanced electrocatalytic activity of
+Pt alloyed with Fe electrodes. Based on the analysis by these authors, the enhanced elec-
+trocatalytic activity is due to increased d-electron vacancy of the Pt surface induced by
+alloying the metal with Fe.16 Furthermore, the efficiency of the Fenton reaction (based on
+Fe3O4 nanoparticles) is expected to decrease as the considerable amount of hydroxyl and
+perhydroxyl radicals formed during reaction are involved in converting Fe3+ back to Fe2+, in
+addition to organic molecules. However, the decomposition reaction catalyzed by platinum-x
+hybrid particles instead of Fenton, eliminate the perhydroxyl formation and the unwanted
+reaction step.6 In 2012, Hsieh and Lin reported the electrocatalytic activity of Fe-Pt nanopar-
+ticles (Fenton-like) for the first time with a maximum efficiency of 90% by the end of the 90
+min reaction time.6 The study of the decomposition of MB based on the Pt/Fe route was not
+pursued by many researchers considering the cost of synthesis involved, despite knowing the
+fact that the issue associated with the cost of production could be eliminated by employing
+a magnetically responsive core-shell particle system decorated with platinum nanoparticles,
+as the particles can be recovered by applying magnetic force externally. Such particles are
+regarded as rough particles with controlled surface deposition. We aim to demonstrate the
+chemical decomposition of MB (an example system of azo dyes) and tetracycline (an exam-
+ple system of pharmaceutical drugs) using nano and micron-sized polystyrene (PS) rough
+particles decorated with platinum as zero charge metal catalyst particles. Since the organ-
+ics mentioned above is chemically deactivated, there is no need to regenerate the catalyst
+particles once the reaction is completed. Furthermore, since the particles can be made mag-
+netically responsive and easily separated by applying magnetic force externally, it is possible
+to carry out the treatment process continuously with less or no manual intervention without
+regeneration. Thus, as outlined above, the method of heterogeneous catalyst particles is a
+cost-effective alternative, as the costs spent are predominantly linked to the capital budget
+3
+
+or one-time investment.
+In the context of the synthesis of rough particles concerning immobilization of metal-
+lic nanoparticles, a common approach has been to decorate the surface of core particles
+with gold (Au) or any other metallic nanoparticles via electrostatic interaction or covalent
+bonding.17–19 These routes demand the core particles be cationic to adsorb/deposit the gold
+nanoparticles (anionic) to make suitable rough colloids. However, it is of our interest to im-
+mobilize platinum (Pt) instead of Au nanoparticles to explore its application in the field of
+contaminant removal. The synthesis route follows the covalent bonding route in which the Pt
+nanoparticles are deposited by the chemical reduction of platinic acid using a reducing agent
+such as sodium borohydride (NaBH4). In 2012, Hsieh and Lin devised Fe-Pt nanoparticles to
+deactivate MB dye to achieve greater reaction extent.6 However, the method offered by Hsieh
+and Lin is slightly complicated and involves heating up to 297◦C. Therefore, it is essential
+to study the heterogeneous catalysts of many kinds to find the suitability of the method
+to tackle various pollutants. Thus, we have chosen to extend the studies beyond Fe-based
+supports to get insight into the degradation kinetics against other organics removal. For
+instance, the as-synthesized Pt-based rough particles have shown good catalytic activity in
+decomposing hydrogen peroxide, which helps degrade methylene blue (MB) and tetracycline
+(TC). Our suggested technique achieves a near-100% degrading efficiency within 10 to 40
+minutes at lower concentrations (200 ppm), making it a viable alternative to Fenton-based
+AOPs.
+The remainder of the manuscript is structured as follows: To provide the reader a bet-
+ter understanding, we first address the synthesis, characterization, and application of Pt-
+decorated PS rough particles for wastewater treatment. Consequently, we show the proof-
+of-concept (POC) by using magnetically-responsive rough particles (MR-RP) for the appli-
+cation. In addition to the application of discolouration or deactivation, this POC portion
+addresses the separation of catalyst particles under an externally applied magnetic force.
+4
+
+Materials and Methods
+Materials
+We used Invitrogen™ grade amidine functionalized polystyrene (PS) latex particles purchased
+from Thermo Fisher Scientific, India, to prepare catalytic-enabled rough particles for treating
+the wastewater containing specific contaminants. We used polystyrene particles of different
+sizes for carrying out the decomposition reaction. The size of the particles employed, as
+per the manufacturer’s specifications, are 1.0, 0.5, and 0.02 µm. However, the size of the
+particles determined based on image analysis using a scanning electron microscope (SEM)
+and dynamic light scattering (DLS) technique are 1.0 ± 0.1, 0.579 ± 0.006, and 0.022 ±
+0.001 µm. An electrophoretic study was carried out to determine the particles’ potential at
+the boundary between the shear and diffuse plane, i.e., Zeta potential (ζ). The ζ potential
+of the particles determined for PS particles of size 1.0, 0.6 (rounded-off) and 0.02 µm in 1
+mM NaCl electrolyte medium were 46 ± 3, 55.8 ± 3.2 and 42.7 ± 1 mV, respectively. To
+prepare a core-shell assembly of magnetically-responsive rough particles, we employed iron
+oxide (IO) nanoparticles purchased from Sigma Aldrich Chemicals Pvt. Ltd., India. Polydi-
+allyldimethylammonium chloride (DADMAC), gifted by Dr. Sarang Gumfekar, IIT Ropar,
+was used to modify the IO particles by imparting positive sites to the surface. Chloropla-
+tinic acid hexahydrate (H2PtCl6.6H2O), used as precursor solution to deposit platinum (Pt)
+nanoparticles on the PS surface, was procured from Sigma-Aldrich Chemicals Pvt. Ltd.,
+India. Sodium borohydride (NaBH4), obtained from Sigma-Aldrich Chemicals Pvt. Ltd.,
+was used as a reducing agent. Hydrogen peroxide (30 w/v %), used in the decomposition
+reaction mixture to generate hydroxy radical, was procured from Sigma-Aldrich Chemicals
+Pvt.
+Ltd., India.
+The model pollutants used to demonstrate the deactivation kinetics,
+such as methylene blue and tetracycline, were procured from Sigma-Aldrich Chemicals Pvt.
+Ltd., India. A hydrogen peroxide test strip (MQuant® Peroxide-Test) received from Sigma-
+Aldrich Chemicals Pvt. Ltd., India, was used to quantitatively predict the catalytic-assisted
+5
+
+decomposition kinetics under the presence of rough colloids. All reagents received were ana-
+lytical grades and used without any further purification. We used deionized water obtained
+from a Smart2Pure™ water purification system (Make: Thermo Fisher Scientific, Model:
+Smart2Pure 12 UV/UF) for all experiments.
+Synthesis of PS rough particles decorated with Pt
+First, we prepared a chloroplatinic acid hexahydrate solution using DI water such that the
+precursor concentration in the stock solution was 10 mM. We devised a wet-chemical method
+to achieve a uniform deposition of Pt nanoparticles on the PS surface in an endless array
+of monolayer fashion. The deposition of Pt nanoparticles was realized by using the method
+reported by our group that involves the deposition of gold (Au) nanoparticles.20 We used a
+slightly different strategy, including a wet-chemical deposition by nucleating Pt nanoparticles
+instead of Au on the PS surface. Since both the precursors of Au and Pt bear a net negative
+charge, favouring the attachment of ions over the positive sites of PS due to net electrostatic
+attraction, the modified protocol helped us synthesize Pt-decorated PS particles in a single
+step.
+Unlike the work of Lu et al., our modified protocol is devoid of pre-deposition of
+Au nanoparticles as seeds to catalyze the reduction reaction of PtCl2−
+6
+ions.
+Scheme 1
+depicts the schematic description of making rough particles and their application towards
+wastewater treatment.
+As explained in Scheme 1A, a known concentration of platinum
+precursor (0.5 mM), PS particles (0.5 mg/mL) are mixed using a magnetic stirrer for up to
+30 min before adding a reducing agent, sodium borohydride, to the mixture. Subsequently,
+after the equilibration of the mixture for about 30 min, the chemical reduction reaction is
+induced by adding 5 mL of 10 mM sodium borohydride solution completely into the beaker
+containing the reaction mixture.
+The reaction is allowed to continue until the solution
+becomes clear. To ensure the reaction is complete, we have used sodium borohydride ten
+times in excess. Additionally, we have given sufficient time for reaction, i.e., 4 hr. We have
+maintained the reaction time of four hours in all the experiments that involve the deposition
+6
+
+of Pt on the PS surface. Note: The reaction mixture becomes transparent and clear when the
+chemical reduction process continues for up to four hours. Further, we allowed the samples
+to centrifugation to separate the modified PS particles by removing the supernatant and
+washing them several times. The particles recovered in this way are further utilized to study
+the kinetics of MB and TC decomposition. Table 1 presents the size and zeta potential (ζ)
+of the bare PS and the rough PS particles.
+Selection of target pollutants
+We chose to work with two representative pollutants from the family of azo dyes and an-
+tibiotic drugs, such as MB and TC, respectively, for the entire decomposition studies. We
+endeavoured to understand the role of rough particles in the chemical deactivation of these
+target pollutants. The initial pollutants concentration was maintained constant at 15 µM
+throughout. We studied the degradation of these pollutants in batch and semi-batch condi-
+tions to replicate the real-time process of large-scale operation.
+Heterogeneous batch degradation process
+All degradation experiments that involve batch-wise operations were performed in different
+initial volumes, such as 25, 50 and 100 mL. The rough particle concentrations employed
+were 200, 100, and 50 ppm, respectively. For the studies, we prepare the required mixture
+using the rough particles at appropriate concentrations along with the target pollutants
+with known initial concentration, i.e. 15 µM, to induce the decomposition reaction in the
+presence of a known amount of H2O2, 5 vol% per the total volume mentioned above. The
+reaction is set to proceed for the desired time by adding the radical generator, i.e., H2O2.
+We continuously stirred the reaction mixture using the stirrer at a programmed speed of
+140 RPM. The entire reaction was conducted at room temperature, 25◦C, throughout the
+desired duration.
+7
+
+Heterogeneous semi-batch degradation process
+All degradation experiments that involved semi-batch operations were performed in different
+initial dosing volume at a uniform interval of 10 min until completion. The rough particles
+concentration employed were 200, 100, and 50 PPM, respectively. For the studies, we prepare
+the required mixture using the rough particles at appropriate concentrations along with the
+target pollutants with known initial concentration, i.e. 15 µM, to induce the decomposition
+reaction in the presence of a known amount of H2O2, 5 vol% per the total initial volume. The
+reaction is initiated to proceed for the desired length by adding the radical generator, i.e.,
+H2O2 at a desired dosing volume. Using the stirrer, we stirred the reaction mixture nonstop
+at a programmed speed of 140 RPM. The entire decomposition reaction was conducted at
+room temperature, 25◦C, throughout the desired duration.
+Scheme 1B and E depicts the schematic description showing the catalytic action of PS
+rough particles and the process of discoloration of MB due to contact with the modified PS
+particles of varying sizes, respectively.
+Hydrogen peroxide decomposition
+To understand the kinetics of H2O2 decomposition, we decoupled the decomposition reactions
+of MB and TC by eliminating them in the lab-scale reactor. Subsequently, in a separate
+set of experimental studies, the reaction flasks containing H2O2 and rough particles were
+employed to conduct studies at different lengths of time. We recorded the concentration vs
+time of H2O2 as the reaction proceeds to understand the effect of contacting patterns on the
+catalyst-assisted H2O2 decomposition reaction.
+Characterization
+The zeta potential of bare PS and rough PS particles were measured based on the elec-
+trophoretic light scattering (ELS) technique using Zetasizer procured from Malvern Instru-
+8
+
+ments, Model: Zetasizer Nano ZSP. The average diameter of PS of size 0.6 µm and 1.0
+µm were determined using a high-resolution scanning electron microscope (HRSEM), Make:
+FEI, USA and Model: Inspect F50. The hydrodynamic radius of PS of size 22 nm was
+deduced using the dynamic light scattering (DLS), Make: Malvern Instruments and Model:
+Zetasizer Nano ZSP. The kinetics of decomposition reactions of MB and TC were studied by
+capturing concentration as a function of decomposition times of MB and TC using UV-VIS
+spectrophotometers (Make: Hach company US, Model: DR3900 & Make: PerkinElmer Inc.,
+USA, Model: LAMBDA 365)
+Scheme 1: Schematic description showing the process of synthesizing rough particles and
+their application towards deactivation of the target pollutants. A) Schematic illustration
+showing the synthesis methodology of making rough particles. B) Schematic illustration
+showing the application of rough particles in treating wastewater containing the target pol-
+lutants. C and D) FESEM and EDX spectrum of 1 µm rough particles modified with Pt
+nanoparticles, respectively. E) Vial pictures showing the discoloration of MB dye in the
+presence of rough particles.
+Results & Discussion
+FESEM was used to determine the size of PS particles and the morphological changes in
+PS rough particles due to the reaction. EDX was performed concurrently to identify the
+9
+
+A
+B
+Fpeed=140rpm
+Temperature=25°C
+NaBH4
+t=4 hr
+H,02
+NH
+t(min)
+H,PtCl
+H,PtCl6
+NaBH
+PS
+PS-Pt
+functionalized
+(Core-shell)
+At%
+E
+c
+Element
+wt%
+D
+657K
+H,02
+HO + HO←
+9.45
+v90
+584K
+90.55
+9E66
+511K
+438K
+365K
+292K
+219K
+146K
+Pt-coated rough particle
+73K
+Pt
+%
+1.3
+2.6
+3.9
+5.2
+6.5Table 1: Average size and zeta potential of the PS particles
+Type
+Avg. Size (µm)
+Zeta potential (mV)
+Bare
+Modified
+PS
+0.02 ± 0.009
+42.7 ± 1.0
+-33 ± 0.7
+0.6 ± 0.006
+55.8 ± 3.2
+-33 ± 0.3
+1.0 ± 0.1
+46 ± 3
+-35 ± 0.4
+IO
+0.06 ± 0.001
+-2.7 ± 1.8
+33.2 ± 2.2
+presence of Pt nanoparticles. Figure 1 displays the surface morphology of surface-engineered
+PS rough particles decorated with Pt nanoparticles in the form of islands (Figures 1A and
+1B) and the EDX spectrum used to validate the deposition of Pt nanoparticles on the surface
+of PS (Figures 1C and 1D). As depicted in Figures 1C and 1D, the distinctive peaks at 1.5
+KeV indicate the existence of Pt nanoparticles.
+Next, we discuss the performance of rough particles in batch and semi-batch conditions.
+This study highlights the engineering processes that can substantially impact when transi-
+tioning between real-time applications. Figure 2 refers to the decomposition reaction cat-
+alyzed by rough particles (Pt/PS) in the presence of a moderate concentration of 5% H2O2.
+Note: The absorbance maxima at 358 nm from each UV-Vis spectral plot were utilised to
+determine the concentration as a function of time. The concentration of particles and the
+pollutants used for the comparison study are 100 ppm and 15 µM, respectively. According to
+the data, the highest TC elimination may be accomplished when the reaction is conducted
+under semi-batch conditions. On the other hand, when similar decomposition kinetics was
+run under batch circumstances, the catalyst-assisted heterogeneous response produced a high
+reaction rate for MB removal, i.e., 100% elimination of MB in 10 minutes. These results
+intrigued us to set up reaction conditions appropriately. We, therefore, systematically con-
+ducted the heterogeneous catalyst-assisted reactions using the rough particles of desired sizes
+and concentrations.
+10
+
+Figure 1: Structural characterization of modified PS particles. A and B) FESEM images of
+1.0 and 0.6 µm, respectively. C and D) EDX spectrum of 1.0 and 0.6 µm, respectively. Scale
+bars correspond to 500 nm.
+11
+
+A
+B
+603K
+c
+657K
+D
+536K
+584K
+469K
+511K
+402K
+438K
+335K
+365K
+268K
+292K
+201K
+219K
+134K
+146K
+Pt
+Pt
+67K
+73K
+Pt
+C
+Pt
+Pt
+1.3
+2.6
+3.9
+5.2
+6.5
+7.8
+9.1
+1.3
+2.6
+3.9
+5.2
+6.5
+7.8
+9.1Figure 2: Decomposition of TC and MB with time. Removal of TC when carried out in
+batch (A) and semi-batch (B) setup. Removal of MB when carried out in batch (C) and
+semi-batch (D) setup. The concentration of rough particles, MB, TC, and H2O2 employed
+are 100 ppm, 15 µM, 15 µM, and 5%, respectively.
+12
+
+100
+A
+100
+B
+80
+80-
+60
+H
+60 -
+40
+40-
+20
+20-
+0
+0-
+0
+10
+20
+30
+40
+50
+60
+70
+80
+0
+10
+20
+30
+40
+50
+60
+70
+80
+100
+c
+100-
+D
+田
+日
+日
+日
+日
+T
+80-
+80
+Efficiency (%)
+60
+60-
+40-
+40-
+20.
+20
+0-
+0
+10
+20
+30
+40
+50
+60
+70
+80
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Time (min)
+Time (min)To this end, we demonstrate the performance of rough particles in semi-batch conditions
+to determine the optimum parameters. Since the semi-batch operation was effective for TC,
+we conducted all experiments related to TC removal using a semi-batch set-up to find the
+optimal point. Figure 3 displays the kinetics of the decomposition reaction of TC carried out
+in semi-batch operation. By comparing Figures 2 and 3, the kinetic data suggest that pollu-
+tants’ concentration has an inverse effect on the reaction rate associated with TC removal.
+Therefore, the semi-batch catalyzed reaction favours maximal elimination at a higher rate.
+In contrast to batch mode, this condition improved the reaction rate because TC concentra-
+tions were kept as low as possible due to drop-wise addition. We infer from Figure 3 that the
+maximum efficiency is directly proportional to particle concentration. Table 2 summarizes
+the performance of rough particles employed to deactivate TC. As shown in Figure 3 and
+Table 2, the rough particles with a size of 0.6 µm and concentration 200 ppm along with
+a dosing volume of 0.416 mL H2O2 exhibited the best performance among the given set of
+rough particles and the experimental conditions.
+Table 2: Summary of performances of the rough particles over removal of TC
+Type
+Size (µm)
+Initial TC conc. (µM)
+Particles conc. (ppm)
+Dosing volume (mL)
+Max. Efficiency (%)
+Time (min)
+Pt-PS
+1
+15
+100
+1.25
+100
+50
+0.6
+200
+0.416
+100
+40
+0.02
+40
+1.66
+65
+100
+Further, we discuss the performance of rough particles in batch conditions. Since the
+batch operation was effective for MB, we ran all experiments related to MB removal using
+a batch mode to determine the optimal point. Figure 4 displays the kinetics of the decom-
+position reaction of MB carried out in batch operation. Note: The absorbance maxima at
+664 nm from each UV-Vis spectral plot were utilized to determine the concentration as a
+function of time. We infer that the performances of the rough particles of size 1.0 and 0.6
+µm, respectively, overlap significantly over time. Barring 0.02 µm, the rate of decomposi-
+tion data shows direct dependency on the concentration of rough particles. Therefore, the
+batch-catalyzed reaction favours the maximal elimination rate at a higher concentration of
+13
+
+Figure 3: Effect of dosing volume of H2O2 on the decomposition of TC with time under
+semi-batch mode at regular interval. A-C) Removal of TC at different dosing volume of
+0.416, 0.624, and 0.833 mL of H2O2, respectively. D-F) Removal of TC at different dosing
+volume of 0.833, 1.25, and 1.66 mL of H2O2, respectively. G-I) Removal of TC at different
+dosing volume of 1.66, 2.5, and 3.33 mL of H2O2, respectively. The concentration of rough
+particles used are 200 ppm (A-C), 100 ppm (D-F), and 50 ppm (G-I), respectively. The
+initial concentration of TC employed for the study is 15 µM, throughout.
+14
+
+0.02μm
+0.02μm
+B
+0.02μm
+c
+100-
+100-
+100
+-0.6μm
+-0.6 μm
+-0.6μm
+-1um
+-1μm
+-1μm
+80
+80
+80-
+(%)
+60
+60
+60.
+40-
+40
+40
+TC
+20-
+20
+20
+0-
+0.
+0
+0
+102030
+40
+50
+60
+70
+80
+90100110
+0
+10
+20
+30
+40
+50
+60
+70
+80
+0
+10
+20
+30
+40
+50
+60
+0.02μm
+-0.02μm
+-0.02μm
+100-
+D
+100-
+F
+-0.6μm
+100
+0.6μm
+0.6μm
+1um
+-1um
+-1um
+80
+80-
+80-
+removal (%)
+60
+60
+60-
+40
+40
+40
+20-
+20-
+20-
+0
+0
+010
+2030
+40
+50
+60
+70
+80
+90
+100110
+0
+10
+20
+30
+40
+50
+60
+70
+80
+0
+10
+20
+30
+40
+50
+60
+0.02μm
++0.02μm
+G
+H
+-0.02μm
+100
+-0.6μm
+100
+0.6 μm
+100-
+0.6μm
+-
+-1um
+-1 μm
+-1μm
+80-
+80-
+80-
+(%) I
+60-
+60-
+60-
+40-
+40.
+40-
+TC
+20-
+20-
+20-
+0-
+01020
+30
+40
+50
+60
+70
+80
+90100110
+0
+10
+20
+30
+40
+50
+60
+70
+80
+0
+10
+20
+30
+40
+50
+60
+Time (min)
+Time (min)
+Time (min)rough particles. In contrast to the semi-batch mode, this condition improved the reaction
+rate because the concentration of MB was kept high at the start. Table 3 summarizes the
+performance of rough particles employed to deactivate MB. As shown in Figure 4 and Table
+3, the rough particles with a size of 1 and 0.6 µm and concentration 200 ppm at a concen-
+tration of 5% H2O2 exhibited the best performance among the given set of rough particles
+and the experimental conditions.
+Figure 4: Removal of MB catalyzed by varying sizes of rough particles at a concentration of
+A) 200 ppm, B) 100 ppm, and C) 50 ppm.
+Table 3: Summary of performances of rough particles against MB removal at 5% H2O2
+Type
+Size (µm)
+Initial MB conc. (µM)
+Particle conc. (ppm)
+Max. Efficiency (%)
+Time (min)
+PS
+1 & 0.6
+15
+200
+100
+10
+100
+100
+20
+50
+100
+40
+Besides % of elimination, the order of the reaction and rate constant also provide good
+insight into the kinetics of the Pt-PS-assisted reactions. Langmuir-Hinshelwood (LH) kinetics
+is the most popular model for describing the kinetics of heterogeneous catalytic processes
+which is described as shown in Eq. 1 below,22
+r = −dC
+dt = krKC
+1 + KC
+(1)
+where r, kr, K, and C refer to the rate of reaction that changes with time, limiting
+rate constant of reaction at maximum coverage under the given experimental conditions,
+15
+
+A
+B
+100
+100-
+100
+80-
+80-
+80-
+0.02μm
+0.02μm
+(%)
+- 0.6 μm
+-0.02μm
+0.6 μm
+(%)
+0.6μm
+-1μm
+60
+1um
+Efficiency
+60
+Efficiency
+1um
+40
+40
+20-
+20-
+0 -
+70
+Time (min)
+Time(min)
+Time (min)equilibrium constant for adsorption of the substrate onto the catalyst, and concentration at
+anytime t during degradation, respectively.
+Most of the time, researchers approximated Eq. 1 to the first order, i.e., n=1, by assuming
+KC << 1. Thus, when −ln
+�
+C
+C0
+�
+is plotted on the ordinate and time is plotted on the abscissa,
+the slope of the straight line is the product of kr and K. However, if the value of the slope
+found is >> 1, then the assumption that KC << 1 becomes invalid.
+That means the
+subsequent kinetics will be in zero order. Figure 5 displays the pseudo-first-order kinetics
+corresponding to the heterogeneous reactions catalyzed by the rough particles of varying
+sizes. The rate constants for the reaction catalyzed by 0.6 and 1.0 µm particles at 50 ppm
+are 0.048 and 0.032 min−1, respectively. The reported values are in the same order as that
+of the performance of the Fe-Pt nanoparticles reported by Hsieh and Lin for 5 ppm.6 Due
+to variations in the concentration of the rough particles, the reported values in our case are
+1.4 to 2.0 times higher than the literature value.6 It is also noteworthy that the process
+catalyzed by the rough particles at a concentration of 50 ppm completes (100%) at the end
+of the 40-minute reaction time. By 90 minutes, the absorbance maximum near 665 nm for
+the Fe-Pt particles exhibited by Hsieh and Lin had dropped by just 90 %.6
+Before moving on to the following section and exhibiting the proof-of-concept employ-
+ing magnetically responsive particles, we propose probable causes for the disparity in per-
+formance between the different-sized rough particles. It is generally understood that the
+nanoparticles offer a more significant surface area (SA) for a given concentration in a batch
+reactor. For instance, we anticipate that the smaller rough particles would expose Pt sig-
+nificantly due to the more significant number of particles than the same concentration of
+bigger rough particles. By referring to Table 1 for zeta potential values and EDX analysis of
+the rough particles of varying sizes, we cancel out any variation associated with the quality
+of the Pt deposition. Consequently, the SA of rough particles should increase as their size
+decreases, i.e. 1.0 µm > 0.6 µm > 0.02 µm. In contrast, according to Figures 4 and 5, we
+observed a kinetics that is an inverse function of particle size when examining the decom-
+16
+
+Figure 5: Pseudo first-order kinetic plot for the deactivation of MB in the presence of rough
+particles of size 0.6 and 1.0 µm.
+17
+
+3
+1 μm
+0.6 μm
+2 
+-ln (C/Co)
+1.
+0
+0
+10
+20
+30
+40
+Time (min)position of MB using the rough particles under batch conditions. We attribute this unusual
+behaviour to the crowding effect of bubbles and the buoyancy effect of the nanoparticles
+caused due to the attachment of bubbles. As a result, 1) particles diffuse rapidly towards
+the interface, decreasing the contact time with a key reactant, and 2) the catalyst’s power is
+diminished due to a layer of bubbles around the particles obstructing the active sites. Figure
+6 displays the peroxide decomposition reaction as a function of time in minutes. Except
+for 0.6 and 1.0 µm, we found a marked difference in performance concerning 0.02 µm. The
+differences in peroxide breakdown kinetics found using the detector strips could be linked to
+the aforementioned factors. The video showing the rapid generation of the bubbles and the
+movement of the nanoparticles from bulk to the interface can be found in Supplementary
+Information (SI).
+Proof-of-concept
+Thus far, we have demonstrated a straightforward methodology to make the rough particles
+by modifying the surface of the PS using the Pt nanoparticles through the wet-chemical route
+and their performances in decontamination application. Here, we show that the magnetically-
+responsive rough particles (MR-RP) can be made by decorating platinum-coated PS nanopar-
+ticles with iron oxide nanoparticles like a core-shell structure to accomplish chemical deac-
+tivation and particle recovery in a single step.
+The benefits of MR-RP include ease of
+fabrication of Pt nanoparticles as the deposition takes place by attractive electrostatic forces
+between Pt precursor (negatively charged) and the positively charged PS particles. This
+combination helps us achieve a uniform deposition like a monolayer of islands distributed
+in space throughout without needing stabilizers like oleic acid and oleyl amine. Secondly,
+unlike the method proposed by Hsieh and Lin,6 the technique does not demand heating at
+high temperature, i.e., 297◦C, and the entire process can be completed in room temperature.
+To this end, we demonstrate the discolourization of MB using MR-RP. Scheme 2 de-
+scribes the process of making MR-RP using a wet-chemical route. Since the process follows
+18
+
+Figure 6: H2O2 decomposition using the rough particles of varying sizes.
+19
+
+60000
+- 0.02 μm
+= 0.6 μm
+50000
+1 μm
+40000
+(udd) ZOH
+30000
+20000
+10000
+0
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Time (min)the modifications in bulk, we can eliminate the shortcomings such as scale-up and yield.
+As shown in Scheme 2, the attractive forces between IO and Polydiallyldimethylammonium
+chloride (PolyDADMAC) allow the binding of the polymers on the surface of IO nanopar-
+ticles. Subsequently, the suspension containing Pt-modified PS nanoparticles of size 0.02
+µm was introduced into the polymer-coated IO mixture. This addition results in the self-
+assembly of negatively charged Pt-modified PS and positively charged polymer-modified IO
+leading to the formation of core-shell particles. Note: Since the sufficient concentration of
+PS nanoparticles was considered during the wet-chemical deposition and a suitable ratio of
+Pt-decorated PS and polymer-decorated IO was chosen before mixing, there were very little
+or no free Pt nanoparticles found in the supernatant solution (Please see supplementary
+video-2 shown in the supplementary information). We demonstrate a start-to-finish proof-
+of-concept of a two-step process (removal of MB + recovery of MR-RP) achieved by the
+proposed surface-engineered nanoparticles via a real-time video (Please see supplementary
+video-3 shown in the supplementary information).
+Scheme 2: Schematic description showing the process of synthesizing magnetically-responsive
+rough particles using the wet-chemical method.
+20
+
+Sonication (1hr)
+Soaking (12 hr)
+PolvDADMAC
+Iron oxide nanoparticles
+Polymer grafted
+Iron oxide nanoparticles
+Mixing (300 RPM)
+1 hr
+Platinum coated polystyrene nanoparticles
+Platinum coated
+Polymer
+grafted
+coating on iron oxide nanoparticles
+Polystyrene nanoparticles
+Iron oxide nanoparticlesGeneral Remarks
+We comment on the proposed method’s applicability and shortcomings in this section. The
+proposed method works well against pollutants that accept electrons for chemical deactiva-
+tion. For instance, the process would not fit well for the methyl orange dye, a contaminant
+that would give up electrons while decomposing. We tested the applicability of the rough
+particles against methyl orange and found that the decomposition rate is relatively slow
+compared to the reaction of its counterparts. Thus, we understand that the proposed rough
+particles show catalytic power and facilitate electron transfer from free radicals to pollutants
+over time. Secondly, the proposed method requires PS particles to be positively charged to
+attract the precursor ions, which are negatively charged and induce nucleation of platinum
+nanoparticles on the surface of PS. Therefore, the method is unsuitable for making rough
+particles if the terminal end groups of PS are negatively charged.
+Conclusion
+We established a facile approach for producing Pt-studded PS-based rough particles with
+good catalytic activity in a single step. The proposed method offers an exciting route to
+make catalyst particles on desired supports at room temperature. As the existing process re-
+quires chemical precipitation followed by heating the reaction mixture to 297◦C, our proposed
+approach could serve as an excellent alternative to existing Fenton-based heterogeneous par-
+ticles studded with Pt metal nanoparticles. Our study revealed that the contacting patterns
+play a significant role in determining the performance of the rough particles over the decom-
+position of any given pollutant. For instance, the catalytic action of PS-Pt rough particles
+was visible when tetracycline decomposition was performed in semi-batch rather than batch
+settings. Conversely, batch-wise operations recorded the maximal output of rough particles
+in eliminating methylene blue. We studied the impact of particle sizes, concentrations and
+dosing rate of hydrogen peroxide on the catalytic activity of rough particles. To cite an ex-
+21
+
+ample, the desired operating parameters to achieve 100% efficiency vary for different rough
+particles of size 0.6 µm (Optimum concentration = 200 ppm, dosing volume = 0.416 mL) and
+1µm (Optimum concentration = 100 ppm, dosing volume = 1.25 mL). The heterogeneous
+reaction modelled by Langmuir-Hinshelwood kinetics indicates that the catalytic reaction
+follows the pseudo-first-order process with rate constants of 0.048 and 0.032 min−1 for the
+reaction catalyzed by 0.6 and 1.0 µm-sized rough particles, respectively. Using a well-known
+layer-by-layer technique, we presented a straightforward methodology to achieve a core-shell
+assembly of magnetically-responsive rough particles (MR-RP). In a batch reactor config-
+uration, we demonstrated proof-of-concept for the decomposition of methylene blue using
+MR-RP. Since we can deactivate pollutants and recover catalyst particles in a single cycle,
+the proposed technique would be effective in real-time and large-scale applications without
+incurring operating expenses.
+Acknowledgement
+We thank Dr Neethu Thomas (Postdoctoral Researcher) and Prof. Parasuraman Swami-
+nathan, Department of Metallurgical and Materials Engineering, IIT Madras, India, for
+helping us with the FESEM combined with the EDX analysis of our samples. MS thanks
+IIT Ropar for providing the seed grant (ISIRD phase II) to set up a laboratory.
+References
+(1) Visa, M.; Chelaru, A.-M. Hydrothermally modified fly ash for heavy metals and dyes
+removal in advanced wastewater treatment. Applied Surface Science 2014, 303, 14–22.
+(2) Liu, L.; Chen, Z.; Zhang, J.; Shan, D.; Wu, Y.; Bai, L.; Wang, B. Treatment of in-
+dustrial dye wastewater and pharmaceutical residue wastewater by advanced oxidation
+22
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+processes and its combination with nanocatalysts: A review. Journal of Water Process
+Engineering 2021, 42, 102122.
+(3) Minero, C.; Lucchiari, M.; Vione, D.; Maurino, V. Fe (III)-enhanced sonochemical
+degradation of methylene blue in aqueous solution. Environmental science & technology
+2005, 39, 8936–8942.
+(4) Peter, A.; Mihaly-Cozmuta, A.; Nicula, C.; Mihaly-Cozmuta, L.; Jastrzębska, A.; Ol-
+szyna, A.; Baia, L. UV light-assisted degradation of methyl orange, methylene blue,
+phenol, salicylic acid, and rhodamine B: photolysis versus photocatalyis. Water, Air,
+& Soil Pollution 2017, 228, 1–12.
+(5) Mohammed, H. A.; Khaleefa, S. A.; Basheer, M. I. Photolysis of methylene blue dye
+using an advanced oxidation process (ultraviolet light and hydrogen peroxide). Journal
+of Engineering and Sustainable Development 2021, 25.
+(6) Hsieh, S.; Lin, P.-Y. FePt nanoparticles as heterogeneous Fenton-like catalysts for hy-
+drogen peroxide decomposition and the decolorization of methylene blue. Journal of
+Nanoparticle Research 2012, 14, 1–10.
+(7) Saleh, R.; Taufik, A. Degradation of methylene blue and congo-red dyes using Fenton,
+photo-Fenton, sono-Fenton, and sonophoto-Fenton methods in the presence of iron
+(II, III) oxide/zinc oxide/graphene (Fe3O4/ZnO/graphene) composites. Separation and
+Purification Technology 2019, 210, 563–573.
+(8) Kirchon, A.; Zhang, P.; Li, J.; Joseph, E. A.; Chen, W.; Zhou, H.-C. Effect of isomor-
+phic metal substitution on the fenton and photo-fenton degradation of methylene blue
+using Fe-based metal–organic frameworks. ACS applied materials & interfaces 2020,
+12, 9292–9299.
+(9) Ikhlaqa, A.; Javedb, F.; Niaza, A.; Munirc, H. M. S.; Qid, F. Combined UV catalytic
+ozonation process on iron loaded peanut shell ash for the removal of methylene blue
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+from aqueous solution. DESALINATION AND WATER TREATMENT 2020, 200,
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+(10) Adelin, M.; Gunawan, G.; Nur, M.; Haris, A.; Widodo, D.; Suyati, L. Ozonation of
+methylene blue and its fate study using LC-MS/MS. Journal of Physics: Conference
+Series. 2020; p 012079.
+(11) Qian, L.-L.; Blatov, V. A.; Wang, Z.-X.; Ding, J.-G.; Zhu, L.-M.; Li, K.; Li, B.-L.;
+Wu, B. Sonochemical synthesis and characterization of four nanostructural nickel co-
+ordination polymers and photocatalytic degradation of methylene blue. Ultrasonics
+sonochemistry 2019, 56, 213–228.
+(12) Liu, Y.; He, X.; Fu, Y.; Dionysiou, D. D. Degradation kinetics and mechanism of
+oxytetracycline by hydroxyl radical-based advanced oxidation processes. Chemical En-
+gineering Journal 2016, 284, 1317–1327.
+(13) Pignatello, J. J.; Oliveros, E.; MacKay, A. Advanced oxidation processes for organic
+contaminant destruction based on the Fenton reaction and related chemistry. Critical
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+(14) Giordano, G.; Perathoner, S.; Centi, G.; De Rosa, S.; Granato, T.; Katovic, A.; Si-
+ciliano, A.; Tagarelli, A.; Tripicchio, F. Wet hydrogen peroxide catalytic oxidation of
+olive oil mill wastewaters using Cu-zeolite and Cu-pillared clay catalysts. Catalysis
+Today 2007, 124, 240–246, Advanced Catalytic Oxidation Processes.
+(15) Ai, Z.; Lu, L.; Li, J.; Zhang, L.; Qiu, J.; Wu, M. Fe@ Fe2O3 core- shell nanowires as
+iron reagent. 1. Efficient degradation of rhodamine B by a novel sono-Fenton process.
+The Journal of Physical Chemistry C 2007, 111, 4087–4093.
+(16) Toda, T.; Igarashi, H.; Uchida, H.; Watanabe, M. Enhancement of the Electroreduction
+of Oxygen on Pt Alloys with Fe, Ni, and Co. Journal of The Electrochemical Society
+1999, 146, 3750–3756.
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+(17) Dokoutchaev, A.; James, J. T.; Koene, S. C.; Pathak, S.; Prakash, G. S.; Thomp-
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+librium position of colloidal particles at fluid–water interfaces by deposition of nanopar-
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+method to hollow bimetallic nanostructures. Langmuir 2003, 19, 3074–3077.
+(22) Kumar, K. V.; Porkodi, K.; Rocha, F. Langmuir–Hinshelwood kinetics – A theoretical
+study. Catalysis Communications 2008, 9, 82–84.
+25
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf,len=676
+page_content='Removal of industrial dye and pharmaceutical  product using the nano and micron-sized PS  rough particles studded with Pt nanoparticles  Faizan Khan, Chandra Shekhar, Tarak Mondal, and Manigandan Sabapathy∗  Department of Chemical Engineering, Indian Institute of Technology Ropar,  Rupnagar-140001, Punjab, India  E-mail: mani@iitrpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='in  Abstract  We show that the rough particles studded with platinum nanoparticles can be fabri-  cated straightforwardly and in a single step at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' These rough particles  displayed a good catalytic power (100% removal efficiency) against a model industrial  dye (methylene blue) and pharmaceutical residue (tetracycline) within a reasonable  time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Further, we illustrate the effects of particle size, concentration, and contact  patterns on the performance of rough catalytic particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The semi-batch conditions  favoured the complete decomposition of tetracycline within 40 min, but the batch-wise  operation offered a good contacting pattern for methylene blue yielding a maximal out-  put within 10 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The kinetics of the heterogeneous catalytic process modelled by  Langmuir-Hinshelwood kinetics predicts that the given methylene blue decomposition  reaction induced by the rough particles follows the pseudo-first-order kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The rate  constants for the reaction catalyzed by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='0 µm-sized rough particles are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='048  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='032 min−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Furthermore, we established the proof-of-concept using  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='03891v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='chem-ph]  10 Jan 2023  magnetically-responsive rough particles for real-time applications, including decontam-  ination and recovery of catalyst particles via an externally applied magnetic field in one  cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Our proposed method helps achieve a near-100% degrading efficiency within 10  to 40 min at minimal catalytic particle concentration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=', 200 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Since we can turn  the rough particles into super-paramagnetic, we can recover and reuse them for several  wastewater treatment cycles without incurring any running costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Introduction  The rapid rise of contamination in the groundwater due to industrial dyes and pharma-  ceutical products generated from textile and pharma industries has been a most pressing  issue globally that it has urged the intervention of policymakers, researchers, and engineers  around the globe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='1,2 Although the contamination of several azo-based dyes and pharmaceu-  tical residues is supposed to be a matter of concern, the primary issue that remains to be  tackled is establishing efficient removal of the contaminants from the wastewater as it poses  a severe threat to ecology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' In recent years, much attention has been given to the advanced  oxidation processes (AOPs)3 over traditional water treatment methods such as mechanical  separation, coagulation, and filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The various techniques of AOPs include 1) photoly-  sis,4,5 2) Fenton reaction,6,7 3) photo-Fenton,7,8 4) photo-catalysis,4,9 5) ozonation,9,10 and  6) sonochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='7,11 Despite the advancement in these techniques being well received under  AOPs, each method has merits and demerits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' For instance, high energy demand, strict pH  range (Fenton),12 high H2O2 consumption (Fenton/photo-Fenton), and cost of implementa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Among several AOPs, Fenton reaction-based methods are found to be the most efficient  one for the removal of organic pollutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='13 Nevertheless, this method is not much attractive  due to a few shortcomings such as 1) tendency to aggregate and sediment against gravity,  2) process condition that demands narrow pH range of 2-3 to be maintained, and 3) high  peroxide consumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' On the contrary, heterogeneous Fenton-like catalyst-assisted reac-  2  tion based on iron oxides,13 transition metal oxides,14 and the alloy made up of zero-charge  metal/metal-iron oxides15 are promising candidates owing to its increased electrocatalytic  activity even at higher pH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Toda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' reported the enhanced electrocatalytic activity of  Pt alloyed with Fe electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Based on the analysis by these authors, the enhanced elec-  trocatalytic activity is due to increased d-electron vacancy of the Pt surface induced by  alloying the metal with Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='16 Furthermore, the efficiency of the Fenton reaction (based on  Fe3O4 nanoparticles) is expected to decrease as the considerable amount of hydroxyl and  perhydroxyl radicals formed during reaction are involved in converting Fe3+ back to Fe2+, in  addition to organic molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' However, the decomposition reaction catalyzed by platinum-x  hybrid particles instead of Fenton, eliminate the perhydroxyl formation and the unwanted  reaction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 In 2012, Hsieh and Lin reported the electrocatalytic activity of Fe-Pt nanopar-  ticles (Fenton-like) for the first time with a maximum efficiency of 90% by the end of the 90  min reaction time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 The study of the decomposition of MB based on the Pt/Fe route was not  pursued by many researchers considering the cost of synthesis involved, despite knowing the  fact that the issue associated with the cost of production could be eliminated by employing  a magnetically responsive core-shell particle system decorated with platinum nanoparticles,  as the particles can be recovered by applying magnetic force externally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Such particles are  regarded as rough particles with controlled surface deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' We aim to demonstrate the  chemical decomposition of MB (an example system of azo dyes) and tetracycline (an exam-  ple system of pharmaceutical drugs) using nano and micron-sized polystyrene (PS) rough  particles decorated with platinum as zero charge metal catalyst particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Since the organ-  ics mentioned above is chemically deactivated, there is no need to regenerate the catalyst  particles once the reaction is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Furthermore, since the particles can be made mag-  netically responsive and easily separated by applying magnetic force externally, it is possible  to carry out the treatment process continuously with less or no manual intervention without  regeneration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Thus, as outlined above, the method of heterogeneous catalyst particles is a  cost-effective alternative, as the costs spent are predominantly linked to the capital budget  3  or one-time investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  In the context of the synthesis of rough particles concerning immobilization of metal-  lic nanoparticles, a common approach has been to decorate the surface of core particles  with gold (Au) or any other metallic nanoparticles via electrostatic interaction or covalent  bonding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='17–19 These routes demand the core particles be cationic to adsorb/deposit the gold  nanoparticles (anionic) to make suitable rough colloids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' However, it is of our interest to im-  mobilize platinum (Pt) instead of Au nanoparticles to explore its application in the field of  contaminant removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The synthesis route follows the covalent bonding route in which the Pt  nanoparticles are deposited by the chemical reduction of platinic acid using a reducing agent  such as sodium borohydride (NaBH4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' In 2012, Hsieh and Lin devised Fe-Pt nanoparticles to  deactivate MB dye to achieve greater reaction extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 However, the method offered by Hsieh  and Lin is slightly complicated and involves heating up to 297◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Therefore, it is essential  to study the heterogeneous catalysts of many kinds to find the suitability of the method  to tackle various pollutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Thus, we have chosen to extend the studies beyond Fe-based  supports to get insight into the degradation kinetics against other organics removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' For  instance, the as-synthesized Pt-based rough particles have shown good catalytic activity in  decomposing hydrogen peroxide, which helps degrade methylene blue (MB) and tetracycline  (TC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Our suggested technique achieves a near-100% degrading efficiency within 10 to 40  minutes at lower concentrations (200 ppm), making it a viable alternative to Fenton-based  AOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  The remainder of the manuscript is structured as follows: To provide the reader a bet-  ter understanding, we first address the synthesis, characterization, and application of Pt-  decorated PS rough particles for wastewater treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Consequently, we show the proof-  of-concept (POC) by using magnetically-responsive rough particles (MR-RP) for the appli-  cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' In addition to the application of discolouration or deactivation, this POC portion  addresses the separation of catalyst particles under an externally applied magnetic force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  4  Materials and Methods  Materials  We used Invitrogen™ grade amidine functionalized polystyrene (PS) latex particles purchased  from Thermo Fisher Scientific, India, to prepare catalytic-enabled rough particles for treating  the wastewater containing specific contaminants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' We used polystyrene particles of different  sizes for carrying out the decomposition reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The size of the particles employed, as  per the manufacturer’s specifications, are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='5, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' However, the size of the  particles determined based on image analysis using a scanning electron microscope (SEM)  and dynamic light scattering (DLS) technique are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='579 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='006, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='022 ±  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='001 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' An electrophoretic study was carried out to determine the particles’ potential at  the boundary between the shear and diffuse plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=', Zeta potential (ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The ζ potential  of the particles determined for PS particles of size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 (rounded-off) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02 µm in 1  mM NaCl electrolyte medium were 46 ± 3, 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='8 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='2 and 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='7 ± 1 mV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' To  prepare a core-shell assembly of magnetically-responsive rough particles, we employed iron  oxide (IO) nanoparticles purchased from Sigma Aldrich Chemicals Pvt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=', India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Polydi-  allyldimethylammonium chloride (DADMAC), gifted by Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Sarang Gumfekar, IIT Ropar,  was used to modify the IO particles by imparting positive sites to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Chloropla-  tinic acid hexahydrate (H2PtCl6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6H2O), used as precursor solution to deposit platinum (Pt)  nanoparticles on the PS surface, was procured from Sigma-Aldrich Chemicals Pvt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=',  India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Sodium borohydride (NaBH4), obtained from Sigma-Aldrich Chemicals Pvt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=',  was used as a reducing agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Hydrogen peroxide (30 w/v %), used in the decomposition  reaction mixture to generate hydroxy radical, was procured from Sigma-Aldrich Chemicals  Pvt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=', India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  The model pollutants used to demonstrate the deactivation kinetics,  such as methylene blue and tetracycline, were procured from Sigma-Aldrich Chemicals Pvt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=', India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' A hydrogen peroxide test strip (MQuant® Peroxide-Test) received from Sigma-  Aldrich Chemicals Pvt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=', India, was used to quantitatively predict the catalytic-assisted  5  decomposition kinetics under the presence of rough colloids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' All reagents received were ana-  lytical grades and used without any further purification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' We used deionized water obtained  from a Smart2Pure™ water purification system (Make: Thermo Fisher Scientific, Model:  Smart2Pure 12 UV/UF) for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Synthesis of PS rough particles decorated with Pt  First, we prepared a chloroplatinic acid hexahydrate solution using DI water such that the  precursor concentration in the stock solution was 10 mM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' We devised a wet-chemical method  to achieve a uniform deposition of Pt nanoparticles on the PS surface in an endless array  of monolayer fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The deposition of Pt nanoparticles was realized by using the method  reported by our group that involves the deposition of gold (Au) nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='20 We used a  slightly different strategy, including a wet-chemical deposition by nucleating Pt nanoparticles  instead of Au on the PS surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Since both the precursors of Au and Pt bear a net negative  charge, favouring the attachment of ions over the positive sites of PS due to net electrostatic  attraction, the modified protocol helped us synthesize Pt-decorated PS particles in a single  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Unlike the work of Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=', our modified protocol is devoid of pre-deposition of  Au nanoparticles as seeds to catalyze the reduction reaction of PtCl2−  6  ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Scheme 1  depicts the schematic description of making rough particles and their application towards  wastewater treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  As explained in Scheme 1A, a known concentration of platinum  precursor (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='5 mM), PS particles (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='5 mg/mL) are mixed using a magnetic stirrer for up to  30 min before adding a reducing agent, sodium borohydride, to the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Subsequently,  after the equilibration of the mixture for about 30 min, the chemical reduction reaction is  induced by adding 5 mL of 10 mM sodium borohydride solution completely into the beaker  containing the reaction mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  The reaction is allowed to continue until the solution  becomes clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' To ensure the reaction is complete, we have used sodium borohydride ten  times in excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Additionally, we have given sufficient time for reaction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=', 4 hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' We have  maintained the reaction time of four hours in all the experiments that involve the deposition  6  of Pt on the PS surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Note: The reaction mixture becomes transparent and clear when the  chemical reduction process continues for up to four hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Further, we allowed the samples  to centrifugation to separate the modified PS particles by removing the supernatant and  washing them several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The particles recovered in this way are further utilized to study  the kinetics of MB and TC decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Table 1 presents the size and zeta potential (ζ)  of the bare PS and the rough PS particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Selection of target pollutants  We chose to work with two representative pollutants from the family of azo dyes and an-  tibiotic drugs, such as MB and TC, respectively, for the entire decomposition studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' We  endeavoured to understand the role of rough particles in the chemical deactivation of these  target pollutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The initial pollutants concentration was maintained constant at 15 µM  throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' We studied the degradation of these pollutants in batch and semi-batch condi-  tions to replicate the real-time process of large-scale operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Heterogeneous batch degradation process  All degradation experiments that involve batch-wise operations were performed in different  initial volumes, such as 25, 50 and 100 mL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The rough particle concentrations employed  were 200, 100, and 50 ppm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' For the studies, we prepare the required mixture  using the rough particles at appropriate concentrations along with the target pollutants  with known initial concentration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' 15 µM, to induce the decomposition reaction in the  presence of a known amount of H2O2, 5 vol% per the total volume mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The  reaction is set to proceed for the desired time by adding the radical generator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=', H2O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  We continuously stirred the reaction mixture using the stirrer at a programmed speed of  140 RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The entire reaction was conducted at room temperature, 25◦C, throughout the  desired duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  7  Heterogeneous semi-batch degradation process  All degradation experiments that involved semi-batch operations were performed in different  initial dosing volume at a uniform interval of 10 min until completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The rough particles  concentration employed were 200, 100, and 50 PPM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' For the studies, we prepare  the required mixture using the rough particles at appropriate concentrations along with the  target pollutants with known initial concentration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' 15 µM, to induce the decomposition  reaction in the presence of a known amount of H2O2, 5 vol% per the total initial volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The  reaction is initiated to proceed for the desired length by adding the radical generator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=',  H2O2 at a desired dosing volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Using the stirrer, we stirred the reaction mixture nonstop  at a programmed speed of 140 RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The entire decomposition reaction was conducted at  room temperature, 25◦C, throughout the desired duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Scheme 1B and E depicts the schematic description showing the catalytic action of PS  rough particles and the process of discoloration of MB due to contact with the modified PS  particles of varying sizes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Hydrogen peroxide decomposition  To understand the kinetics of H2O2 decomposition, we decoupled the decomposition reactions  of MB and TC by eliminating them in the lab-scale reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Subsequently, in a separate  set of experimental studies, the reaction flasks containing H2O2 and rough particles were  employed to conduct studies at different lengths of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' We recorded the concentration vs  time of H2O2 as the reaction proceeds to understand the effect of contacting patterns on the  catalyst-assisted H2O2 decomposition reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Characterization  The zeta potential of bare PS and rough PS particles were measured based on the elec-  trophoretic light scattering (ELS) technique using Zetasizer procured from Malvern Instru-  8  ments, Model: Zetasizer Nano ZSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The average diameter of PS of size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 µm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='0  µm were determined using a high-resolution scanning electron microscope (HRSEM), Make:  FEI, USA and Model: Inspect F50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The hydrodynamic radius of PS of size 22 nm was  deduced using the dynamic light scattering (DLS), Make: Malvern Instruments and Model:  Zetasizer Nano ZSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The kinetics of decomposition reactions of MB and TC were studied by  capturing concentration as a function of decomposition times of MB and TC using UV-VIS  spectrophotometers (Make: Hach company US, Model: DR3900 & Make: PerkinElmer Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=',  USA, Model: LAMBDA 365)  Scheme 1: Schematic description showing the process of synthesizing rough particles and  their application towards deactivation of the target pollutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' A) Schematic illustration  showing the synthesis methodology of making rough particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' B) Schematic illustration  showing the application of rough particles in treating wastewater containing the target pol-  lutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' C and D) FESEM and EDX spectrum of 1 µm rough particles modified with Pt  nanoparticles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' E) Vial pictures showing the discoloration of MB dye in the  presence of rough particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Results & Discussion  FESEM was used to determine the size of PS particles and the morphological changes in  PS rough particles due to the reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' EDX was performed concurrently to identify the  9  A  B  Fpeed=140rpm  Temperature=25°C  NaBH4  t=4 hr  H,02  NH  t(min)  H,PtCl  H,PtCl6  NaBH  PS  PS-Pt  functionalized  (Core-shell)  At%  E  c  Element  wt%  D  657K  H,02  HO + HO←  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='45  v90  584K  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='55  9E66  511K  438K  365K  292K  219K  146K  Pt-coated rough particle  73K  Pt  %  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='2  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='5Table 1: Average size and zeta potential of the PS particles  Type  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Size (µm)  Zeta potential (mV)  Bare  Modified  PS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='009  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='0  33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='006  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='8 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='2  33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='1  46 ± 3  35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='4  IO  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='001  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='8  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='2  presence of Pt nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Figure 1 displays the surface morphology of surface-engineered  PS rough particles decorated with Pt nanoparticles in the form of islands (Figures 1A and  1B) and the EDX spectrum used to validate the deposition of Pt nanoparticles on the surface  of PS (Figures 1C and 1D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' As depicted in Figures 1C and 1D, the distinctive peaks at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='5  KeV indicate the existence of Pt nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Next, we discuss the performance of rough particles in batch and semi-batch conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  This study highlights the engineering processes that can substantially impact when transi-  tioning between real-time applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Figure 2 refers to the decomposition reaction cat-  alyzed by rough particles (Pt/PS) in the presence of a moderate concentration of 5% H2O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Note: The absorbance maxima at 358 nm from each UV-Vis spectral plot were utilised to  determine the concentration as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The concentration of particles and the  pollutants used for the comparison study are 100 ppm and 15 µM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' According to  the data, the highest TC elimination may be accomplished when the reaction is conducted  under semi-batch conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' On the other hand, when similar decomposition kinetics was  run under batch circumstances, the catalyst-assisted heterogeneous response produced a high  reaction rate for MB removal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=', 100% elimination of MB in 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' These results  intrigued us to set up reaction conditions appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' We, therefore, systematically con-  ducted the heterogeneous catalyst-assisted reactions using the rough particles of desired sizes  and concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  10  Figure 1: Structural characterization of modified PS particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' A and B) FESEM images of  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' C and D) EDX spectrum of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Scale  bars correspond to 500 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  11  A  B  603K  c  657K  D  536K  584K  469K  511K  402K  438K  335K  365K  268K  292K  201K  219K  134K  146K  Pt  Pt  67K  73K  Pt  C  Pt  Pt  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='2  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='8  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='2  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='8  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='1Figure 2: Decomposition of TC and MB with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Removal of TC when carried out in  batch (A) and semi-batch (B) setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Removal of MB when carried out in batch (C) and  semi-batch (D) setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The concentration of rough particles, MB, TC, and H2O2 employed  are 100 ppm, 15 µM, 15 µM, and 5%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  12  100  A  100  B  80  80-  60  H  60 -  40  40-  20  20-  0  0-  0  10  20  30  40  50  60  70  80  0  10  20  30  40  50  60  70  80  100  c  100-  D  田  日  日  日  日  T  80-  80  Efficiency (%)  60  60-  40-  40-  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  20  0-  0  10  20  30  40  50  60  70  80  0  10  20  30  40  50  60  70  80  Time (min)  Time (min)To this end, we demonstrate the performance of rough particles in semi-batch conditions  to determine the optimum parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Since the semi-batch operation was effective for TC,  we conducted all experiments related to TC removal using a semi-batch set-up to find the  optimal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Figure 3 displays the kinetics of the decomposition reaction of TC carried out  in semi-batch operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' By comparing Figures 2 and 3, the kinetic data suggest that pollu-  tants’ concentration has an inverse effect on the reaction rate associated with TC removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Therefore, the semi-batch catalyzed reaction favours maximal elimination at a higher rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  In contrast to batch mode, this condition improved the reaction rate because TC concentra-  tions were kept as low as possible due to drop-wise addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' We infer from Figure 3 that the  maximum efficiency is directly proportional to particle concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Table 2 summarizes  the performance of rough particles employed to deactivate TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' As shown in Figure 3 and  Table 2, the rough particles with a size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 µm and concentration 200 ppm along with  a dosing volume of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='416 mL H2O2 exhibited the best performance among the given set of  rough particles and the experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Table 2: Summary of performances of the rough particles over removal of TC  Type  Size (µm)  Initial TC conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' (µM)  Particles conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' (ppm)  Dosing volume (mL)  Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Efficiency (%)  Time (min)  Pt-PS  1  15  100  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='25  100  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='416  100  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02  40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='66  65  100  Further, we discuss the performance of rough particles in batch conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Since the  batch operation was effective for MB, we ran all experiments related to MB removal using  a batch mode to determine the optimal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Figure 4 displays the kinetics of the decom-  position reaction of MB carried out in batch operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Note: The absorbance maxima at  664 nm from each UV-Vis spectral plot were utilized to determine the concentration as a  function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' We infer that the performances of the rough particles of size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6  µm, respectively, overlap significantly over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Barring 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02 µm, the rate of decomposi-  tion data shows direct dependency on the concentration of rough particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Therefore, the  batch-catalyzed reaction favours the maximal elimination rate at a higher concentration of  13  Figure 3: Effect of dosing volume of H2O2 on the decomposition of TC with time under  semi-batch mode at regular interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' A-C) Removal of TC at different dosing volume of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='416, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='624, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='833 mL of H2O2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' D-F) Removal of TC at different dosing  volume of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='833, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='25, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='66 mL of H2O2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' G-I) Removal of TC at different  dosing volume of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='66, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='5, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='33 mL of H2O2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The concentration of rough  particles used are 200 ppm (A-C), 100 ppm (D-F), and 50 ppm (G-I), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The  initial concentration of TC employed for the study is 15 µM, throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02μm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02μm  B  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02μm  c  100-  100-  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6μm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 μm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6μm  1um  1μm  1μm  80  80  80-  (%)  60  60  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  40-  40  40  TC  20-  20  20  0-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  0  0  102030  40  50  60  70  80  90100110  0  10  20  30  40  50  60  70  80  0  10  20  30  40  50  60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02μm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02μm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02μm  100-  D  100-  F  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6μm  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6μm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6μm  1um  1um  1um  80  80-  80-  removal (%)  60  60  60-  40  40  40  20-  20-  20-  0  0  010  2030  40  50  60  70  80  90  100110  0  10  20  30  40  50  60  70  80  0  10  20  30  40  50  60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02μm  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02μm  G  H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02μm  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6μm  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 μm  100-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6μm 1um  1 μm  1μm  80-  80-  80-  (%) I  60-  60-  60-  40-  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  40-  TC  20-  20-  20-  0-  01020  30  40  50  60  70  80  90100110  0  10  20  30  40  50  60  70  80  0  10  20  30  40  50  60  Time (min)  Time (min)  Time (min)rough particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' In contrast to the semi-batch mode, this condition improved the reaction  rate because the concentration of MB was kept high at the start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Table 3 summarizes the  performance of rough particles employed to deactivate MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' As shown in Figure 4 and Table  3, the rough particles with a size of 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 µm and concentration 200 ppm at a concen-  tration of 5% H2O2 exhibited the best performance among the given set of rough particles  and the experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Figure 4: Removal of MB catalyzed by varying sizes of rough particles at a concentration of  A) 200 ppm, B) 100 ppm, and C) 50 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Table 3: Summary of performances of rough particles against MB removal at 5% H2O2  Type  Size (µm)  Initial MB conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' (µM)  Particle conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' (ppm)  Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Efficiency (%)  Time (min)  PS  1 & 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6  15  200  100  10  100  100  20  50  100  40  Besides % of elimination, the order of the reaction and rate constant also provide good  insight into the kinetics of the Pt-PS-assisted reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Langmuir-Hinshelwood (LH) kinetics  is the most popular model for describing the kinetics of heterogeneous catalytic processes  which is described as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' 1 below,22  r = −dC  dt = krKC  1 + KC  (1)  where r, kr, K, and C refer to the rate of reaction that changes with time, limiting  rate constant of reaction at maximum coverage under the given experimental conditions,  15  A  B  100  100-  100  80-  80-  80-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02μm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02μm  (%)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 μm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02μm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 μm  (%)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6μm  1μm  60  1um  Efficiency  60  Efficiency  1um  40  40  20-  20-  0 -  70  Time (min)  Time(min)  Time (min)equilibrium constant for adsorption of the substrate onto the catalyst, and concentration at  anytime t during degradation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Most of the time, researchers approximated Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' 1 to the first order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=', n=1, by assuming  KC << 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Thus, when −ln  �  C  C0  �  is plotted on the ordinate and time is plotted on the abscissa,  the slope of the straight line is the product of kr and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' However, if the value of the slope  found is >> 1, then the assumption that KC << 1 becomes invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  That means the  subsequent kinetics will be in zero order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Figure 5 displays the pseudo-first-order kinetics  corresponding to the heterogeneous reactions catalyzed by the rough particles of varying  sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The rate constants for the reaction catalyzed by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='0 µm particles at 50 ppm  are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='048 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='032 min−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The reported values are in the same order as that  of the performance of the Fe-Pt nanoparticles reported by Hsieh and Lin for 5 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 Due  to variations in the concentration of the rough particles, the reported values in our case are  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='4 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='0 times higher than the literature value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 It is also noteworthy that the process  catalyzed by the rough particles at a concentration of 50 ppm completes (100%) at the end  of the 40-minute reaction time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' By 90 minutes, the absorbance maximum near 665 nm for  the Fe-Pt particles exhibited by Hsieh and Lin had dropped by just 90 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6  Before moving on to the following section and exhibiting the proof-of-concept employ-  ing magnetically responsive particles, we propose probable causes for the disparity in per-  formance between the different-sized rough particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' It is generally understood that the  nanoparticles offer a more significant surface area (SA) for a given concentration in a batch  reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' For instance, we anticipate that the smaller rough particles would expose Pt sig-  nificantly due to the more significant number of particles than the same concentration of  bigger rough particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' By referring to Table 1 for zeta potential values and EDX analysis of  the rough particles of varying sizes, we cancel out any variation associated with the quality  of the Pt deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Consequently, the SA of rough particles should increase as their size  decreases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='0 µm > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 µm > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' In contrast, according to Figures 4 and 5, we  observed a kinetics that is an inverse function of particle size when examining the decom-  16  Figure 5: Pseudo first-order kinetic plot for the deactivation of MB in the presence of rough  particles of size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='0 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  17  3  1 μm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 μm  2  ln (C/Co)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  0  0  10  20  30  40  Time (min)position of MB using the rough particles under batch conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' We attribute this unusual  behaviour to the crowding effect of bubbles and the buoyancy effect of the nanoparticles  caused due to the attachment of bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' As a result, 1) particles diffuse rapidly towards  the interface, decreasing the contact time with a key reactant, and 2) the catalyst’s power is  diminished due to a layer of bubbles around the particles obstructing the active sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Figure  6 displays the peroxide decomposition reaction as a function of time in minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Except  for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='0 µm, we found a marked difference in performance concerning 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The  differences in peroxide breakdown kinetics found using the detector strips could be linked to  the aforementioned factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The video showing the rapid generation of the bubbles and the  movement of the nanoparticles from bulk to the interface can be found in Supplementary  Information (SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Proof-of-concept  Thus far, we have demonstrated a straightforward methodology to make the rough particles  by modifying the surface of the PS using the Pt nanoparticles through the wet-chemical route  and their performances in decontamination application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Here, we show that the magnetically-  responsive rough particles (MR-RP) can be made by decorating platinum-coated PS nanopar-  ticles with iron oxide nanoparticles like a core-shell structure to accomplish chemical deac-  tivation and particle recovery in a single step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  The benefits of MR-RP include ease of  fabrication of Pt nanoparticles as the deposition takes place by attractive electrostatic forces  between Pt precursor (negatively charged) and the positively charged PS particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' This  combination helps us achieve a uniform deposition like a monolayer of islands distributed  in space throughout without needing stabilizers like oleic acid and oleyl amine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Secondly,  unlike the method proposed by Hsieh and Lin,6 the technique does not demand heating at  high temperature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=', 297◦C, and the entire process can be completed in room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  To this end, we demonstrate the discolourization of MB using MR-RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Scheme 2 de-  scribes the process of making MR-RP using a wet-chemical route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Since the process follows  18  Figure 6: H2O2 decomposition using the rough particles of varying sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  19  60000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02 μm  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 μm  50000  1 μm  40000  (udd) ZOH  30000  20000  10000  0  0  10  20  30  40  50  60  70  80  Time (min)the modifications in bulk, we can eliminate the shortcomings such as scale-up and yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  As shown in Scheme 2, the attractive forces between IO and Polydiallyldimethylammonium  chloride (PolyDADMAC) allow the binding of the polymers on the surface of IO nanopar-  ticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Subsequently, the suspension containing Pt-modified PS nanoparticles of size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='02  µm was introduced into the polymer-coated IO mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' This addition results in the self-  assembly of negatively charged Pt-modified PS and positively charged polymer-modified IO  leading to the formation of core-shell particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Note: Since the sufficient concentration of  PS nanoparticles was considered during the wet-chemical deposition and a suitable ratio of  Pt-decorated PS and polymer-decorated IO was chosen before mixing, there were very little  or no free Pt nanoparticles found in the supernatant solution (Please see supplementary  video-2 shown in the supplementary information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' We demonstrate a start-to-finish proof-  of-concept of a two-step process (removal of MB + recovery of MR-RP) achieved by the  proposed surface-engineered nanoparticles via a real-time video (Please see supplementary  video-3 shown in the supplementary information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Scheme 2: Schematic description showing the process of synthesizing magnetically-responsive  rough particles using the wet-chemical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  20  Sonication (1hr)  Soaking (12 hr)  PolvDADMAC  Iron oxide nanoparticles  Polymer grafted  Iron oxide nanoparticles  Mixing (300 RPM)  1 hr  Platinum coated polystyrene nanoparticles  Platinum coated  Polymer  grafted  coating on iron oxide nanoparticles  Polystyrene nanoparticles  Iron oxide nanoparticlesGeneral Remarks  We comment on the proposed method’s applicability and shortcomings in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The  proposed method works well against pollutants that accept electrons for chemical deactiva-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' For instance, the process would not fit well for the methyl orange dye, a contaminant  that would give up electrons while decomposing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' We tested the applicability of the rough  particles against methyl orange and found that the decomposition rate is relatively slow  compared to the reaction of its counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Thus, we understand that the proposed rough  particles show catalytic power and facilitate electron transfer from free radicals to pollutants  over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Secondly, the proposed method requires PS particles to be positively charged to  attract the precursor ions, which are negatively charged and induce nucleation of platinum  nanoparticles on the surface of PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Therefore, the method is unsuitable for making rough  particles if the terminal end groups of PS are negatively charged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Conclusion  We established a facile approach for producing Pt-studded PS-based rough particles with  good catalytic activity in a single step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The proposed method offers an exciting route to  make catalyst particles on desired supports at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' As the existing process re-  quires chemical precipitation followed by heating the reaction mixture to 297◦C, our proposed  approach could serve as an excellent alternative to existing Fenton-based heterogeneous par-  ticles studded with Pt metal nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Our study revealed that the contacting patterns  play a significant role in determining the performance of the rough particles over the decom-  position of any given pollutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' For instance, the catalytic action of PS-Pt rough particles  was visible when tetracycline decomposition was performed in semi-batch rather than batch  settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Conversely, batch-wise operations recorded the maximal output of rough particles  in eliminating methylene blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' We studied the impact of particle sizes, concentrations and  dosing rate of hydrogen peroxide on the catalytic activity of rough particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' To cite an ex-  21  ample, the desired operating parameters to achieve 100% efficiency vary for different rough  particles of size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 µm (Optimum concentration = 200 ppm, dosing volume = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='416 mL) and  1µm (Optimum concentration = 100 ppm, dosing volume = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='25 mL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' The heterogeneous  reaction modelled by Langmuir-Hinshelwood kinetics indicates that the catalytic reaction  follows the pseudo-first-order process with rate constants of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='048 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='032 min−1 for the  reaction catalyzed by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='6 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='0 µm-sized rough particles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Using a well-known  layer-by-layer technique, we presented a straightforward methodology to achieve a core-shell  assembly of magnetically-responsive rough particles (MR-RP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' In a batch reactor config-  uration, we demonstrated proof-of-concept for the decomposition of methylene blue using  MR-RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Since we can deactivate pollutants and recover catalyst particles in a single cycle,  the proposed technique would be effective in real-time and large-scale applications without  incurring operating expenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  Acknowledgement  We thank Dr Neethu Thomas (Postdoctoral Researcher) and Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Parasuraman Swami-  nathan, Department of Metallurgical and Materials Engineering, IIT Madras, India, for  helping us with the FESEM combined with the EDX analysis of our samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' MS thanks  IIT Ropar for providing the seed grant (ISIRD phase II) to set up a laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
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+page_content=' UV light-assisted degradation of methyl orange, methylene blue,  phenol, salicylic acid, and rhodamine B: photolysis versus photocatalyis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Water, Air,  & Soil Pollution 2017, 228, 1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  (5) Mohammed, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
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+page_content=' Khaleefa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
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+page_content=' Journal  of Engineering and Sustainable Development 2021, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
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+page_content='  (7) Saleh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
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+page_content=' Degradation of methylene blue and congo-red dyes using Fenton,  photo-Fenton, sono-Fenton, and sonophoto-Fenton methods in the presence of iron  (II, III) oxide/zinc oxide/graphene (Fe3O4/ZnO/graphene) composites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Separation and  Purification Technology 2019, 210, 563–573.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  (8) Kirchon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
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+page_content='  (9) Ikhlaqa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
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+page_content=' Ultrasonics  sonochemistry 2019, 56, 213–228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
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+page_content=' Degradation kinetics and mechanism of  oxytetracycline by hydroxyl radical-based advanced oxidation processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Chemical En-  gineering Journal 2016, 284, 1317–1327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
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+page_content=' MacKay, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Advanced oxidation processes for organic  contaminant destruction based on the Fenton reaction and related chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Critical  reviews in environmental science and technology 2006, 36, 1–84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content='  (14) Giordano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
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+page_content=' Granato, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
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+page_content=' Katovic, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
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+page_content=' Si-  ciliano, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
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+page_content=' Tagarelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
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+page_content=' Wet hydrogen peroxide catalytic oxidation of  olive oil mill wastewaters using Cu-zeolite and Cu-pillared clay catalysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
+page_content=' Catalysis  Today 2007, 124, 240–246, Advanced Catalytic Oxidation Processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE2T4oBgHgl3EQfcAds/content/2301.03891v1.pdf'}
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+arXiv:2301.11713v1  [math.CO]  27 Jan 2023
+Dispersed graph labellings
+William J. Martin∗1 and Douglas R. Stinson†2,3
+1Department of Mathematical Sciences, Worcester Polytechnic Institute,
+Worcester MA, 01609, USA
+2David R. Cheriton School of Computer Science, University of Waterloo,
+Waterloo ON, N2L 3G1, Canada
+3School of Mathematics and Statistics, Carleton University, Ottawa, Ontario,
+K1S 5B6, Canada
+January 30, 2023
+Abstract
+A k-dispersed labelling of a graph G on n vertices is a labelling of the vertices
+of G by the integers 1, . . . , n such that d(i, i + 1) ≥ k for 1 ≤ i ≤ n − 1. DL(G)
+denotes the maximum value of k such that G has a k-dispersed labelling. In this
+paper, we study upper and lower bounds on DL(G). Computing DL(G) is NP-
+hard. However, we determine the exact values of DL(G) for cycles, paths, grids,
+hypercubes and complete binary trees. We also give a product construction and
+we prove a degree-based bound.
+1
+Introduction
+Many graph labelling problems have been studied over the years, starting with the
+graceful labellings introduced by Rosa. Gallian’s dynamic survey [3] is an excel-
+lent starting point for this area of research. We assume standard graph-theoretic
+terminology throughout this paper, e.g., as defined in [1].
+Let G be a graph having vertex set V , where |V | = n. Let d(u, v) denote the
+distance between any two vertices u and v in G. It is easy to observe that G has a
+hamiltonian path if and only if there is a labelling of the vertices with the integers
+1, . . . , n such that d(i, i + 1) = 1 for 1 ≤ i ≤ n − 1. Here we consider a labelling
+problem motivated by the requirement that consecutively labelled vertices should
+be far apart. Thus we define a k-dispersed labelling to be a labelling of the vertices
+of G by the integers 1, . . . , n such that d(i, i+1) ≥ k for 1 ≤ i ≤ n−1. Equivalently,
+∗W.J. Martin’s research is supported by NSF DMS Award #1808376.
+†D.R. Stinson’s research is supported by NSERC discovery grant RGPIN-03882.
+1
+
+for a graph G = (V, E) with |V | = n, we could define a k-dispersed labelling to be
+a bijection φ : {1, . . . , n} → V such that d(φ(i), φ(i + 1)) ≥ k for 1 ≤ i ≤ n − 1.
+Although it is not a main topic of this paper, we could also consider a “circular”
+variant of the above definition. We define a k-circular-dispersed labelling to be a
+k-dispersed labelling that satisfies the additional property that d(n, 1) ≥ k. Equiva-
+lently, a k-circular-dispersed labelling could be defined to be a bijection φ : Zn → V
+such that d(φ(i), φ(i + 1)) ≥ k for 0 ≤ i ≤ n − 1 (in this definition, for convenience,
+the vertices are labelled with the elements of Zn).
+We note that most of the labelling problems discussed in [3] do not involve
+distances between the vertices in a graph. One exception is the problem of radio
+labellings [3, §7.4].
+Let DL(G) denote the maximum value of k such that G has a k-dispersed la-
+belling and let DL◦(G) denote the maximum value of k such that G has a k-circular-
+dispersed labelling. Here are two easy preliminary lemmas that we state without
+proof.
+Lemma 1.1. If H is a spanning subgraph of G (obtained by removing edges but no
+vertices), then DL(H) ≥ DL(G) and DL◦(H) ≥ DL◦(G).
+Lemma 1.2. DL◦(G) ≤ DL(G).
+Our first real result relates the values DL(G) to the distance k graph of G. The
+distance k graph of G, denoted Gk, is the graph in which two vertices x and y are
+joined by an edge if d(x, y) = k. Clearly G1 = G. Denote G∗
+k−1 = G1 ∪ · · · ∪ Gk−1
+and let Hk = (G∗
+k−1)c. So two vertices of Hk are adjacent if the distance between
+them (in G) is at least k.
+Theorem 1.3. For a graph G, DL(G) ≥ k if and only if Hk contains a hamiltonian
+path. Further, DL◦(G) ≥ k if and only if Hk contains a hamiltonian cycle.
+Proof. Suppose that G is a graph on vertex set V , where |V | = n. Suppose G has a
+k-dispersed labelling, say α, and define P = (1, . . . , n). For any i, 1 ≤ i ≤ n − 1, it
+holds that d(i, i + 1) ≥ k (because α is a k-dipsersed labelling) and hence {i, i + 1}
+is an edge of Hk. It follows that P is a hamiltonian path in Hk.
+The proof of the converse result is similar, as is the proof of the corresponding
+result for DL◦(G).
+Corollary 1.4. Computing DL(G) is NP-hard.
+Proof. Suppose that O is an oracle that computes DL(G) in polynomial time. We
+can use O to solve the NP-complete hamiltonian path problem as follows. Given
+a graph G, run O on Gc. Observe that the H2(Gc) = G. So O(Gc) ≥ 2 if and only
+if G has a hamiltonian path.
+The rest of the paper is organized as follows. In Section 2, we determine the
+exact value of DL(G) for cycles, paths, grids, hypercubes and complete binary trees.
+For these classes of graphs, DL(G) = r(G) or r(G) − 1, where r(G) is the radius
+2
+
+of the graph G.
+In Section 3, we prove a product construction: we show that
+DL◦(G □ H) ≥ DL◦(G) + DL◦(H), provided that the number of vertices in G and H
+is relatively prime. In Section 4, we prove a degree-based lower bound on DL(G).
+Finally, in Section 5, we list a few interesting open questions.
+2
+Computing DL(G) for some classes of graphs
+We first consider cycles, for which the values DL(G) can easily be determined.
+Theorem 2.1. Let Cn denote a cycle of length n. Then DL(Cn) = (n − 1)/2 if n
+is odd, and DL(Cn) = (n − 2)/2 if n is even.
+Proof. The maximum distance between two points of Cn is n/2 if n is even and
+(n − 1)/2 if n is odd.
+First, suppose that n is odd and let k = (n − 1)/2. The graph Hk is a single
+(hamiltonian) cycle of length n, so DL(Cn) ≥ k follows from Theorem 1.3. Also,
+Hk+1 is the empty graph, so DL(Cn) ≤ k.
+Next, suppose that n is even and let k = n/2. Hk consists of k disjoint edges, so
+Hk is not hamiltonian and therefore DL(Cn) ≤ k − 1. We now study the structure
+of the graph Hk−1, which is a cubic graph. We consider two subcases.
+First, suppose n ≡ 0 mod 4. Here, the edges in Hk−1 that are not in Hk form a
+hamiltonian cycle, so we are done. If n ≡ 2 mod 4, then Hk−1 is a prism; the edges
+in Hk−1 that are not in Hk form two disjoint cycles of length n/2. It is an easy
+exercise to verify that the prism contains a hamiltonian path. Thus DL(Cn) ≥ k − 1
+when n is even, and the proof is complete.
+The eccentricity of a vertex v is the quantity ǫ(v) = max{d(v, u) : u ∈ V }. The
+radius of a graph G, denoted r(G), is the minimum eccentricity of any vertex, i.e.,
+r(G) = min{ǫ(v) : v ∈ V }.
+Theorem 2.2. For a graph G, DL(G) ≤ r(G).
+Proof. Suppose that G has a k-dispersed labelling.
+Let i be a vertex such that
+ǫ(i) = r(G). If i < n, then let j = i + 1; if i = n, then let j = i − 1. We must
+have d(i, j) ≥ k since the labelling is k-dispersed. However, d(i, j) ≤ ǫ(i) = r(G).
+Therefore, since k ≥ DL(G), it follows that DL(G) ≤ r(G).
+We computed DL(Cn) for all cycles Cn in Theorem 2.1. It is easy to verify that
+r(Cn) = n/2 if n is even and r(Cn) = (n−1)/2 if n is even. Hence, DL(Cn) = r(Cn)
+if n is odd and DL(Cn) = r(Cn) − 1 if n is even.
+We will require some additional related definitions for later use. A vertex is
+v ∈ V is uniquely eccentric if there is a unique vertex u such that d(u, v) = ǫ(v). A
+vertex v ∈ V is a central vertex if ǫ(v) = r(G).
+3
+
+19
+20
+21
+22
+23
+24
+13
+14
+15
+16
+17
+18
+7
+8
+9
+10
+11
+12
+1
+2
+3
+4
+5
+6
+Figure 1: The graph L4,6
+2.1
+Paths
+A simple class of graphs to consider are the paths. Let Pm denote the path having
+m edges and m + 1 vertices. It is easy to check that the radius of a path is given by
+the following formula:
+r(Pm) =
+�
+m
+2
+if m is even
+m+1
+2
+if m is odd.
+Theorem 2.3. DL(Pm) = r(Pm) for any path Pm.
+Proof. First, suppose m is even. An m
+2 -dispersed labelling is as follows:
+2 4
+· · ·
+m 1 3
+· · ·
+m + 1.
+For odd m, an m+1
+2 -dispersed labelling is as follows:
+2 4
+· · ·
+m + 1 1 3
+· · ·
+m.
+2.2
+Grids
+As another, more complicated class of graphs, we consider the m × n grid graphs
+(or lattice graphs), which we denote by Lm,n. The graph L4,6 is depicted in Figure
+1. The following lemma will be useful.
+Lemma 2.4. Suppose a graph G contains three central vertices, each of which is
+uniquely eccentric. Then DL(G) ≤ r(G) − 1.
+Proof. Suppose that G contains n vertices that are labelled 1, . . . , n. Further, sup-
+pose that DL(G) = r(G). At least one of the three hypothesized central vertices
+must receive a label i, where 2 ≤ i ≤ n − 1. Consider the vertices labelled i − 1 and
+i+1. Since vertex i is uniquely eccentric, either d(i−1, i) > r(G) or d(i, i+1) > r(G).
+This contradicts the assumption that DL(G) = r(G).
+4
+
+2
+4
+6
+8
+10
+12
+14
+9
+11
+13
+1
+3
+5
+7
+Figure 2: A 4-dispersed labelling of L2,7
+The graph L4,6, has radius r(L4,6) = 5 and there are four central vertices, namely
+vertices 9, 10, 15 and 16. Each of these four central vertices is uniquely eccentric:
+d(9, 24) = 5, d(10, 19) = 5, d(15, 6) = 5 and d(16, 11) = 5. (To illustrate, vertex 24
+is the only vertex that is distance five from vertex 9. A path of length five from 9
+to 24 is indicated by the blackened vertices in Figure 1.) Therefore, from Lemma
+2.4, it follows that DL(L4,6) ≤ 4. We will prove a bit later that DL(L4,6) = 4.
+More generally, we have the following upper bound.
+Theorem 2.5. Suppose m and n are both even. Then DL(Lm,n) ≤ (m + n)/2 − 1.
+Proof. When m and n are even, it is easy to see that r(Lm,n) = (m + n)/2 and this
+graph has four central vertices, each of which is uniquely eccentric. Apply Lemma
+2.4.
+For other cases of m and n, we note the following.
+Lemma 2.6.
+1. Suppose m and n are both odd. Then r(Lm,n) = (m + n)/2 − 1 and Lm,n has
+one central vertex, which is not uniquely eccentric.
+2. Suppose m+n is odd. Then r(Lm,n) = (m+n−1)/2 and Lm,n has two central
+vertices, neither of which is uniquely eccentric.
+Thus Lemma 2.4 cannot be applied in these cases, so we cannot rule out the
+possibility that DL(Lm,n) = r(Lm,n) if at least one of m and n is odd. In fact, we
+will prove in this section that DL(Lm,n) = r(Lm,n) if at least one of m and n is odd;
+and DL(Lm,n) = r(Lm,n) − 1 if m and n are both even.
+First, we solve the case of 2 × n grids.
+Theorem 2.7. DL(L2,n) = r(L2,n) = (n + 1)/2 for all odd n ≥ 3.
+Proof. The first row of L2,n is labelled 2, 4, . . . , 2n and the second row is labelled
+n + 2, n + 4, . . . , 2n − 1, 1, 3, . . . , n.
+Theorem 2.8. DL(L2,n) = r(L2,n) − 1 = n/2 for all even n ≥ 4.
+Proof. The first row of L2,n is labelled 2, 4, . . . , 2n and the second row is labelled
+n + 3, n + 5, . . . , 2n − 1, 1, 3, . . . , n + 1.
+5
+
+2
+4
+6
+8
+10
+12
+14
+16
+11
+13
+15
+1
+3
+5
+7
+9
+Figure 3: A 4-dispersed labelling of L2,8
+See Figures 2 and 3 for illustrations of the constructions in Theorems 2.7 and
+2.8, respectively.
+Suppose m ≥ 4 is even. We can construct optimal labellings of m×n grids from
+optimal labellings of 2 × n grids recursively.
+Theorem 2.9. Suppose m ≥ 2 is even. Then
+DL(Lm,n) =
+�
+m+n−2
+2
+if n is even
+m+n−1
+2
+if n is odd.
+Proof. For odd n, we start with the n+1
+2 -dispersed labelling of L2,n constructed in
+Theorem 2.7; for even n, we start with the n
+2-dispersed labelling of L2,n constructed
+in Theorem 2.8. Let the first row of one of these optimal labellings be denoted A
+and let the second row be denoted B. Then construct the labelling of Lm,n having
+the following m rows:
+A
+A + 2n
+...
+A + (m − 2)n
+B
+B + 2n
+...
+B + (m − 2)n
+See Figure 4 for an illustration of the construction when m = 4 and n = 7.
+It is not hard to prove that the result is an m+n−2
+2
+-dispersed labelling of Lm,n
+when n is even, and an m+n−1
+2
+-dispersed labelling of Lm,n when n is odd. Basically,
+we have interleaved m/2 isomorphic copies of the labellings of L2,n. The interleaving
+increases the distances between consecutively labelled vertices within a particular
+L2,n by m/2 − 1 (this is because, for any two consecutively labelled vertices, one is
+in A and the other is in B). In the case of even n, the minimum distance of n/2 is
+increased to n/2 + m/2 − 1 = (m + n − 2)/2, as desired. For odd n, the minimum
+distance of (n+1)/2 is increased to (n+1)/2+m/2−1 = (m+n−1)/2, as desired.
+It is also necessary to consider the distance between the “last vertex” in one copy
+of L2,n and the “first” vertex in the next copy. The first vertex in a copy of of L2,n
+is the middle element (when n is odd) or the leftmost of the two middle elements
+6
+
+2
+4
+6
+8
+10
+12
+14
+9
+11
+13
+1
+3
+5
+7
+16
+18
+20
+22
+24
+26
+28
+23
+25
+27
+15
+17
+19
+21
+Figure 4: A 4-dispersed labelling of L4,7
+(when n is even) of row B, while the last element is the rightmost vertex in row A.
+The distance between these two vertices is (n + 1)/2 when n is odd, and (n + 2)/2
+when n is even. The interleaving increases the distance by m/2, which is more than
+what is required in the resulting labelling of Lm,n.
+We now study Lm,n for odd m. We can also assume that n is odd, because the
+case of odd m and even n is equivalent to the case where m is even and m is odd,
+and this is covered by Theorem 2.9. We begin with m = 3.
+Theorem 2.10. DL(L3,n) = r(L3,n) = (n + 1)/2 for all odd n ≥ 3.
+Proof. It suffices to present an n+1
+2 -dispersed labelling of L3,n when n is odd. The
+first row of L2,n is labelled
+6, 9, . . . , 3n, 3;
+the second row is labelled
+3n + 7
+2
+, 3n + 13
+2
+, . . . , 3n − 1, 2, 5, . . . , 3n + 1
+2
+;
+and the third row is labelled
+1, 4, . . . , 3n − 2
+To get from vertex j to vertex j + 1, we proceed (n − 1)/2 vertices to the
+right and one row up, wrapping around if necessary. So the distance is at least
+(n − 1)/2 + 1 = (n + 1)/2 (because wrapping around cannot decrease the distance
+between the two vertices).
+For future use, we observe that the vertex labelled 1 is in the bottom left corner
+of the grid. The vertex labelled 2 is the central vertex of the grid). The last vertex
+(i.e., the one labelled 3m) is immediately to the left of the vertex in the top right
+corner of the grid.
+Example 2.1. We construct a 3-dispersed labelling of L3,5 using Theorem 2.10; see
+Figure 5. Therefore, DL(L3,5) = 3.
+7
+
+11
+14
+2
+5
+8
+6
+9
+12
+15
+3
+1
+4
+7
+10
+13
+Figure 5: A 3-dispersed labelling of L3,5
+Theorem 2.11. DL(Lm,n) = r(Lm,n) = (n + m)/2 − 1 for all odd m, n ≥ 3.
+Proof. In view of Theorem 2.10, we can assume that m ≥ 5. Denote m = 3 + 2t;
+then t = (m − 3)/2 and t ≥ 1.
+Let the (n + 1)/2-dispersed labelling of L2,n have rows A and B, as in the proof
+of Theorem 2.9. Also, let the (n + 1)/2-dispersed labelling of L3,n constructed in
+Theorem 2.10 have rows C, D and E. Then construct the labelling of Lm,n having
+the following m rows:
+C
+A + 3n
+A + 5n
+...
+A + (2t + 1)n
+D
+B + 3n
+B + 5n
+...
+B + (2t + 1)n
+E
+See Figure 6 for an illustration of the construction when m = 5 and n = 5.
+We are interleaving the n+1
+2 -dispersed labelling of L3,n with t copies of the n+1
+2 -
+dispersed labelling of L2,n. We need to prove that the distance between consecutively
+labelled vertices (say j and j + 1) is at least (m + n − 2)/2. The proof naturally
+divides into a number of cases:
+case 1 Vertices j and j + 1 are both in the same copy of L2,n.
+case 2 Vertex j is the last vertex in one copy of L2,n and vertex j + 1 is the first
+vertex in the next copy of L2,n.
+case 3 Vertices j and j + 1 are both in the L3,n.
+case 4 Vertex j is the last vertex in the L3,n and vertex j + 1 is the first vertex in
+the first copy of L2,n (i.e., j = 3n).
+8
+
+11
+14
+2
+5
+8
+6
+9
+12
+15
+3
+1
+4
+7
+10
+13
+22
+24
+16
+18
+20
+17
+19
+21
+23
+25
+Figure 6: A 4-dispersed labelling of L5,5
+Case 1 is similar to the analogous case in the proof of Theorem 2.9. The distance
+between vertices j and j + 1 in Lm,n is t larger than the distance in L3,n. So the
+distance is at least
+n + 1
+2
++ t = n + 1 + m − 3
+2
+= m + n − 2
+2
+.
+Case 2 is also similar to the analogous case in the proof of Theorem 2.9.
+For case 3, the distance between vertices j and j + 1 in Lm,n is t larger than the
+distance in L3,n. As in case 1, the distance is at least (m + n − 2)/2.
+For case 4, we use the fact that vertex 3n is in row 1 and column n − 1, and
+vertex 3n + 1 is in row t + 3 and column (n + 1)/2. So the distance is at least
+n − 3
+2
++ t + 2 = n − 3 + m − 3
+2
++ 2 = m + n − 2
+2
+.
+This completes the proof.
+2.3
+Hypercubes
+The n-dimensional hypercube, denoted Qn, is a graph having vertex set V = (Z2)n.
+Two vertices are adjacent in Qn if they differ in exactly one co-ordinate. For v, w ∈
+V , where w = (v1, . . . , vn) and w = (w1, . . . , wn), it is easy to see that
+d(v, w) = |{i : vi ̸= wi}|.
+Here are some other easily verified properties of Qn.
+Lemma 2.12. Let v ∈ (Z2)n and suppose 0 ≤ i ≤ n. Then there are precisely
+�n
+i
+�
+vertices w such that d(v, w) = i in the graph Qn.
+9
+
+Corollary 2.13. For any integer n ≥ 2, r(Qn) = n. Further, every vertex is a
+uniquely eccentric central vertex.
+Lemma 2.4 and Corollary 2.13 immediately imply that DL(Qn) ≤ n − 1 for all
+n ≥ 2. We will prove that DL(Qn) ≥ n−1 by constructing a suitable labelling of the
+vertices of Qn. It suffices to find a permutation of the n-tuples in V , say v1, . . . , v2n,
+such that d(vj, vj+1) ≥ n−1 for 1 ≤ j ≤ 2n −1. In the resulting dispersed labelling,
+vertex vj is labelled j for 1 ≤ j ≤ 2n.
+We construct the permutations recursively. In order for the recursive construc-
+tion to work, we require some additional properties. Hence, for each n ≥ 2, we
+will construct a particular permutation Πn = (v1, . . . , v2n) satisfying the following
+properties.
+1. d(vj, vj+1) ≥ n − 1 for 1 ≤ j ≤ 2n − 1,
+2. v1 = (0, . . . , 0),
+3. v2n−1 = (1, . . . , 1, 0),
+4. v2n−1+1 = (0, . . . , 0, 1),
+5. v2n = (1, . . . , 1).
+Construction 2.14. We describe how to construct the permutations Πn. To begin,
+when n = 2, Π2 is the following permutation:
+(0, 0), (1, 0), (0, 1), (1, 1).
+Now we construct Πn+1 from Πn.
+For notational convenience, we denote Πn =
+(v1, . . . , v2n) and Πn+1 = (w1, . . . , w2n+1).
+• For 1 ≤ j ≤ 2n−1, define
+wj =
+�
+(vj, 0)
+if j is odd
+(vj, 1)
+if j is even.
+• For 2n−1 + 1 ≤ j ≤ 2n, define
+wj =
+�
+(vj, 1)
+if j is odd
+(vj, 0)
+if j is even.
+• For 2n + 1 ≤ j ≤ 2n + 2n−1, define
+wj =
+�
+(vj−2n, 1)
+if j is odd
+(vj−2n, 0)
+if j is even.
+10
+
+• For 2n + 2n−1 + 1 ≤ j ≤ 2n+1, define
+wj =
+�
+(vj−2n, 0)
+if j is odd
+(vj−2n, 1)
+if j is even.
+Example 2.2. Π3 is the following permutation:
+(0, 0, 0), (1, 0, 1), (0, 1, 1), (1, 1, 0), (0, 0, 1), (1, 0, 0), (0, 1, 0), (1, 1, 1).
+Theorem 2.15. For all n ≥ 2, DL(Qn) = n − 1.
+Proof. We have already observed that DL(Qn) ≤ n − 1, so it suffices to prove that
+DL(Qn) ≥ n−1. This follows by showing that each of the permutations Πn obtained
+from Construction 2.14 satisfies the properties 1–5 enumerated above, which we
+prove by induction on n.
+First, we observe that each Πn is a permutation. This is because every n-tuple
+in Πn is appended with both a 0 and a 1 in the construction of Πn+1.
+For n = 2, the stated properties can be verified easily.
+Now, suppose that
+properties 1–5 hold for a particular Πn, where n ≥ 2. We will show that properties
+1–5 hold for Πn+1. We use the notation from Construction 2.14.
+When j ̸= 2n, wj and wj+1 are obtained from two consecutive vj’s, which are
+assumed to have distance at least n − 1, by induction. In most cases, one of wj
+and wj+1 is appended with a 1 and the other is appended with a 0. In these cases,
+d(wj, wj+1) ≥ n.
+The only cases where wj and wj+1 are appended with the same symbol are
+when j = 2n−1, 2n or 2n + 2n−1. When j = 2n−1 or 2n + 2n−1, properties 3 and 4
+ensure that wj and wj+1 are obtained from two consecutive vj’s that have distance
+n. When j = 2n, wj and wj+1 are obtained from v2n and v1 (resp.), which have
+distance n due to properties 2 and 5. Hence, property 1 is satisfied for Πn+1.
+It is easy to verify that properties 2–5 hold for Πn+1, so the proof is complete.
+We note that there is an alternative approach that gives an easier proof of
+Theorem 2.15 in the case where n is even. It is well-known that, for n even, the
+distance n − 1 graph of Qn is isomorphic to Qn (see, e.g., [2, p. 265, Remark (i)]).
+One such isomorphism is obtained by flipping the bits of all odd-weight n-tuples. It
+is also well-known that Qn has a hamiltonian cycle; the binary reflected gray code is
+one example (see, e.g., [4]). Hence it immediately follows that DL◦(Qn) = DL(Qn) =
+n − 1 if n is even.
+The distance n − 1 graph of Qn is disconnected if n is odd, so this particular
+approach will not work.
+2.4
+Complete Binary Trees
+The radius of the complete binary tree of depth n is n. We will show that the
+complete binary tree of depth n admits an n-dispersed labelling.
+11
+
+ε
+0
+1
+00
+01
+10
+11
+000
+001
+010
+011
+100
+101
+110
+111
+1
+10
+3
+12
+14
+5
+7
+2
+4
+6
+8
+9
+11
+13
+15
+Figure 7: A 3-dispersed labelling of the complete binary tree of depth three.
+Let T have vertex set {ε} ∪ {0, 1} ∪ · · · ∪ {0, 1}n where ε denotes the empty
+string. We have an edge edge joining x to y if one can be obtained from the other
+by deleting the rightmost bit. Let |x| denote the length of the binary string x. A
+3-dispersed labelling of a complete binary tree of depth three is given in Figure 7.
+Our labelling is as follows. The root node, ε, receives label 1. The “left” nodes—
+those vertices whose strings begin with zero—receive even labels, with the rule that
+labels 2, 4, 6, . . . , 2n are used on leaf nodes and the remaining even labels occur on
+nodes of degree three. The “right” nodes—those vertices whose strings begin with
+one—receive odd labels, with the rule that labels 2n + 1, 2n + 3, . . . , 2n+1 − 1 are
+used for leaf nodes and the remaining odd labels ℓ ≥ 3 occur on nodes of degree
+three.
+A concrete rule for such a labelling is given as follows, where N(x) denotes the
+integer whose binary representation is x. We set
+ℓ(x) =
+
+
+
+
+
+2N(x) + 1
+if x1 = 1;
+2N(x) + 2
+if x1 = 0 and x is a leaf node;
+2N(x) + 2n + 2|x|
+if x1 = 0 and x is not a leaf node.
+One may verify by inspection that each label 0 < i < 2n+1 is used once. Among any
+consecutive pair of labels i, i + 1, one of these appears at a leaf node and the path
+from this node to the other one passes through the root of the tree and therefore
+has length at least n.
+Therefore we have proven the following result.
+Theorem 2.16. Let Tn denote the complete binary tree of depth n. Then DL(Tn) =
+n.
+Proof. The labelling described above shows that DL(Tn) ≥ n. Since Tn has radius
+n, it follows immediately that DL(Tn) = n.
+12
+
+3
+A product construction
+The Cartesian product of graphs G = (V (G), E(G)) and H = (V (H), E(H)), de-
+noted G □ H, is the graph with vertex set V (G) × V (H) and edge set
+E(G □ H) =
+�
+{(u, v), (u′, v′)} : u = u′, {v, v′} ∈ E(H) or {u, u′} ∈ E(G), v = v′ �
+.
+Suppose G is a graph on m vertices having a k-circular-dispersed labelling and H
+is a graph on n vertices having a k′-circular-dispersed labelling. When gcd (m, n) =
+1, we will show how to combine these two labellings to give a (k + k′)-circular-
+dispersed labelling of the product graph G □ H.
+Theorem 3.1. Let G and H be graphs with |V (G)| = m, |V (H)| = n and gcd (m, n) =
+1. Then DL◦(G □ H) ≥ DL◦(G) + DL◦(H).
+Proof. Denote k = DL◦(G) and k′ = DL◦(H).
+Suppose ℓG : Zm → V (G) is a
+k-circular-dispersed labelling of G and ℓH : Zn → V (H) is a k′-circular-dispersed
+labelling of H. For 0 ≤ i ≤ mn − 1, define
+ℓ(i) = (ℓG(i mod m), ℓH(i mod n)).
+Since gcd (m, n) = 1, ℓ is a bijection from Zmn to the vertices of G □ H.
+The
+distance between two vertices in G □ H is just the sum of the distances between
+the corresponding projections (i.e., to the first and second coordinates) in G and
+H.
+Hence, for any i, where 0 ≤ i ≤ mn − 1, The distance between ℓ(i) and
+ℓ(i + 1 mod mn) in G □ H is the sum
+dG(i mod m, i + 1 mod m) + dH(i mod n, i + 1 mod n) ≥ k + k′,
+as desired.
+We illustrate the product construction by returning to grids. Our goal is not to
+compute all the values DL◦(Lm,n), but rather to illustrate how Theorem 3.1 can be
+applied to a specific family of graphs.
+Recall that Lm,n is an m by n grid and Pm−1 is a path having m vertices. It is
+easy to see that Pm−1 □ Pn−1 is isomorphic to Lm,n. We can obtain lower bounds
+on DL◦(Lm,n) if we have lower bounds on DL◦(Pm−1) and DL◦(Pn−1). Therefore,
+we first look at circular-dispersed labellings of paths.
+Lemma 3.2. Suppose a graph G = (V, E) contains two central vertices, say u and
+v, both of which are uniquely eccentric. Suppose also that {u, v} ∈ E and suppose
+r(G) ≥ 2. Then DL◦(G) ≤ r(G) − 1.
+Proof. Suppose that |V | = n and DL◦(G) = r(G). Vertices u and v must receive the
+labels 1 and n since they are uniquely eccentric. However, d(u, v) = 1 ≤ r(G) − 1,
+so it follows that DL◦(G) ≤ r(G) − 1.
+13
+
+Lemma 3.3. Let Pm be the path with m edges (and m + 1 vertices). If m is even,
+then DL◦(Pm) = DL(Pm) = m
+2 . If m ≥ 3 is odd, then DL◦(Pm) = DL(Pm)−1 = m−1
+2 .
+Proof. We already have shown that DL(Pm) = r(Pm) in Theorem 2.3. For m even,
+the labelling
+2 4
+· · ·
+m 1 3
+· · ·
+m + 1
+given in Theorem 2.3 has d(1, m + 1) = m
+2 . Hence, DL◦(Pm) = DL(Pm) = m
+2 when
+m is even.
+However, when m ≥ 3 is odd, Pm contains two uniquely eccentric central vertices
+that are adjacent, so Lemma 3.2 asserts that DL◦(Pm) ≤ r(Pm)−1 = (m−1)/2. To
+achieve a m−1
+2 -circular dispersed labelling of Pm, we start with the above labelling
+of Pm−1 (on m vertices) and attach a new vertex labelled m + 1 to the left end of
+the path.
+Theorem 3.4. Suppose gcd(m, n) = 1, m ≥ 3 and n ≥ 3. Then
+DL◦(Lm,n) ≥
+�
+m+n
+2
+− 1
+if m, n are both odd
+m+n−3
+2
+if m + n is odd.
+Proof. This is an immediate application of Theorem 3.2 and Lemma 3.3.
+We do not consider the case where both m and n are even in Theorem 3.4. This
+is because gcd(m, n) ≥ 2 when m and n are even and hence the hypotheses cannot
+be satisfied.
+Corollary 3.5. Suppose gcd(m, n) = 1, m ≥ 3 is odd and n ≥ 3 is odd. Then
+DL◦(Lm,n) = m + n
+2
+− 1.
+Proof. From Theorem 2.11, we have DL(Lm,n) = m+n
+2
+−1 when n ≥ 3 and m ≥ 3 are
+odd. Lemma 1.2 shows that DL◦(Lm,n) ≤ DL(Lm,n). Finally, Theorem 3.4 shows
+that DL◦(Lm,n) ≥ m+n
+2
+− 1 if n ≥ 3 and m ≥ 3 are odd and gcd(m, n) = 1.
+4
+Degree-based bounds
+We now consider a degree-based lower bound on DL(G). Suppose G has n vertices.
+For x ∈ V and i ≥ 0, let
+ηi(x) = |{y ∈ V : d(x, y) = i}| ,
+and
+κ(x) = max
+�
+j :
+j
+�
+i=0
+ηi(x) ≤ n/2
+�
+.
+For example, in Qm, it is easy to see that ηi(x) =
+�m
+i
+�
+for each vertex x and
+κ(x) = ⌊m−1
+2 ⌋.
+14
+
+Lemma 4.1. DL(G) ≥ min{1 + κ(x) : x ∈ V (G)}.
+Proof. For any vertex x, the there are at least n/2 vertices y such that d(x, y) ≥ 1+
+κ(x). Therefore, by Dirac’s Theorem (e.g., see [1, Theorem 18.4]), Hk is hamiltonian
+for k = min{1 + κ(x) : x ∈ V (G)}. Now apply Theorem 1.3.
+Theorem 4.2. If G has n vertices and maximum degree ∆ > 2, then
+DL(G) ≥ 1 + log∆−1
+�
+1 + (n − 2)(∆ − 2)
+2∆
+�
+.
+(1)
+Proof. For any x ∈ V (G) and i ≥ 1, ni(x) ≤ ∆(∆−1)i−1. In order to apply Lemma
+4.1, we compute the largest value of ℓ such that
+1 + ∆ + ∆(∆ − 1) + · · · + ∆(∆ − 1)ℓ ≤ n/2.
+This inequality is equivalent to
+1 + ∆(1 − (∆ − 1)ℓ+1)
+2 − ∆
+≤ n
+2.
+Elementary algebra tells us that this can be rewritten as
+ℓ + 1 ≤ log∆−1
+�
+1 + (n − 2)(∆ − 2)
+2∆
+�
+.
+Since κ(x) ≥ ℓ + 1, it therefore follows from Lemma 4.1 that DL(G) ≥ ℓ + 2 and
+hence the inequality (1) holds.
+Corollary 4.3. If G has n vertices and maximum degree three, then
+DL(G) ≥ log2
+�n + 4
+6
+�
++ 1.
+Proof. Apply Theorem 4.2 with ∆ = 3.
+For example, if n = 3×2t−4 and ∆ = 3, then Corollary 4.3 says that DL(G) ≥ t.
+5
+Open questions
+There are many interesting open questions. We list a few now.
+1. Which graphs G have DL(G) = r(G)?
+2. We note that DL(G) ∈ {r(G), r(G)−1} for all the graphs considered in Section
+2. We ask if there is an infinite family of graphs G1, G2, . . . such that
+lim
+i→∞(r(G) − DL(G)) = ∞.
+Actually, at this point, we do not have any examples of graphs G with DL(G) ≤
+r(G) − 2.
+15
+
+3. Is there an efficient algorithm to find “good” dispersed labellings of graphs
+(i.e., labellings where DL(G) is “close to” r(G)) for certain classes of graphs
+(e.g., trees)?
+4. Is there a generalization of the product construction (Theorem 3.1) that han-
+dles cases where gcd(m, n) > 1?
+References
+[1] J.A. Bondy and U.S.R. Murty. Graph Theory, Springer, 2008.
+[2] A.E. Brouwer, A.M. Cohen and A. Neumaier. Distance-Regular Graphs,
+Springer, 1989.
+[3] J.A. Gallian. Graph labeling, version 25. Electron. J. Combin., Article #DS6,
+Dec. 2, 2022.
+[4] C. Savage. A survey of combinatorial Gray codes. SIAM Review 39 (1997),
+605–629.
+16
+
diff --git a/l9FKT4oBgHgl3EQfDy1k/content/tmp_files/load_file.txt b/l9FKT4oBgHgl3EQfDy1k/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..ca81c71bc716555c2efd5ce020c31bf8abc47528
--- /dev/null
+++ b/l9FKT4oBgHgl3EQfDy1k/content/tmp_files/load_file.txt
@@ -0,0 +1,608 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf,len=607
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='11713v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='CO]  27 Jan 2023  Dispersed graph labellings  William J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Martin∗1 and Douglas R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Stinson†2,3  1Department of Mathematical Sciences, Worcester Polytechnic Institute,  Worcester MA, 01609, USA  2David R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Cheriton School of Computer Science, University of Waterloo,  Waterloo ON, N2L 3G1, Canada  3School of Mathematics and Statistics, Carleton University, Ottawa, Ontario,  K1S 5B6, Canada  January 30, 2023  Abstract  A k-dispersed labelling of a graph G on n vertices is a labelling of the vertices  of G by the integers 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , n such that d(i, i + 1) ≥ k for 1 ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' DL(G)  denotes the maximum value of k such that G has a k-dispersed labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' In this  paper, we study upper and lower bounds on DL(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Computing DL(G) is NP-  hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' However, we determine the exact values of DL(G) for cycles, paths, grids,  hypercubes and complete binary trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We also give a product construction and  we prove a degree-based bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  1  Introduction  Many graph labelling problems have been studied over the years, starting with the  graceful labellings introduced by Rosa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Gallian’s dynamic survey [3] is an excel-  lent starting point for this area of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We assume standard graph-theoretic  terminology throughout this paper, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=', as defined in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Let G be a graph having vertex set V , where |V | = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Let d(u, v) denote the  distance between any two vertices u and v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' It is easy to observe that G has a  hamiltonian path if and only if there is a labelling of the vertices with the integers  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , n such that d(i, i + 1) = 1 for 1 ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Here we consider a labelling  problem motivated by the requirement that consecutively labelled vertices should  be far apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Thus we define a k-dispersed labelling to be a labelling of the vertices  of G by the integers 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , n such that d(i, i+1) ≥ k for 1 ≤ i ≤ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Equivalently,  ∗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Martin’s research is supported by NSF DMS Award #1808376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  †D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Stinson’s research is supported by NSERC discovery grant RGPIN-03882.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  1  for a graph G = (V, E) with |V | = n, we could define a k-dispersed labelling to be  a bijection φ : {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , n} → V such that d(φ(i), φ(i + 1)) ≥ k for 1 ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Although it is not a main topic of this paper, we could also consider a “circular”  variant of the above definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We define a k-circular-dispersed labelling to be a  k-dispersed labelling that satisfies the additional property that d(n, 1) ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Equiva-  lently, a k-circular-dispersed labelling could be defined to be a bijection φ : Zn → V  such that d(φ(i), φ(i + 1)) ≥ k for 0 ≤ i ≤ n − 1 (in this definition, for convenience,  the vertices are labelled with the elements of Zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  We note that most of the labelling problems discussed in [3] do not involve  distances between the vertices in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' One exception is the problem of radio  labellings [3, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Let DL(G) denote the maximum value of k such that G has a k-dispersed la-  belling and let DL◦(G) denote the maximum value of k such that G has a k-circular-  dispersed labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Here are two easy preliminary lemmas that we state without  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' If H is a spanning subgraph of G (obtained by removing edges but no  vertices), then DL(H) ≥ DL(G) and DL◦(H) ≥ DL◦(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' DL◦(G) ≤ DL(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Our first real result relates the values DL(G) to the distance k graph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The  distance k graph of G, denoted Gk, is the graph in which two vertices x and y are  joined by an edge if d(x, y) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Clearly G1 = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Denote G∗  k−1 = G1 ∪ · · · ∪ Gk−1  and let Hk = (G∗  k−1)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' So two vertices of Hk are adjacent if the distance between  them (in G) is at least k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' For a graph G, DL(G) ≥ k if and only if Hk contains a hamiltonian  path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Further, DL◦(G) ≥ k if and only if Hk contains a hamiltonian cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Suppose that G is a graph on vertex set V , where |V | = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Suppose G has a  k-dispersed labelling, say α, and define P = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' For any i, 1 ≤ i ≤ n − 1, it  holds that d(i, i + 1) ≥ k (because α is a k-dipsersed labelling) and hence {i, i + 1}  is an edge of Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' It follows that P is a hamiltonian path in Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  The proof of the converse result is similar, as is the proof of the corresponding  result for DL◦(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Computing DL(G) is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Suppose that O is an oracle that computes DL(G) in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We  can use O to solve the NP-complete hamiltonian path problem as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Given  a graph G, run O on Gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Observe that the H2(Gc) = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' So O(Gc) ≥ 2 if and only  if G has a hamiltonian path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' In Section 2, we determine the  exact value of DL(G) for cycles, paths, grids, hypercubes and complete binary trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  For these classes of graphs, DL(G) = r(G) or r(G) − 1, where r(G) is the radius  2  of the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  In Section 3, we prove a product construction: we show that  DL◦(G □ H) ≥ DL◦(G) + DL◦(H), provided that the number of vertices in G and H  is relatively prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' In Section 4, we prove a degree-based lower bound on DL(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Finally, in Section 5, we list a few interesting open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  2  Computing DL(G) for some classes of graphs  We first consider cycles, for which the values DL(G) can easily be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Let Cn denote a cycle of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Then DL(Cn) = (n − 1)/2 if n  is odd, and DL(Cn) = (n − 2)/2 if n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The maximum distance between two points of Cn is n/2 if n is even and  (n − 1)/2 if n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  First, suppose that n is odd and let k = (n − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The graph Hk is a single  (hamiltonian) cycle of length n, so DL(Cn) ≥ k follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Also,  Hk+1 is the empty graph, so DL(Cn) ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Next, suppose that n is even and let k = n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Hk consists of k disjoint edges, so  Hk is not hamiltonian and therefore DL(Cn) ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We now study the structure  of the graph Hk−1, which is a cubic graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We consider two subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  First, suppose n ≡ 0 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Here, the edges in Hk−1 that are not in Hk form a  hamiltonian cycle, so we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' If n ≡ 2 mod 4, then Hk−1 is a prism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' the edges  in Hk−1 that are not in Hk form two disjoint cycles of length n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' It is an easy  exercise to verify that the prism contains a hamiltonian path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Thus DL(Cn) ≥ k − 1  when n is even, and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  The eccentricity of a vertex v is the quantity ǫ(v) = max{d(v, u) : u ∈ V }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The  radius of a graph G, denoted r(G), is the minimum eccentricity of any vertex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=',  r(G) = min{ǫ(v) : v ∈ V }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' For a graph G, DL(G) ≤ r(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Suppose that G has a k-dispersed labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Let i be a vertex such that  ǫ(i) = r(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' If i < n, then let j = i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' if i = n, then let j = i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We must  have d(i, j) ≥ k since the labelling is k-dispersed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' However, d(i, j) ≤ ǫ(i) = r(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Therefore, since k ≥ DL(G), it follows that DL(G) ≤ r(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  We computed DL(Cn) for all cycles Cn in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' It is easy to verify that  r(Cn) = n/2 if n is even and r(Cn) = (n−1)/2 if n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Hence, DL(Cn) = r(Cn)  if n is odd and DL(Cn) = r(Cn) − 1 if n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  We will require some additional related definitions for later use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' A vertex is  v ∈ V is uniquely eccentric if there is a unique vertex u such that d(u, v) = ǫ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' A  vertex v ∈ V is a central vertex if ǫ(v) = r(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  3  19  20  21  22  23  24  13  14  15  16  17  18  7  8  9  10  11  12  1  2  3  4  5  6  Figure 1: The graph L4,6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='1  Paths  A simple class of graphs to consider are the paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Let Pm denote the path having  m edges and m + 1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' It is easy to check that the radius of a path is given by  the following formula:  r(Pm) =  �  m  2  if m is even  m+1  2  if m is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' DL(Pm) = r(Pm) for any path Pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' First, suppose m is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' An m  2 -dispersed labelling is as follows:  2 4  · ·  m 1 3  · ·  m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  For odd m, an m+1  2 -dispersed labelling is as follows:  2 4  · ·  m + 1 1 3  · ·  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='2  Grids  As another, more complicated class of graphs, we consider the m × n grid graphs  (or lattice graphs), which we denote by Lm,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The graph L4,6 is depicted in Figure  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The following lemma will be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Suppose a graph G contains three central vertices, each of which is  uniquely eccentric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Then DL(G) ≤ r(G) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Suppose that G contains n vertices that are labelled 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Further, sup-  pose that DL(G) = r(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' At least one of the three hypothesized central vertices  must receive a label i, where 2 ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Consider the vertices labelled i − 1 and  i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Since vertex i is uniquely eccentric, either d(i−1, i) > r(G) or d(i, i+1) > r(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  This contradicts the assumption that DL(G) = r(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  4  2  4  6  8  10  12  14  9  11  13  1  3  5  7  Figure 2: A 4-dispersed labelling of L2,7  The graph L4,6, has radius r(L4,6) = 5 and there are four central vertices, namely  vertices 9, 10, 15 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Each of these four central vertices is uniquely eccentric:  d(9, 24) = 5, d(10, 19) = 5, d(15, 6) = 5 and d(16, 11) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' (To illustrate, vertex 24  is the only vertex that is distance five from vertex 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' A path of length five from 9  to 24 is indicated by the blackened vertices in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=') Therefore, from Lemma  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='4, it follows that DL(L4,6) ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We will prove a bit later that DL(L4,6) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  More generally, we have the following upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Suppose m and n are both even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Then DL(Lm,n) ≤ (m + n)/2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' When m and n are even, it is easy to see that r(Lm,n) = (m + n)/2 and this  graph has four central vertices, each of which is uniquely eccentric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Apply Lemma  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  For other cases of m and n, we note the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Suppose m and n are both odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Then r(Lm,n) = (m + n)/2 − 1 and Lm,n has  one central vertex, which is not uniquely eccentric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Suppose m+n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Then r(Lm,n) = (m+n−1)/2 and Lm,n has two central  vertices, neither of which is uniquely eccentric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Thus Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='4 cannot be applied in these cases, so we cannot rule out the  possibility that DL(Lm,n) = r(Lm,n) if at least one of m and n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' In fact, we  will prove in this section that DL(Lm,n) = r(Lm,n) if at least one of m and n is odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  and DL(Lm,n) = r(Lm,n) − 1 if m and n are both even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  First, we solve the case of 2 × n grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' DL(L2,n) = r(L2,n) = (n + 1)/2 for all odd n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The first row of L2,n is labelled 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , 2n and the second row is labelled  n + 2, n + 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , 2n − 1, 1, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' DL(L2,n) = r(L2,n) − 1 = n/2 for all even n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The first row of L2,n is labelled 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , 2n and the second row is labelled  n + 3, n + 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , 2n − 1, 1, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  5  2  4  6  8  10  12  14  16  11  13  15  1  3  5  7  9  Figure 3: A 4-dispersed labelling of L2,8  See Figures 2 and 3 for illustrations of the constructions in Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='7 and  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Suppose m ≥ 4 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We can construct optimal labellings of m×n grids from  optimal labellings of 2 × n grids recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Suppose m ≥ 2 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Then  DL(Lm,n) =  �  m+n−2  2  if n is even  m+n−1  2  if n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' For odd n, we start with the n+1  2 -dispersed labelling of L2,n constructed in  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' for even n, we start with the n  2-dispersed labelling of L2,n constructed  in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Let the first row of one of these optimal labellings be denoted A  and let the second row be denoted B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Then construct the labelling of Lm,n having  the following m rows:  A  A + 2n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  A + (m − 2)n  B  B + 2n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  B + (m − 2)n  See Figure 4 for an illustration of the construction when m = 4 and n = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  It is not hard to prove that the result is an m+n−2  2  dispersed labelling of Lm,n  when n is even, and an m+n−1  2  dispersed labelling of Lm,n when n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Basically,  we have interleaved m/2 isomorphic copies of the labellings of L2,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The interleaving  increases the distances between consecutively labelled vertices within a particular  L2,n by m/2 − 1 (this is because, for any two consecutively labelled vertices, one is  in A and the other is in B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' In the case of even n, the minimum distance of n/2 is  increased to n/2 + m/2 − 1 = (m + n − 2)/2, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' For odd n, the minimum  distance of (n+1)/2 is increased to (n+1)/2+m/2−1 = (m+n−1)/2, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  It is also necessary to consider the distance between the “last vertex” in one copy  of L2,n and the “first” vertex in the next copy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The first vertex in a copy of of L2,n  is the middle element (when n is odd) or the leftmost of the two middle elements  6  2  4  6  8  10  12  14  9  11  13  1  3  5  7  16  18  20  22  24  26  28  23  25  27  15  17  19  21  Figure 4: A 4-dispersed labelling of L4,7  (when n is even) of row B, while the last element is the rightmost vertex in row A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  The distance between these two vertices is (n + 1)/2 when n is odd, and (n + 2)/2  when n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The interleaving increases the distance by m/2, which is more than  what is required in the resulting labelling of Lm,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  We now study Lm,n for odd m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We can also assume that n is odd, because the  case of odd m and even n is equivalent to the case where m is even and m is odd,  and this is covered by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We begin with m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' DL(L3,n) = r(L3,n) = (n + 1)/2 for all odd n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' It suffices to present an n+1  2 -dispersed labelling of L3,n when n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The  first row of L2,n is labelled  6, 9, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , 3n, 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  the second row is labelled  3n + 7  2  , 3n + 13  2  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , 3n − 1, 2, 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , 3n + 1  2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  and the third row is labelled  1, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , 3n − 2  To get from vertex j to vertex j + 1, we proceed (n − 1)/2 vertices to the  right and one row up, wrapping around if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' So the distance is at least  (n − 1)/2 + 1 = (n + 1)/2 (because wrapping around cannot decrease the distance  between the two vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  For future use, we observe that the vertex labelled 1 is in the bottom left corner  of the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The vertex labelled 2 is the central vertex of the grid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The last vertex  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=', the one labelled 3m) is immediately to the left of the vertex in the top right  corner of the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We construct a 3-dispersed labelling of L3,5 using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' see  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Therefore, DL(L3,5) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  7  11  14  2  5  8  6  9  12  15  3  1  4  7  10  13  Figure 5: A 3-dispersed labelling of L3,5  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' DL(Lm,n) = r(Lm,n) = (n + m)/2 − 1 for all odd m, n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' In view of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='10, we can assume that m ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Denote m = 3 + 2t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  then t = (m − 3)/2 and t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Let the (n + 1)/2-dispersed labelling of L2,n have rows A and B, as in the proof  of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Also, let the (n + 1)/2-dispersed labelling of L3,n constructed in  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='10 have rows C, D and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Then construct the labelling of Lm,n having  the following m rows:  C  A + 3n  A + 5n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  A + (2t + 1)n  D  B + 3n  B + 5n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  B + (2t + 1)n  E  See Figure 6 for an illustration of the construction when m = 5 and n = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  We are interleaving the n+1  2 -dispersed labelling of L3,n with t copies of the n+1  2 -  dispersed labelling of L2,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We need to prove that the distance between consecutively  labelled vertices (say j and j + 1) is at least (m + n − 2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The proof naturally  divides into a number of cases:  case 1 Vertices j and j + 1 are both in the same copy of L2,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  case 2 Vertex j is the last vertex in one copy of L2,n and vertex j + 1 is the first  vertex in the next copy of L2,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  case 3 Vertices j and j + 1 are both in the L3,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  case 4 Vertex j is the last vertex in the L3,n and vertex j + 1 is the first vertex in  the first copy of L2,n (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=', j = 3n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  8  11  14  2  5  8  6  9  12  15  3  1  4  7  10  13  22  24  16  18  20  17  19  21  23  25  Figure 6: A 4-dispersed labelling of L5,5  Case 1 is similar to the analogous case in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The distance  between vertices j and j + 1 in Lm,n is t larger than the distance in L3,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' So the  distance is at least  n + 1  2  + t = n + 1 + m − 3  2  = m + n − 2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Case 2 is also similar to the analogous case in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  For case 3, the distance between vertices j and j + 1 in Lm,n is t larger than the  distance in L3,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' As in case 1, the distance is at least (m + n − 2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  For case 4, we use the fact that vertex 3n is in row 1 and column n − 1, and  vertex 3n + 1 is in row t + 3 and column (n + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' So the distance is at least  n − 3  2  + t + 2 = n − 3 + m − 3  2  + 2 = m + n − 2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='3  Hypercubes  The n-dimensional hypercube, denoted Qn, is a graph having vertex set V = (Z2)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Two vertices are adjacent in Qn if they differ in exactly one co-ordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' For v, w ∈  V , where w = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , vn) and w = (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , wn), it is easy to see that  d(v, w) = |{i : vi ̸= wi}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Here are some other easily verified properties of Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Let v ∈ (Z2)n and suppose 0 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Then there are precisely  �n  i  �  vertices w such that d(v, w) = i in the graph Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  9  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' For any integer n ≥ 2, r(Qn) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Further, every vertex is a  uniquely eccentric central vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='4 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='13 immediately imply that DL(Qn) ≤ n − 1 for all  n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We will prove that DL(Qn) ≥ n−1 by constructing a suitable labelling of the  vertices of Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' It suffices to find a permutation of the n-tuples in V , say v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , v2n,  such that d(vj, vj+1) ≥ n−1 for 1 ≤ j ≤ 2n −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' In the resulting dispersed labelling,  vertex vj is labelled j for 1 ≤ j ≤ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  We construct the permutations recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' In order for the recursive construc-  tion to work, we require some additional properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Hence, for each n ≥ 2, we  will construct a particular permutation Πn = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , v2n) satisfying the following  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' d(vj, vj+1) ≥ n − 1 for 1 ≤ j ≤ 2n − 1,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' v1 = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , 0),  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' v2n−1 = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , 1, 0),  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' v2n−1+1 = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , 0, 1),  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' v2n = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We describe how to construct the permutations Πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' To begin,  when n = 2, Π2 is the following permutation:  (0, 0), (1, 0), (0, 1), (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Now we construct Πn+1 from Πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  For notational convenience, we denote Πn =  (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , v2n) and Πn+1 = (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , w2n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  For 1 ≤ j ≤ 2n−1, define  wj =  �  (vj, 0)  if j is odd  (vj, 1)  if j is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  For 2n−1 + 1 ≤ j ≤ 2n, define  wj =  �  (vj, 1)  if j is odd  (vj, 0)  if j is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  For 2n + 1 ≤ j ≤ 2n + 2n−1, define  wj =  �  (vj−2n, 1)  if j is odd  (vj−2n, 0)  if j is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  10  For 2n + 2n−1 + 1 ≤ j ≤ 2n+1, define  wj =  �  (vj−2n, 0)  if j is odd  (vj−2n, 1)  if j is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Π3 is the following permutation:  (0, 0, 0), (1, 0, 1), (0, 1, 1), (1, 1, 0), (0, 0, 1), (1, 0, 0), (0, 1, 0), (1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' For all n ≥ 2, DL(Qn) = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We have already observed that DL(Qn) ≤ n − 1, so it suffices to prove that  DL(Qn) ≥ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' This follows by showing that each of the permutations Πn obtained  from Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='14 satisfies the properties 1–5 enumerated above, which we  prove by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  First, we observe that each Πn is a permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' This is because every n-tuple  in Πn is appended with both a 0 and a 1 in the construction of Πn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  For n = 2, the stated properties can be verified easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Now, suppose that  properties 1–5 hold for a particular Πn, where n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We will show that properties  1–5 hold for Πn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We use the notation from Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  When j ̸= 2n, wj and wj+1 are obtained from two consecutive vj’s, which are  assumed to have distance at least n − 1, by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' In most cases, one of wj  and wj+1 is appended with a 1 and the other is appended with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' In these cases,  d(wj, wj+1) ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  The only cases where wj and wj+1 are appended with the same symbol are  when j = 2n−1, 2n or 2n + 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' When j = 2n−1 or 2n + 2n−1, properties 3 and 4  ensure that wj and wj+1 are obtained from two consecutive vj’s that have distance  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' When j = 2n, wj and wj+1 are obtained from v2n and v1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' ), which have  distance n due to properties 2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Hence, property 1 is satisfied for Πn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  It is easy to verify that properties 2–5 hold for Πn+1, so the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  We note that there is an alternative approach that gives an easier proof of  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='15 in the case where n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' It is well-known that, for n even, the  distance n − 1 graph of Qn is isomorphic to Qn (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=', [2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' 265, Remark (i)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  One such isomorphism is obtained by flipping the bits of all odd-weight n-tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' It  is also well-known that Qn has a hamiltonian cycle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' the binary reflected gray code is  one example (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=', [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Hence it immediately follows that DL◦(Qn) = DL(Qn) =  n − 1 if n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  The distance n − 1 graph of Qn is disconnected if n is odd, so this particular  approach will not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='4  Complete Binary Trees  The radius of the complete binary tree of depth n is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We will show that the  complete binary tree of depth n admits an n-dispersed labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  11  ε  0  1  00  01  10  11  000  001  010  011  100  101  110  111  1  10  3  12  14  5  7  2  4  6  8  9  11  13  15  Figure 7: A 3-dispersed labelling of the complete binary tree of depth three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Let T have vertex set {ε} ∪ {0, 1} ∪ · · · ∪ {0, 1}n where ε denotes the empty  string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We have an edge edge joining x to y if one can be obtained from the other  by deleting the rightmost bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Let |x| denote the length of the binary string x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' A  3-dispersed labelling of a complete binary tree of depth three is given in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Our labelling is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The root node, ε, receives label 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The “left” nodes—  those vertices whose strings begin with zero—receive even labels, with the rule that  labels 2, 4, 6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , 2n are used on leaf nodes and the remaining even labels occur on  nodes of degree three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The “right” nodes—those vertices whose strings begin with  one—receive odd labels, with the rule that labels 2n + 1, 2n + 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' , 2n+1 − 1 are  used for leaf nodes and the remaining odd labels ℓ ≥ 3 occur on nodes of degree  three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  A concrete rule for such a labelling is given as follows, where N(x) denotes the  integer whose binary representation is x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We set  ℓ(x) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  2N(x) + 1  if x1 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  2N(x) + 2  if x1 = 0 and x is a leaf node;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  2N(x) + 2n + 2|x|  if x1 = 0 and x is not a leaf node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  One may verify by inspection that each label 0 < i < 2n+1 is used once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Among any  consecutive pair of labels i, i + 1, one of these appears at a leaf node and the path  from this node to the other one passes through the root of the tree and therefore  has length at least n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Therefore we have proven the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Let Tn denote the complete binary tree of depth n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Then DL(Tn) =  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' The labelling described above shows that DL(Tn) ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Since Tn has radius  n, it follows immediately that DL(Tn) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  12  3  A product construction  The Cartesian product of graphs G = (V (G), E(G)) and H = (V (H), E(H)), de-  noted G □ H, is the graph with vertex set V (G) × V (H) and edge set  E(G □ H) =  �  {(u, v), (u′, v′)} : u = u′, {v, v′} ∈ E(H) or {u, u′} ∈ E(G), v = v′ �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Suppose G is a graph on m vertices having a k-circular-dispersed labelling and H  is a graph on n vertices having a k′-circular-dispersed labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' When gcd (m, n) =  1, we will show how to combine these two labellings to give a (k + k′)-circular-  dispersed labelling of the product graph G □ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Let G and H be graphs with |V (G)| = m, |V (H)| = n and gcd (m, n) =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Then DL◦(G □ H) ≥ DL◦(G) + DL◦(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Denote k = DL◦(G) and k′ = DL◦(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Suppose ℓG : Zm → V (G) is a  k-circular-dispersed labelling of G and ℓH : Zn → V (H) is a k′-circular-dispersed  labelling of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' For 0 ≤ i ≤ mn − 1, define  ℓ(i) = (ℓG(i mod m), ℓH(i mod n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Since gcd (m, n) = 1, ℓ is a bijection from Zmn to the vertices of G □ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  The  distance between two vertices in G □ H is just the sum of the distances between  the corresponding projections (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=', to the first and second coordinates) in G and  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Hence, for any i, where 0 ≤ i ≤ mn − 1, The distance between ℓ(i) and  ℓ(i + 1 mod mn) in G □ H is the sum  dG(i mod m, i + 1 mod m) + dH(i mod n, i + 1 mod n) ≥ k + k′,  as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  We illustrate the product construction by returning to grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Our goal is not to  compute all the values DL◦(Lm,n), but rather to illustrate how Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='1 can be  applied to a specific family of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Recall that Lm,n is an m by n grid and Pm−1 is a path having m vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' It is  easy to see that Pm−1 □ Pn−1 is isomorphic to Lm,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We can obtain lower bounds  on DL◦(Lm,n) if we have lower bounds on DL◦(Pm−1) and DL◦(Pn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Therefore,  we first look at circular-dispersed labellings of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Suppose a graph G = (V, E) contains two central vertices, say u and  v, both of which are uniquely eccentric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Suppose also that {u, v} ∈ E and suppose  r(G) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Then DL◦(G) ≤ r(G) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Suppose that |V | = n and DL◦(G) = r(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Vertices u and v must receive the  labels 1 and n since they are uniquely eccentric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' However, d(u, v) = 1 ≤ r(G) − 1,  so it follows that DL◦(G) ≤ r(G) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  13  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Let Pm be the path with m edges (and m + 1 vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' If m is even,  then DL◦(Pm) = DL(Pm) = m  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' If m ≥ 3 is odd, then DL◦(Pm) = DL(Pm)−1 = m−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We already have shown that DL(Pm) = r(Pm) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' For m even,  the labelling  2 4  · ·  m 1 3  · ·  m + 1  given in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='3 has d(1, m + 1) = m  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Hence, DL◦(Pm) = DL(Pm) = m  2 when  m is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  However, when m ≥ 3 is odd, Pm contains two uniquely eccentric central vertices  that are adjacent, so Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='2 asserts that DL◦(Pm) ≤ r(Pm)−1 = (m−1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' To  achieve a m−1  2 -circular dispersed labelling of Pm, we start with the above labelling  of Pm−1 (on m vertices) and attach a new vertex labelled m + 1 to the left end of  the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Suppose gcd(m, n) = 1, m ≥ 3 and n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Then  DL◦(Lm,n) ≥  �  m+n  2  − 1  if m, n are both odd  m+n−3  2  if m + n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' This is an immediate application of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='2 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  We do not consider the case where both m and n are even in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' This  is because gcd(m, n) ≥ 2 when m and n are even and hence the hypotheses cannot  be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Suppose gcd(m, n) = 1, m ≥ 3 is odd and n ≥ 3 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Then  DL◦(Lm,n) = m + n  2  − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='11, we have DL(Lm,n) = m+n  2  −1 when n ≥ 3 and m ≥ 3 are  odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='2 shows that DL◦(Lm,n) ≤ DL(Lm,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Finally, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='4 shows  that DL◦(Lm,n) ≥ m+n  2  − 1 if n ≥ 3 and m ≥ 3 are odd and gcd(m, n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  4  Degree-based bounds  We now consider a degree-based lower bound on DL(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Suppose G has n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  For x ∈ V and i ≥ 0, let  ηi(x) = |{y ∈ V : d(x, y) = i}| ,  and  κ(x) = max  �  j :  j  �  i=0  ηi(x) ≤ n/2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  For example, in Qm, it is easy to see that ηi(x) =  �m  i  �  for each vertex x and  κ(x) = ⌊m−1  2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  14  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' DL(G) ≥ min{1 + κ(x) : x ∈ V (G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' For any vertex x, the there are at least n/2 vertices y such that d(x, y) ≥ 1+  κ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Therefore, by Dirac’s Theorem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=', see [1, Theorem 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='4]), Hk is hamiltonian  for k = min{1 + κ(x) : x ∈ V (G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Now apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' If G has n vertices and maximum degree ∆ > 2, then  DL(G) ≥ 1 + log∆−1  �  1 + (n − 2)(∆ − 2)  2∆  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  (1)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' For any x ∈ V (G) and i ≥ 1, ni(x) ≤ ∆(∆−1)i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' In order to apply Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='1, we compute the largest value of ℓ such that  1 + ∆ + ∆(∆ − 1) + · · · + ∆(∆ − 1)ℓ ≤ n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  This inequality is equivalent to  1 + ∆(1 − (∆ − 1)ℓ+1)  2 − ∆  ≤ n  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Elementary algebra tells us that this can be rewritten as  ℓ + 1 ≤ log∆−1  �  1 + (n − 2)(∆ − 2)  2∆  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Since κ(x) ≥ ℓ + 1, it therefore follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='1 that DL(G) ≥ ℓ + 2 and  hence the inequality (1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' If G has n vertices and maximum degree three, then  DL(G) ≥ log2  �n + 4  6  �  + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='2 with ∆ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  For example, if n = 3×2t−4 and ∆ = 3, then Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='3 says that DL(G) ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  5  Open questions  There are many interesting open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We list a few now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Which graphs G have DL(G) = r(G)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We note that DL(G) ∈ {r(G), r(G)−1} for all the graphs considered in Section  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' We ask if there is an infinite family of graphs G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' such that  lim  i→∞(r(G) − DL(G)) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  Actually, at this point, we do not have any examples of graphs G with DL(G) ≤  r(G) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  15  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Is there an efficient algorithm to find “good” dispersed labellings of graphs  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=', labellings where DL(G) is “close to” r(G)) for certain classes of graphs  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=', trees)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content=' Is there a generalization of the product construction (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='1) that han-  dles cases where gcd(m, n) > 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='  References  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
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+page_content='  16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf'}
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+arXiv:2301.04350v1  [cs.CG]  11 Jan 2023
+Maximum Centre-Disjoint Mergeable Disks
+Ali Gholami Rudi∗
+Abstract
+Given a set of disks on the plane, the goal of the problem studied
+in this paper is to choose a subset of these disks such that none of
+its members contains the centre of any other. Each disk not in this
+subset must be merged with one of its nearby disks that is, increasing
+the latter’s radius. We prove that this problem is NP-hard. We also
+present polynomial-time algorithms for the special case in which the
+centres of all disks are on a line.
+Keywords: Merging disks, map labelling, rotating maps, NP-hardness, geometric
+independent set.
+1
+Introduction
+A motivating example for the problem studied in this paper is the following
+about drawing text labels on a digital map that can be rotated: suppose
+there are a number of points on the map that represent map features. To
+each of these feature points a text label is assigned that describe the feature,
+like the name a junction.
+When the map is rotated by the user, these
+labels must remain horizontal for the sake of readability, and therefore,
+they are rotated in the reverse direction around their feature point. Labels
+are difficult to read if they overlap, and therefore, only a non-overlapping
+subset of the labels are drawn on the map.
+If a label cannot be drawn
+because it overlaps with other labels, the text of its label must be appended
+to a nearby label that is drawn. The goal is to draw the maximum number
+of labels on the map such that none of them overlap when rotating the
+map. This is demonstrated in Figure 1. Part (a) shows four feature points
+and their labels.
+Part (b) shows the map when it is rotated 45 degrees
+counterclockwise; instead of rotating the map, the labels are equivalently
+rotated 45 degrees clockwise. Obviously the two labels on the left side of
+the map overlap. Part (c) shows what happens when these labels are merged.
+The remaining three labels never overlap when the map is rotated.
+∗Department of Electrical and Computer Engineering, Babol Noshirvani University of
+Technology, Babol, Mazandaran, Iran. Email: gholamirudi@nit.ac.ir.
+1
+
+Christopher
+Tim
+Steven
+Nathaniel
+ChristopherTim
+Steven
+Nathaniel
+Christopher & Tim
+Steven
+Nathaniel
+(a)
+(b)
+(c)
+Figure 1: An example rotating map; (a) the initial configuration, (b) after
+rotating the map 45 degrees counterclockwise, (c) merging two of the labels.
+Placing as many labels as possible on a map (known as map labelling)
+is a classical optimization problem in cartography and graph drawing [1].
+For static maps, i.e. maps whose contents does not change, the problem
+of placing labels on a map can be stated as an instance of geometric in-
+dependent set problem (sometimes also called packing for fixed geometric
+objects): given a set of geometric objects, the goal is to find its largest non-
+intersecting subset. In the weighted version, each object also has a weight
+and the goal is to find a non-intersecting subset of the maximum possible
+weight.
+A geometric intersection graph, with a vertex for each object and an edge
+between intersecting objects, converts this geometric problem to the classi-
+cal maximum independent set for graphs, which is NP-hard and difficult to
+approximate even within a factor of n1−ǫ, where n is the number of vertices
+and ǫ is any non-zero positive constant [2]. Although the geometric version
+remains NP-hard even for unit disks [3], it is easier to approximate, and sev-
+eral polynomial-time approximation schemes (PTAS) have been presented
+for this problem [4, 5, 6, 7, 8].
+Dynamic maps allow zooming, panning, or rotation, and labelling in such
+maps seems more challenging. Most work on labelling dynamic maps con-
+sider zooming and panning operations [10]. Gemsa et al. [11] were the first
+to formally study labelling rotating maps. With the goal of maximising the
+total duration in which labels are visible without intersecting other labels,
+they proved the problem to be NP-hard and presented a 1/4-approximation
+algorithm and a PTAS, with the presence of restrictions on the distribution
+of labels on the map. Heuristic algorithms and Integer Linear Programming
+(ILP) formulations have also been presented for this problem [12, 13]. Note
+that in these problems, invisible labels do not get merged with visible labels.
+Yokosuka and Imai [14] examined a variant of this problem, in which all of
+the labels are always present in the solution and the goal is to maximise
+their size.
+A related problem is crushing disks [15], in which a set of prioritized
+2
+
+disks are given as input, whose radii grow over time, as map labels do
+when zooming in. When two disks touch, the one with the lower priority
+disappears. The radii of the disks grow linearly, and when a disk disappears,
+the radius of the other disk does not change. The goal is to find the order in
+which disks disappear and the process finishes when only one disk remains.
+In this paper, we investigate a problem similar to geometric independent
+set for a set of disks, except that i) the disks in the output must be centre-
+disjoint (none of them can contain the centre of another) but they may
+overlap, ii) each disk that does not appear in the output must be merged
+with a disk, containing its centre, that does. When a disk is merged with
+another, the radius of the latter is increased by the radius of the former. Also
+to preserve the locality of the merges, a disk A can be merged with another
+disk B, only if all disks closer to B than A (considering the distance between
+disk centres) are also merged with B, and after merging these closer disks, B
+must contain the centre of A (without this restriction, we presented a PTAS
+in an earlier paper [9]). This problem is formally defined in Section 2.
+To observe how the introductory example at the beginning of this sec-
+tion reduces to this problem, consider the disks in Figure 1-(a). The disk
+centred at each feature point shows the region covered by its label during
+rotation. Only if a disk contains the centre of another, their corresponding
+labels intersect at some point during rotation. As another application of
+this problem, centre-disjoint disks can show the distribution of facilities in
+an area. For instance, Figure 2 shows the distribution of schools in Munich.
+It was obtained by placing a disk of radius 50 meters on each school (the
+coordinates of schools were obtained from OpenStreetMap data). Then, an
+integer program was used to obtain the maximum number of centre-disjoint
+disks in our problem (based on Definition 2.4)1.
+We prove this problem to be NP-hard via a reduction from Planar Mono-
+tone 3-SAT [16]. Note that the centre-disjointness property of disks are used
+in the definition of transmission graphs of a set of disks, in which a vertex is
+assigned to each disk and a directed edge from a disk to another shows that
+the former contains the centre of the latter [17]. These graphs have been
+studied for interesting properties or their recognition [18, 17, 19], but those
+results do not apply to our problem.
+We also study the problem when the centres of input disks are on a line.
+Many difficult problem become less challenging with this restriction. For
+instance, Biniaz et al. [20] study three problems about a set of points and
+disks on a line. For our problem, we present a polynomial-time algorithm
+that incrementally obtains a set of centre-disjoint disks with the maximum
+size.
+This paper is organised as follows. In Section 2 we introduce the notation
+1the
+integer
+program
+used
+to
+obtain
+this
+figure
+is
+available
+at
+https://github.com/nit-ce/mcmd.git
+3
+
+Figure 2: The distribution of schools in Munich; disks corresponding to
+neighbouring schools were merged to obtain larger, centre-disjoint disks.
+used in this paper and formally state the problem. Then, in Section 3 we
+show that the problem studied in this paper is NP-hard. In Section 4, we
+present a polynomial-time algorithm for the 1.5D variant of the problem, in
+which all disk centres are on a line. Finally, in Section 5 we conclude this
+paper.
+2
+Notation and Preliminary Results
+Let D = {d1, d2, . . . , dn} be a set of n disks. The radius of di is denoted as
+ri and sometimes as rdi, and its centre is denoted as pi.
+Definition 2.1. A function φ from D to itself is an assignment, if φ(φ(di))
+is φ(di) for every di in D. According to an assignment φ, the disks in D
+can be either selected or merged: if φ(di) is di, the disk di is selected, and
+otherwise, it is merged. The cardinality of an assignment, denoted as |φ|, is
+the number of selected disks in φ.
+The relation defined by assignments (Definition 2.1) describes disk merges
+in our problem. For any disk di, if we have φ(di) = dj and i ̸= j, it implies
+that di is merged with dj. On the other hand, the relation φ(di) = di implies
+that it is a selected disk (is not merged with any other disk). Since a disk can
+be merged with selected disks only, for any disk di, we have φ(φ(di)) = φ(di).
+4
+
+St.Benno-Viertel
+Maxvorstadt
+Arabenapark
+Altbogenhausen
+Boqenhausen
+ricke
+Mun
+Kreuzvierte
+Haupt
+hof
+Kar
+splatz
+chus)
+Munchen
+Altstadt
+Steinhausen
+Maximilansbrucke
+ackenviertel
+Marienplatz
+Klinikviertel
+sarto
+Leuchtenbergring
+Haidhause
+Ludwigsvorstadt
+Haidhausen
+AmAitenSudlichen
+sarvorstadt
+Sud
+Goetheplatz
+Friedhof
+Rosenheimer
+Platz
+Miunchen
+ost
+B2R
+AmSchlachthof
+Fruhingsonlogen
+Untere
+Werksviertel
+ObereAu
+Welfenstrabe
+Ostfriedhof
+Sendlinger
+Feld
+Untergiesing
+2
+Ramersdort
+udermahlstraBe
+Obergiesing
+Minchen.Ramersdod
+GiesindDefinition 2.2. The aggregate radius of a selected disk di with respect to
+assignment φ, denoted with some misuse of notation as ri(φ), is the sum of
+its radius and that of every disk merged with it, or equivalently,
+ri(φ) =
+�
+φ(dj)=di
+rj.
+Let δi be the sequence of disks in D\{di}, ordered increasingly by the distance
+of their centres from the centre of di, and let δi(j) denote its j-th disk. The
+j-th aggregate radius of di, denoted as ri(j), is defined as its aggregate radius
+if {δi(1), δi(2), . . . , δi(j)} are merged with di.
+We can now define proper assignments (Definition 2.3). In the rest of this
+paper, the distance between two disks is defined as the Euclidean distance
+between their centres.
+Definition 2.3. An assignment φ is proper if it meets the following condi-
+tions.
+1. The disk δi(j) can be merged with di, only if δi(k), for every k where
+1 ≤ k < j, are also merged with di. In other words, all disks closer to
+di than δi(j) are also merged with di.
+2. The disk δi(j) can be merged with di, only if the distance between the
+centre of di and δi(j) is less than ri(j − 1).
+In other words, after
+merging δi(k) for 1 ≤ k < j, di must contain the centre of δi(j).
+3. Selected disks must be centre-disjoint with respect to their aggregate
+radii; i.e. none of them can contain the centre of any other selected
+disk. More precisely, for indices i and j such that i ̸= j, φ(di) = di,
+and φ(dj) = dj, we have |pipj| ≥ ri(φ) + rj(φ).
+Note the first two items in Definition 2.3 ensure the locality of the merges,
+which is especially important in the labeling application mentioned in the
+Introduction.
+Definition 2.4. Given a set of disks, in the Maximum Centre-Disjoint
+Mergeable Disks Problem (MCMD), the goal is to find a proper assignment
+of the maximum possible cardinality.
+Figure 3 shows a configuration of five disks with more than one proper
+assignment. Disk d3 can be merged with d1, after which, d1 would contain
+the centre of d4 and d5, both of which then have to be merged with d1. These
+merges result in d1 containing the centre of d2, which would also be merged.
+Therefore, in this assignment φ1, we have φ1(di) = d1, for 1 ≤ i ≤ 5, and
+its cardinality is one. Alternatively, in assignment φ2 we can merge d3 with
+d2, as the latter contains the centre of the former. The remaining disks
+5
+
+d1
+d2
+d3
+d4
+d5
+Figure 3: An example set of disks with more than one proper assignments.
+d1
+d2
+d3
+d4
+d5
+Figure 4: An example set of disks with no proper assignment.
+are centre-disjoint. Therefore, we have φ2(d1) = d1, φ2(d2) = d2, φ2(d3) =
+d2, φ2(d4) = d4, φ2(d5) = d5, and its cardinality is four. Assignment φ2
+maximises the number of selected disks, and is a solution to MCMD for the
+configuration of disks in Figure 3.
+Not every set of disks has a proper assignment.
+Figure 4 shows an
+example.
+Disk d3 can be merged with either d1 or d2.
+If d3 is merged
+with d1, d5 cannot be merged with d2, because of the second condition
+of proper assignments: d5 can be merged with d2, only if all closer disks
+to d2 are merged with it (but d3 which is closer to d2 than d5 is not).
+Therefore, d5 can be neither merged, nor selected (because its centre is
+contained in d2).
+Similarly, if d3 is merged with d2, d4 can neither be
+merged nor selected. Thus, there exists no proper assignment for these set
+of disks. In Section 3.2 we introduce a variant of MCMD by relaxing the
+second condition of Definition 2.3, in which every instance has a solution.
+3
+Hardness of Maximum Centre-Disjoint Merge-
+able Disks
+Instead of proving that the decision version of MCMD (Definition 3.1) is
+NP-complete, we show that even deciding whether a set of disks has a
+proper assignment (Definition 3.2) is NP-complete (clearly the latter im-
+6
+
+plies the former). To do so, we perform a reduction from the NP-complete
+Planar Monotone 3-SAT (Definition 3.3) [21] to Proper MCMD (Def-
+inition 3.2).
+3.1
+Hardness of MCMD
+Definition 3.1. In the k-MCMD problem, we are given a set of disks and
+we have to decide if there exists a proper assignment of cardinality at least
+k or not.
+Definition 3.2. In the Proper MCMD problem, we are given a set of
+disks and we have to decide if there exists a proper assignment.
+Definition 3.3. Monotone 3-SAT is a variant of 3-SAT, in which all
+variables of each clause are either positive or negative.
+An instance of
+Monotone 3-SAT is called Planar, if it can be modeled as a planar
+bipartite graph with parts V corresponding to variables and C corresponding
+to clauses; each vertex in C is incident to at most three variables, which cor-
+respond to the variables that appear in the clause. Deciding if an instance
+of Planar Monotone 3-SAT is satisfiable is NP-complete [16].
+It can be proved that every instance of Planar Monotone 3-SAT has
+a monotone rectilinear representation (Definition 3.4), and also, if for every
+instance of Planar Monotone 3-SAT its monotone rectilinear represen-
+tation is also given, the problem remains NP-Complete [16].
+Definition 3.4. A monotone rectilinear representation of an instance of
+Planar Monotone 3-SAT is a drawing of the instance with the following
+properties: i) Variable are drawn as disjoint horizontal segments on the x-
+axis, ii) positive clauses are drawn as horizontal segments above the x-axis,
+iii) negative clauses are drawn as horizontal segments below the x-axis, iv)
+an edge is drawn as a vertical segment between a clause segment and the
+segments corresponding to its variable, and v) the drawing is crossing-free.
+Figure 5 shows a monotone rectilinear representation of an instance of
+Planar Monotone 3-SAT with three clauses, in which c1 = v1 ∨ v2 ∨ v3,
+c2 = v1 ∨ v2 ∨ v3, and c3 = ¬v1 ∨ ¬v2 ∨ ¬v4.
+Lemma 3.5. For an instance of Planar Monotone 3-SAT with v vari-
+ables and c clauses, there exists a monotone rectilinear representation on
+a two-dimensional integer grid with c + 1 rows and 3c + v columns, such
+that horizontal segments, which represent variables and clauses, appear on
+horizontal grid lines, and vertical segments appear on vertical grid lines.
+Proof. Let R be a monotone rectilinear representation of a Planar Mono-
+tone 3-SAT instance (such a representation certainly exists [16]). By ex-
+tending horizontal segments of R we get at most c + 1 lines: one for the
+7
+
+v1
+v2
+v3
+v4
+c1
+c2
+c3
+Figure 5: A monotone rectilinear representation of a Planar Monotone
+3-SAT instance with three clauses and four variables.
+variables (the x-axis) and at most c for clauses. Let ℓ1, ℓ2, . . . , ℓm be the
+lines that appear above the x-axis ordered by their y-coordinates. We move
+them (together with the segments appearing on them) so that, ℓi is moved to
+y = i; vertical segments that connect them to a segment on the x-axis may
+need to be shortend or lengthened during the movement. Given that the
+x-coordinate of the end points of horizontal segments, and also the vertical
+order of the segments, do not change, no new intersection is introduced by
+this transformation. The same is done for the lines that appear below the
+x-axis.
+Repeating the same process for vertical segments, we get at most 3c
+vertical lines. We can similarly move these lines and the segments on them
+horizontally so that they appear in order and consecutively on vertical inte-
+ger grid lines. Variables that do not appear in any clause, can be placed in
+at most v additional vertical grid lines. This results in a (c + 1) × (3c + v)
+grid.
+In the proof of Theorem 3.6, we create an instance of Proper MCMD
+from the monotone rectilinear representation an instance of Planar Mono-
+tone 3-SAT. In our construction, we use two types of disks:
+• Normal disks, which by our construction, are always selected (their
+centres can never be inside any other disk). We call them sdisks for
+brevity.
+• Disks of very small radius, which are contained in at least one sdisk,
+and thus, are surely merged in our construction. We call these disks
+mdisks. We assume that the radius of mdisks is so small compared to
+8
+
+m
+A
+B
+A
+B
+(a)
+(b)
+(c)
+Figure 6: Two gadgets, joined at one of their mdisks.
+the radius of sdisks that after merging any number of mdisks with an
+sdisk, the centre of no new disk would enter the sdisk in our config-
+uration. In the instance of Proper MCMD that we construct, each
+sdisk contains at least one mdisk.
+We create a configuration of disks using gadgets, each of which consists of
+some mdisks and sdisks. The mdisks of a gadget are either internal (internal
+mdisks) or can be shared with other gadgets (shared mdisks). Parts (a) and
+(b) of Figure 6 show two gadgets (from each gadget, only an sdisk and an
+mdisk is shown). In Figure 6-(c) these two gadgets are joined at mdisk m.
+In a proper assignment, m is merged either with an sdisk of A or with an
+sdisk of B. With respect to gadget A, if m is merged with A in a proper
+assignment, we say that it is merged in, and otherwise, merged out.
+We use the following gadgets in our construction. The gadgets and the
+distance between shared mdisks of each of them are shown in Figure 7; sdisks
+(denoted as si) are large disks and mdisks (denoted as mi) are small disks
+(shared mdisks are distinguished with a darker colour).
+• Input: It has only one shared mdisk, which can be either merged in or
+merged out.
+• Copy: We use two gadgets for copy in our construction: one with two
+mdisks and one with four (both of them are demonstrated in Figure 7).
+The logic behind both of them is similar and is explained thus. If m1
+is merged in, m2 (also m5 and m6 if present) is merged out, and if m1
+is merged out, m2 (also m5 and m6) is merged in. To see why, note
+that m3 can be merged either with s1 or with s2. If m3 is merged
+with s1, both m1 and m4 must also be merged with s1, because m3
+is farther than both to s1. Since m4 is merged with s1, m2 (also m5
+and m6) cannot be merged with s2 and therefore they must be merged
+out. Similarly, if m3 is merged with s2, m2 (also m5 and m6) must be
+merged with s2 as well, and m1 must be merged out.
+• Disjunction: One or more of its shared mdisks are merged in. Clearly,
+m4 must be merged with s1, s2, or s3. If it is merged with si (i ∈
+9
+
+0.25
+0.75
+0.25
+0.50
+0.25
+0.25
+m1
+m1
+m2
+m3
+m4
+m1
+m2
+m3
+m4
+m5
+m6
+m1
+m2
+m3
+m4
+m1
+m2
+m3
+m4
+s1
+s1
+s2
+s1
+s2
+s1
+s2
+s3
+s1
+s2
+Input
+Copy
+Copy
+Disjunction
+Not
+Figure 7: Gadgets used in the proof of Theorem 3.6.
+{1, 2, 3}), mi must also be merged with si, and mj (j ̸= i) may or may
+not be merged in.
+• Not: Either both m1 and m2 are merge in or both of them are merged
+out. This is because m4 can be merged either with s1 or s2. If it is
+merged with s1, mdisks m1, m2, and m3 must also be merged with
+s1, because m4 is farther than all of them. Otherwise, if m4 is merged
+with s2, mdisk m3 must also be merged with s2 and therefore, none of
+m1 and m2 can be merged with s1, because m3 (which is closer than
+both) is not merged with s1. Thus, m1 and m2 must merge out.
+Note that in 6-mdisk version of Copy gadget, one or two of its mdisks may
+not be shared with any other gadgets. If so, these mdisks must be removed
+from this instance of Copy.
+Theorem 3.6. Proper MCMD is NP-complete.
+Proof. It is trivial to show that Proper MCMD is in NP. To show that it
+is NP-hard, we reduce Planar Monotone 3-SAT to Proper MCMD.
+Let I be an instance of Planar Monotone 3-SAT, with variables V
+and clauses C. Based on Lemma 3.5, there exists a monotone rectilinear
+representation of I on a (|C| + 1) × (3 · |C| + |V |) integer grid. Let R denote
+this representation.
+We create an instance of Proper MCMD from R as follows. The trans-
+formation is demonstrated in Figure 8, which corresponds to the monotone
+rectilinear representation of Figure 5.
+10
+
+v1
+v2
+v3
+v4
+c1
+c2
+c3
+Figure 8: A Proper MCMD instance obtained from the Planar Mono-
+tone 3-SAT instance of Figure 5.
+1. We replace the segment corresponding to a variable in R with an Input
+gadget and a series of Copy gadgets.
+For each intersection of this
+segment with a vertical segment, a 6-mdisk Copy gadget is used.
+2. Let s be a horizontal segment corresponding to a clause in R. Three
+variables appear in the clause, for each of which there is a vertical
+segment that connects s to a variable segment. For the first and last
+intersections, 6-mdisk Copy gadgets are used. For the 2nd intersection,
+we use a Disjunction gadget. These gadgets are connected using two
+chains of Copy gadgets.
+3. For each vertical segment that connects a variable segment to a clause
+segment above the x-axis, we use a chain of Copy gadgets to connect
+the Copy gadget of the variable segment to the Copy or Disjunction
+gadget (if it is the 2nd intersection) of the clause segment. For seg-
+ments that appear below the x-axis, we do likewise, except that we
+place a Not gadget before the chain of Copy gadgets.
+Note that some of the gadgets of Figure 7 need to be rotated or mirrored.
+Also note that based on the sizes shown in Figure 7, shared mdisks always
+appear on grid lines in our construction. Given that the total area of the grid
+is bounded by O(|C|2 |V |2), and on a segment of unit length, at most four
+gadgets can appear, the number of gadgets used in the resulting instance
+of Proper MCMD is at most O(|C|2 |V |2). Thus, the size of the resulting
+Proper MCMD instance is polynomial in terms of the size of the input
+Planar Monotone 3-SAT instance.
+Suppose there is a proper assignment for our Proper MCMD instance.
+We obtain an assignment A of the variables of our Planar Monotone
+11
+
+3-SAT instance as follows.
+We assign one to a variable if the mdisk of
+its corresponding Input gadget is merged out, and assign zero otherwise.
+Consider any clause c in our Planar Monotone 3-SAT instance. Let g
+be the Disjunction gadget corresponding to c.
+If c is a positive clause, a chain of Copy gadgets connects the Input gadget
+of each of the variables that appear in c to g. Therefore, if variable v appears
+in the clause and if the shared mdisk of the Input gadget corresponding to
+v is merged out, the mdisk of the last Copy gadget of its chain is merged in
+inside g. Since, one or more of the shared mdisks of g are merged in, at least
+one of the terms in g is satisfied. Similarly, if c is a negative clause, because
+there is a Not gadget in the chain that connects each variable v of c to its
+Disjunction gadget, if the shared mdisk of the Input gadget corresponding
+to v is merged out, the mdisk of the last Copy gadget of its chain is also
+merged out inside g. Since, one or more of the shared mdisks of g are merged
+in, at least one of the variables in g is not satisfied.
+Therefore, the Planar Monotone 3-SAT instance is satisfied with
+assignment A.
+For the reverse direction, suppose there exists an assignment A of the
+variables, for which all clauses of G are satisfied. We can obtain a proper
+assignment in our Proper MCMD instance as follows. For each variable
+v in V , if v is one, the shared mdisk of the Input gadget corresponding to v
+is merged out, and otherwise, it is merged in. Let c be a positive clause in
+which variable v with value one appears (since c is satisfied in A, variable
+v must exist), and let g be the gadget corresponding to clause c. Since v
+is merged out, the mdisk of the last Copy gadget that connects the gadget
+corresponding to v to g is merged in with respect to g. This implies that
+one of the shared mdisks of the Disjunction gadget of each positive clause is
+merged in. We can similarly show that at least one of the shared mdisks of
+the Disjunction gadgets corresponding to negative caluses are also merged
+in. This yields a proper assignment for the Proper MCMD instance.
+In Corollary 3.7 we show that even if all disks have the same radius, the
+problem remains NP-hard.
+Corollary 3.7. Proper MCMD remains NP-complete, even if all disks
+are required to be of the same radius.
+Proof. We fix the radius of mdisks to r = 0.01. We use the same construction
+as Theorem 3.6, with the difference that we replace each sdisk with a number
+of smaller disks of radius r with the same centre, so that the sum of the radius
+of these smaller disks equals the radius of the sdisk. Since the disks added
+for each sdisk are not centre-disjoint, and their centre cannot be contained
+in some other disk, exactly one of them must be selected and after merging
+others, it reaches the size of the original sdisk. The rest of the proof of
+Theorem 3.6 applies without significant changes.
+12
+
+3.2
+Relaxing Merge Order
+Due to the first condition of proper assignments (Definition 2.3), in a proper
+assignment φ of a set of disks D, a disk di can be merged with another disk
+dj, only if all closer disks to di than dj are also merged with di.
+This
+condition, in addition to the second condition of Definition 2.3, ensures the
+locality of the merges. By requiring this ordering for merges, however, we
+get instances for which there is no solution, such as the one demonstrated
+in Figure 4. For such instances, a solution can be obtained by relaxing this
+condition. In this section, we relax the first condition of Definition 2.3.
+Definition 3.8. In an assignment φ for a set of disks D, let δφ
+i denote the
+sequence of disks assigned to selected disk di, ordered by their distance to di.
+Also, let δφ
+i (j) denote the j-th disk in this sequence.
+Definition 3.9. An assignment φ is uproper (short for unordered proper)
+if it meets the following conditions.
+1. For each pair of possible indices i and j, in which φ(dj) = di, choose k
+such that δφ
+i (k) = dj. The distance between di and dj must be at most
+ri + �k−1
+x=1 rδφ
+i (x). In other words, after merging all closer disks in δφ
+i ,
+di must contain the centre of dj.
+2. Selected disks must be centre-disjoint with respect to their aggregate
+radii; i.e. none of them can contain the centre of any other selected
+disk.
+Definition 3.10. Given a set of disks, the goal in the Relaxed Maximum
+Centre-Disjoint Mergeable Disks Problem ( RMCMD) is to find a uproper
+assignment of the maximum possible cardinality.
+To show that RMCMD is NP-hard, in Theorem 3.11 we reduce the Par-
+tition problem to RMCMD. In Partition, we are given a set of positive
+integers and have to decide if there is a subset, whose sum is half of the sum
+of all numbers in the input list. Partition is known to be NP-complete.
+Theorem 3.11. RMCMD is NP-hard.
+Proof. We reduce Partition to RMCMD. Let A = {a1, a2, . . . , an} be an
+instance of Partition and let s be the sum of the members of A. Also let
+e be any number smaller than one. We create an instance of RMCMD as
+follows.
+1. Add disk d1 of radius 2s and add d2 with the same radius at distance
+3s on the right of d1.
+2. Add d3 at distance 5s/2 + e above d1 with radius s. Similarly, add d4
+at distance 5s/2 + e above d2 with the same radius.
+13
+
+d1
+d2
+d3
+d4
+s
+__5
+2 s + e
+3s
+Figure 9: The construction in the proof of Theorem 3.11.
+3. Add one disk for each member of A in the midpoint of p1 and p2, such
+that the radius of the one corresponding to ai is ai.
+This is demonstrated in Figure 9. Let φ be the solution of this RMCMD
+instance. We show that there is a valid solution to the Partition instance
+if and only if the cardinality of φ is four.
+Suppose X is a subset of A with sum s/2. We obtain an assignment
+from X as follows: every disk corresponding to a member of X is assigned
+to d1 and others are assigned to d2. Since the sum of the members of X is
+s/2, the aggregate radii of both disks are exactly 5s/2. Therefore, the centre
+of d3 and d4 are outside these disks. This yields a uproper assignment of
+cardinality 4.
+For the reverse direction, suppose the cardinality of φ is four (note that it
+cannot be greater). If so, all of d1, d2, d3, and d4 are selected, and therefore,
+the aggregate radii of d1 and d2 are lower than 5s/2 + e. Given that the
+sum of the radii of the disks corresponding to members of A is s, the sum
+of the set of disks assigned to d1 and d2 (and therefore the subsets of A
+corresponding to them) are equal.
+4
+Collinear Disk Centres
+In this section we present a polynomial-time algorithm for solving MCMD
+for a set of disk with collinear centres. Note that even if disk centres are
+collinear, there may exist no proper assignments, as demonstrated in Fig-
+ure 4. In the rest of this section, let λ = ⟨d1, d2, . . . , dn⟩ be a sequence of
+input disks D, ordered by the x-coordinate of their centres.
+14
+
+Definition 4.1. Let φ be an assignment of {d1, d2, . . . , dn} and let φ′ be an
+assignment of {d1, d2, . . . , dx}, such that x ≤ n. φ is an extension of φ′, if
+for every disk di in {d1, d2, . . . , dx}, we have φ(di) = φ′(di). In other words,
+every selected disk in φ′ is also a selected disk in φ, and every merged disk
+in φ′ is also merged with the same disk in φ. Equivalently, when φ is limited
+to {d1, d2, . . . , dx}, φ′ is obtained.
+Definition 4.2. M(x, y, z) denotes the maximum cardinality of a proper
+assignment of X = {d1, d2, . . . , dx}, such that the following conditions are
+met (y ≤ x ≤ z ≤ n).
+1. dy is its right-most selected disk.
+2. dy+1, dy+2, . . . , dx are all merged with dy.
+3. dz is the right-most disk in D, where z ≥ x, whose centre is contained
+in dy considering its aggregate radius.
+Note that by the third condition of Definition 4.2, the centres of dx+1, dx+2, . . . , dz
+are inside dy, but they are not merged with it, because they are outside X
+and not present in the assignment which is limited to set X. Also, note that
+actually the second condition of Definition 4.2 is implied by its first condi-
+tion: since dy is the right-most selected disk, all of the disks that appear on
+the right of dy in X are surely merged. On the other hand, none of these
+disks can be merged with a selected disk dw on the left of dy, because, in
+that case dw would contain the centre of dy and the assignment cannot be
+proper.
+Theorem 4.3. A proper assignment of the maximum cardinality for a set
+of n disks D, in which the centres of all disks are collinear, can be computed
+in polynomial time.
+Proof. Let M be defined as in Definition 4.2. Obviously, maxn
+i=1 M(n, i, n)
+is the cardinality of the solution to this MCMD instance.
+The function M accepts O(n3) different input values. We can compute
+and store the value returned by M in a three dimensional table, which we
+reference also as M. The values of the entries of M are computed incremen-
+tally, as described in Algorithm 4.4. We explain the steps of this algorithm
+in what follows.
+Algorithm 4.4. Find a solution to MCMD for a set of collinear disks.
+1. Compute the sequences δi for 1 ≤ i ≤ n (Definition 2.2).
+2. Initialize every entry of M to −∞.
+3. For every possible value of i from 1 to n and for every possible value
+of j from 0 to n − 1 perform the following steps.
+15
+
+d1
+dt
+dA
+da
+di
+db
+dB
+dn
+Figure 10: Demonstrating the symbols used in Theorem 4.3.
+(a) Check to see if the first j disks of δi can be merged with di, consid-
+ering the conditions of Definition 2.3. If not, skip this iteration
+of this loop, and continue with the next iteration.
+(b) Compute A, a, b, and B: a and b are the left-most and right-
+most disks in λ that are merged with di, respectively. Also, A and
+B are the left-most and right-most disks of D whose centres are
+contained in di, considering its aggregate radius.
+(c) If a = 1 and M(b, i, B) < 1, assign 1 to M(b, i, B).
+(d) If a > 1, for every possible value of t from 1 to A − 1 and for
+every possible value of k from 0 to n − 2 do as follows.
+i. Check if the first k disks of δt can be merged with dt (based
+on Definition 2.3).
+ii. Compute f and g: df is the right-most disk that is merged
+with δt, and dg is the right-most disk of D whose centre is
+contained in dt, considering its aggregate radius.
+iii. If f ≥ a, f ̸= a − 1, or g ≥ i skip this value of k (φL cannot
+be extended to obtain φ).
+iv. Replace the value of M(b, i, B) with the maximum of its value
+and M(a − 1, t, g) + 1.
+4. Compute and return maxn
+i=1 M(n, i, n).
+Steps 1 and 2 of the algorithm initialize M and δi. In Step 3, we consider
+different cases in which di, for 1 ≤ i ≤ n in order, is selected and update the
+value of different entries of M. For every possible value of j from 0 to n−1,
+suppose j disks are merged with di. These disks are the first j disks of δi
+by the first condition of Definition 2.3. Let S denote the set of such disks.
+If this is not possible (the centre of one of these disks is not contained in di,
+16
+
+after merging its previous disks), we skip this value of j, because it fails the
+second condition of Definition 2.3 (Step 3.a). Note that if there exists no
+proper assignment in which j disks are merged with di, a greater number of
+disks cannot be merged with di in any assignment, and we can safely skip
+the remaining values of j and continue the loop of Step 3 by incrementing
+the value of i.
+Let a, b, A, and B be defined as Step 3.b. If a = 1, selecting di and
+merging with it every disk in {d1, d2, . . . , db} \ {di} is a proper assignment
+of the first b disks of λ with cardinality one. Therefore, we update the value
+of M(b, i, B) to be at least one in step 3.c.
+If a > 1, let φ be any assignment of {d1, . . . , db}, in which i) di is se-
+lected, ii) the members of S are merged with di, and iii) the members of
+{dA, . . . , da−1} ∪ {db+1, . . . , dB} are contained in di after merging the mem-
+bers of S with di. By the definition of M, the value of M(b, i, B) cannot be
+smaller than the cardinality of φ. When φ is limited to L = {d1, . . . , da−1},
+it specifies a proper assignment of L. We denote this assignment with φL.
+We compute the value of M(b, i, B) by considering all possible assignments
+for φL and extending them to obtain φ by selecting di.
+Let dt be the right-most selected disk of φL. This is demonstrated in
+Figure 10. The following conditions hold.
+1. We have t < A, because {dA, . . . , da−1} are contained in di in φ, and
+dt cannot be a selected disk if t ≥ A. Therefore, disks {dt+1, . . . , da−1}
+are merged with dt in φL.
+2. Suppose k disks are merged with dt in φL. Let df be the right-most
+vertex of D contained in dt after merging disks in φL. We have f < i
+(otherwise, df would contain the centre of di, and di cannot be selected
+in φ). Also let dg be the right-most vertex of D contained in df. We
+have g < i; otherwise, df would contain the centre of di and φ cannot
+be an extension of φL.
+By trying possible values of t and k that meet these conditions (Step 3.d),
+we find the maximum cardinality of φL, which has been computed in the
+previous steps of this algorithm as M(a − 1, t, g). Since φ is an extension
+of φL by adding exactly one selected disk di, the maximum cardinality of φ
+therefore is at least 1 + M(a − 1, t, g). Thus, we have
+M(b, i, B) ≥ 1 +
+max
+t and k, as above M(a − 1, t, g)
+Step 3.d.iv updates M(b, i, B) to be at least this value.
+Theorem 4.5. The time complexity of computing M for a set of n disks,
+as described in Theorem 4.3, is O(n5).
+17
+
+Proof. We analyse Algorithm 4.4. Constructing δi (step 1) can be done in
+O(n2 log n) and initializing M (step 2) can be done in O(n3). For each pair
+of values for i and j, steps 3.a-c can be performed in O(n). In step 3.d,
+O(n2) possible cases for t and k are considered, and for each of these cases,
+the steps i, ii, iii, and iv can be performed in O(n). Since the loop of step
+3 is repeated O(n2) times, the time complexity of the whole algorithm is
+O(n5).
+5
+Concluding Remarks
+We introduced a variant of geometric independent set for a set of disks,
+such that the disks that do not appear in the output must be merged with
+a nearby disk that does (the problem was stated formally in Section 2). We
+proved that this problem is NP-hard (Theorem 3.6). Also by relaxing one of
+the conditions of the problem, we introduced a less restricted variant, which
+was proved NP-hard as well (Theorem 3.11). We presented a polynomial-
+time algorithm for the case in which disk centres are collinear.
+Several interesting problems call for further investigation, such as: i)
+general approximation algorithms, ii) studying the case in which the number
+of disks that can be merged with a selected disk is bounded by some constant,
+and iii) solving MCMD when disks are pierced by a line.
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+
diff --git a/ltE3T4oBgHgl3EQfKAnG/content/tmp_files/load_file.txt b/ltE3T4oBgHgl3EQfKAnG/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..3f6facfbdb212fec2eb1336069f9a9bd4f9feee1
--- /dev/null
+++ b/ltE3T4oBgHgl3EQfKAnG/content/tmp_files/load_file.txt
@@ -0,0 +1,737 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf,len=736
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='04350v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='CG]  11 Jan 2023  Maximum Centre-Disjoint Mergeable Disks  Ali Gholami Rudi∗  Abstract  Given a set of disks on the plane, the goal of the problem studied  in this paper is to choose a subset of these disks such that none of  its members contains the centre of any other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Each disk not in this  subset must be merged with one of its nearby disks that is, increasing  the latter’s radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We prove that this problem is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We also  present polynomial-time algorithms for the special case in which the  centres of all disks are on a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Keywords: Merging disks, map labelling, rotating maps, NP-hardness, geometric  independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  1  Introduction  A motivating example for the problem studied in this paper is the following  about drawing text labels on a digital map that can be rotated: suppose  there are a number of points on the map that represent map features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' To  each of these feature points a text label is assigned that describe the feature,  like the name a junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  When the map is rotated by the user, these  labels must remain horizontal for the sake of readability, and therefore,  they are rotated in the reverse direction around their feature point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Labels  are difficult to read if they overlap, and therefore, only a non-overlapping  subset of the labels are drawn on the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  If a label cannot be drawn  because it overlaps with other labels, the text of its label must be appended  to a nearby label that is drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The goal is to draw the maximum number  of labels on the map such that none of them overlap when rotating the  map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' This is demonstrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Part (a) shows four feature points  and their labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Part (b) shows the map when it is rotated 45 degrees  counterclockwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' instead of rotating the map, the labels are equivalently  rotated 45 degrees clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Obviously the two labels on the left side of  the map overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Part (c) shows what happens when these labels are merged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  The remaining three labels never overlap when the map is rotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  ∗Department of Electrical and Computer Engineering, Babol Noshirvani University of  Technology, Babol, Mazandaran, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Email: gholamirudi@nit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='ir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  1  Christopher  Tim  Steven  Nathaniel  ChristopherTim  Steven  Nathaniel  Christopher & Tim  Steven  Nathaniel  (a)  (b)  (c)  Figure 1: An example rotating map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' (a) the initial configuration, (b) after  rotating the map 45 degrees counterclockwise, (c) merging two of the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Placing as many labels as possible on a map (known as map labelling)  is a classical optimization problem in cartography and graph drawing [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  For static maps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' maps whose contents does not change, the problem  of placing labels on a map can be stated as an instance of geometric in-  dependent set problem (sometimes also called packing for fixed geometric  objects): given a set of geometric objects, the goal is to find its largest non-  intersecting subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In the weighted version, each object also has a weight  and the goal is to find a non-intersecting subset of the maximum possible  weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  A geometric intersection graph, with a vertex for each object and an edge  between intersecting objects, converts this geometric problem to the classi-  cal maximum independent set for graphs, which is NP-hard and difficult to  approximate even within a factor of n1−ǫ, where n is the number of vertices  and ǫ is any non-zero positive constant [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Although the geometric version  remains NP-hard even for unit disks [3], it is easier to approximate, and sev-  eral polynomial-time approximation schemes (PTAS) have been presented  for this problem [4, 5, 6, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Dynamic maps allow zooming, panning, or rotation, and labelling in such  maps seems more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Most work on labelling dynamic maps con-  sider zooming and panning operations [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Gemsa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' [11] were the first  to formally study labelling rotating maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' With the goal of maximising the  total duration in which labels are visible without intersecting other labels,  they proved the problem to be NP-hard and presented a 1/4-approximation  algorithm and a PTAS, with the presence of restrictions on the distribution  of labels on the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Heuristic algorithms and Integer Linear Programming  (ILP) formulations have also been presented for this problem [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Note  that in these problems, invisible labels do not get merged with visible labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Yokosuka and Imai [14] examined a variant of this problem, in which all of  the labels are always present in the solution and the goal is to maximise  their size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  A related problem is crushing disks [15], in which a set of prioritized  2  disks are given as input, whose radii grow over time, as map labels do  when zooming in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' When two disks touch, the one with the lower priority  disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The radii of the disks grow linearly, and when a disk disappears,  the radius of the other disk does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The goal is to find the order in  which disks disappear and the process finishes when only one disk remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  In this paper, we investigate a problem similar to geometric independent  set for a set of disks, except that i) the disks in the output must be centre-  disjoint (none of them can contain the centre of another) but they may  overlap, ii) each disk that does not appear in the output must be merged  with a disk, containing its centre, that does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' When a disk is merged with  another, the radius of the latter is increased by the radius of the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Also  to preserve the locality of the merges, a disk A can be merged with another  disk B, only if all disks closer to B than A (considering the distance between  disk centres) are also merged with B, and after merging these closer disks, B  must contain the centre of A (without this restriction, we presented a PTAS  in an earlier paper [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' This problem is formally defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  To observe how the introductory example at the beginning of this sec-  tion reduces to this problem, consider the disks in Figure 1-(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The disk  centred at each feature point shows the region covered by its label during  rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Only if a disk contains the centre of another, their corresponding  labels intersect at some point during rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' As another application of  this problem, centre-disjoint disks can show the distribution of facilities in  an area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' For instance, Figure 2 shows the distribution of schools in Munich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  It was obtained by placing a disk of radius 50 meters on each school (the  coordinates of schools were obtained from OpenStreetMap data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Then, an  integer program was used to obtain the maximum number of centre-disjoint  disks in our problem (based on Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='4)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  We prove this problem to be NP-hard via a reduction from Planar Mono-  tone 3-SAT [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Note that the centre-disjointness property of disks are used  in the definition of transmission graphs of a set of disks, in which a vertex is  assigned to each disk and a directed edge from a disk to another shows that  the former contains the centre of the latter [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' These graphs have been  studied for interesting properties or their recognition [18, 17, 19], but those  results do not apply to our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  We also study the problem when the centres of input disks are on a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Many difficult problem become less challenging with this restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' For  instance, Biniaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' [20] study three problems about a set of points and  disks on a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' For our problem, we present a polynomial-time algorithm  that incrementally obtains a set of centre-disjoint disks with the maximum  size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  This paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In Section 2 we introduce the notation  1the  integer  program  used  to  obtain  this  figure  is  available  at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='com/nit-ce/mcmd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='git  3  Figure 2: The distribution of schools in Munich;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' disks corresponding to  neighbouring schools were merged to obtain larger, centre-disjoint disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  used in this paper and formally state the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Then, in Section 3 we  show that the problem studied in this paper is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In Section 4, we  present a polynomial-time algorithm for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='5D variant of the problem, in  which all disk centres are on a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Finally, in Section 5 we conclude this  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  2  Notation and Preliminary Results  Let D = {d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , dn} be a set of n disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The radius of di is denoted as  ri and sometimes as rdi, and its centre is denoted as pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' A function φ from D to itself is an assignment, if φ(φ(di))  is φ(di) for every di in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' According to an assignment φ, the disks in D  can be either selected or merged: if φ(di) is di, the disk di is selected, and  otherwise, it is merged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The cardinality of an assignment, denoted as |φ|, is  the number of selected disks in φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  The relation defined by assignments (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='1) describes disk merges  in our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' For any disk di, if we have φ(di) = dj and i ̸= j, it implies  that di is merged with dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' On the other hand, the relation φ(di) = di implies  that it is a selected disk (is not merged with any other disk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Since a disk can  be merged with selected disks only, for any disk di, we have φ(φ(di)) = φ(di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  4  St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Benno-Viertel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Maxvorstadt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Arabenapark  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Altbogenhausen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Boqenhausen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='ricke  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Mun  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Kreuzvierte  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Haupt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='hof  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Kar  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='splatz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='chus)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Munchen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Altstadt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Steinhausen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Maximilansbrucke  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='ackenviertel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Marienplatz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Klinikviertel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='sarto  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Leuchtenbergring  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Haidhause  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Ludwigsvorstadt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Haidhausen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='AmAitenSudlichen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='sarvorstadt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Sud  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Goetheplatz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Friedhof  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Rosenheimer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Platz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Miunchen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='ost  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='B2R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='AmSchlachthof  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Fruhingsonlogen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Untere  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Werksviertel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='ObereAu  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Welfenstrabe  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Ostfriedhof  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Sendlinger  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Feld  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Untergiesing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Ramersdort  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='udermahlstraBe  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Obergiesing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Minchen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='Ramersdod  GiesindDefinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The aggregate radius of a selected disk di with respect to  assignment φ, denoted with some misuse of notation as ri(φ), is the sum of  its radius and that of every disk merged with it, or equivalently,  ri(φ) =  �  φ(dj)=di  rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Let δi be the sequence of disks in D\\{di}, ordered increasingly by the distance  of their centres from the centre of di, and let δi(j) denote its j-th disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The  j-th aggregate radius of di, denoted as ri(j), is defined as its aggregate radius  if {δi(1), δi(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , δi(j)} are merged with di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  We can now define proper assignments (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In the rest of this  paper, the distance between two disks is defined as the Euclidean distance  between their centres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' An assignment φ is proper if it meets the following condi-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The disk δi(j) can be merged with di, only if δi(k), for every k where  1 ≤ k < j, are also merged with di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In other words, all disks closer to  di than δi(j) are also merged with di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The disk δi(j) can be merged with di, only if the distance between the  centre of di and δi(j) is less than ri(j − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  In other words, after  merging δi(k) for 1 ≤ k < j, di must contain the centre of δi(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Selected disks must be centre-disjoint with respect to their aggregate  radii;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' none of them can contain the centre of any other selected  disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' More precisely, for indices i and j such that i ̸= j, φ(di) = di,  and φ(dj) = dj, we have |pipj| ≥ ri(φ) + rj(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Note the first two items in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='3 ensure the locality of the merges,  which is especially important in the labeling application mentioned in the  Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Given a set of disks, in the Maximum Centre-Disjoint  Mergeable Disks Problem (MCMD), the goal is to find a proper assignment  of the maximum possible cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Figure 3 shows a configuration of five disks with more than one proper  assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Disk d3 can be merged with d1, after which, d1 would contain  the centre of d4 and d5, both of which then have to be merged with d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' These  merges result in d1 containing the centre of d2, which would also be merged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Therefore, in this assignment φ1, we have φ1(di) = d1, for 1 ≤ i ≤ 5, and  its cardinality is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Alternatively, in assignment φ2 we can merge d3 with  d2, as the latter contains the centre of the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The remaining disks  5  d1  d2  d3  d4  d5  Figure 3: An example set of disks with more than one proper assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  d1  d2  d3  d4  d5  Figure 4: An example set of disks with no proper assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  are centre-disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Therefore, we have φ2(d1) = d1, φ2(d2) = d2, φ2(d3) =  d2, φ2(d4) = d4, φ2(d5) = d5, and its cardinality is four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Assignment φ2  maximises the number of selected disks, and is a solution to MCMD for the  configuration of disks in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Not every set of disks has a proper assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Figure 4 shows an  example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Disk d3 can be merged with either d1 or d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  If d3 is merged  with d1, d5 cannot be merged with d2, because of the second condition  of proper assignments: d5 can be merged with d2, only if all closer disks  to d2 are merged with it (but d3 which is closer to d2 than d5 is not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Therefore, d5 can be neither merged, nor selected (because its centre is  contained in d2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Similarly, if d3 is merged with d2, d4 can neither be  merged nor selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Thus, there exists no proper assignment for these set  of disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='2 we introduce a variant of MCMD by relaxing the  second condition of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='3, in which every instance has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  3  Hardness of Maximum Centre-Disjoint Merge-  able Disks  Instead of proving that the decision version of MCMD (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='1) is  NP-complete, we show that even deciding whether a set of disks has a  proper assignment (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='2) is NP-complete (clearly the latter im-  6  plies the former).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' To do so, we perform a reduction from the NP-complete  Planar Monotone 3-SAT (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='3) [21] to Proper MCMD (Def-  inition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='1  Hardness of MCMD  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In the k-MCMD problem, we are given a set of disks and  we have to decide if there exists a proper assignment of cardinality at least  k or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In the Proper MCMD problem, we are given a set of  disks and we have to decide if there exists a proper assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Monotone 3-SAT is a variant of 3-SAT, in which all  variables of each clause are either positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  An instance of  Monotone 3-SAT is called Planar, if it can be modeled as a planar  bipartite graph with parts V corresponding to variables and C corresponding  to clauses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' each vertex in C is incident to at most three variables, which cor-  respond to the variables that appear in the clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Deciding if an instance  of Planar Monotone 3-SAT is satisfiable is NP-complete [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  It can be proved that every instance of Planar Monotone 3-SAT has  a monotone rectilinear representation (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='4), and also, if for every  instance of Planar Monotone 3-SAT its monotone rectilinear represen-  tation is also given, the problem remains NP-Complete [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' A monotone rectilinear representation of an instance of  Planar Monotone 3-SAT is a drawing of the instance with the following  properties: i) Variable are drawn as disjoint horizontal segments on the x-  axis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' ii) positive clauses are drawn as horizontal segments above the x-axis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  iii) negative clauses are drawn as horizontal segments below the x-axis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' iv)  an edge is drawn as a vertical segment between a clause segment and the  segments corresponding to its variable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' and v) the drawing is crossing-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Figure 5 shows a monotone rectilinear representation of an instance of  Planar Monotone 3-SAT with three clauses, in which c1 = v1 ∨ v2 ∨ v3,  c2 = v1 ∨ v2 ∨ v3, and c3 = ¬v1 ∨ ¬v2 ∨ ¬v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' For an instance of Planar Monotone 3-SAT with v vari-  ables and c clauses, there exists a monotone rectilinear representation on  a two-dimensional integer grid with c + 1 rows and 3c + v columns, such  that horizontal segments, which represent variables and clauses, appear on  horizontal grid lines, and vertical segments appear on vertical grid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Let R be a monotone rectilinear representation of a Planar Mono-  tone 3-SAT instance (such a representation certainly exists [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' By ex-  tending horizontal segments of R we get at most c + 1 lines: one for the  7  v1  v2  v3  v4  c1  c2  c3  Figure 5: A monotone rectilinear representation of a Planar Monotone  3-SAT instance with three clauses and four variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  variables (the x-axis) and at most c for clauses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Let ℓ1, ℓ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , ℓm be the  lines that appear above the x-axis ordered by their y-coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We move  them (together with the segments appearing on them) so that, ℓi is moved to  y = i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' vertical segments that connect them to a segment on the x-axis may  need to be shortend or lengthened during the movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Given that the  x-coordinate of the end points of horizontal segments, and also the vertical  order of the segments, do not change, no new intersection is introduced by  this transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The same is done for the lines that appear below the  x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Repeating the same process for vertical segments, we get at most 3c  vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We can similarly move these lines and the segments on them  horizontally so that they appear in order and consecutively on vertical inte-  ger grid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Variables that do not appear in any clause, can be placed in  at most v additional vertical grid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' This results in a (c + 1) × (3c + v)  grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  In the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='6, we create an instance of Proper MCMD  from the monotone rectilinear representation an instance of Planar Mono-  tone 3-SAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In our construction, we use two types of disks:  Normal disks, which by our construction, are always selected (their  centres can never be inside any other disk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We call them sdisks for  brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Disks of very small radius, which are contained in at least one sdisk,  and thus, are surely merged in our construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We call these disks  mdisks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We assume that the radius of mdisks is so small compared to  8  m  A  B  A  B  (a)  (b)  (c)  Figure 6: Two gadgets, joined at one of their mdisks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  the radius of sdisks that after merging any number of mdisks with an  sdisk, the centre of no new disk would enter the sdisk in our config-  uration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In the instance of Proper MCMD that we construct, each  sdisk contains at least one mdisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  We create a configuration of disks using gadgets, each of which consists of  some mdisks and sdisks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The mdisks of a gadget are either internal (internal  mdisks) or can be shared with other gadgets (shared mdisks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Parts (a) and  (b) of Figure 6 show two gadgets (from each gadget, only an sdisk and an  mdisk is shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In Figure 6-(c) these two gadgets are joined at mdisk m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  In a proper assignment, m is merged either with an sdisk of A or with an  sdisk of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' With respect to gadget A, if m is merged with A in a proper  assignment, we say that it is merged in, and otherwise, merged out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  We use the following gadgets in our construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The gadgets and the  distance between shared mdisks of each of them are shown in Figure 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' sdisks  (denoted as si) are large disks and mdisks (denoted as mi) are small disks  (shared mdisks are distinguished with a darker colour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Input: It has only one shared mdisk, which can be either merged in or  merged out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Copy: We use two gadgets for copy in our construction: one with two  mdisks and one with four (both of them are demonstrated in Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  The logic behind both of them is similar and is explained thus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' If m1  is merged in, m2 (also m5 and m6 if present) is merged out, and if m1  is merged out, m2 (also m5 and m6) is merged in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' To see why, note  that m3 can be merged either with s1 or with s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' If m3 is merged  with s1, both m1 and m4 must also be merged with s1, because m3  is farther than both to s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Since m4 is merged with s1, m2 (also m5  and m6) cannot be merged with s2 and therefore they must be merged  out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Similarly, if m3 is merged with s2, m2 (also m5 and m6) must be  merged with s2 as well, and m1 must be merged out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Disjunction: One or more of its shared mdisks are merged in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Clearly,  m4 must be merged with s1, s2, or s3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' If it is merged with si (i ∈  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='25  m1  m1  m2  m3  m4  m1  m2  m3  m4  m5  m6  m1  m2  m3  m4  m1  m2  m3  m4  s1  s1  s2  s1  s2  s1  s2  s3  s1  s2  Input  Copy  Copy  Disjunction  Not  Figure 7: Gadgets used in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  {1, 2, 3}), mi must also be merged with si, and mj (j ̸= i) may or may  not be merged in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Not: Either both m1 and m2 are merge in or both of them are merged  out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' This is because m4 can be merged either with s1 or s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' If it is  merged with s1, mdisks m1, m2, and m3 must also be merged with  s1, because m4 is farther than all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Otherwise, if m4 is merged  with s2, mdisk m3 must also be merged with s2 and therefore, none of  m1 and m2 can be merged with s1, because m3 (which is closer than  both) is not merged with s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Thus, m1 and m2 must merge out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Note that in 6-mdisk version of Copy gadget, one or two of its mdisks may  not be shared with any other gadgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' If so, these mdisks must be removed  from this instance of Copy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Proper MCMD is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' It is trivial to show that Proper MCMD is in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' To show that it  is NP-hard, we reduce Planar Monotone 3-SAT to Proper MCMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Let I be an instance of Planar Monotone 3-SAT, with variables V  and clauses C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Based on Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='5, there exists a monotone rectilinear  representation of I on a (|C| + 1) × (3 · |C| + |V |) integer grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Let R denote  this representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  We create an instance of Proper MCMD from R as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The trans-  formation is demonstrated in Figure 8, which corresponds to the monotone  rectilinear representation of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  10  v1  v2  v3  v4  c1  c2  c3  Figure 8: A Proper MCMD instance obtained from the Planar Mono-  tone 3-SAT instance of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We replace the segment corresponding to a variable in R with an Input  gadget and a series of Copy gadgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  For each intersection of this  segment with a vertical segment, a 6-mdisk Copy gadget is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Let s be a horizontal segment corresponding to a clause in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Three  variables appear in the clause, for each of which there is a vertical  segment that connects s to a variable segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' For the first and last  intersections, 6-mdisk Copy gadgets are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' For the 2nd intersection,  we use a Disjunction gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' These gadgets are connected using two  chains of Copy gadgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' For each vertical segment that connects a variable segment to a clause  segment above the x-axis, we use a chain of Copy gadgets to connect  the Copy gadget of the variable segment to the Copy or Disjunction  gadget (if it is the 2nd intersection) of the clause segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' For seg-  ments that appear below the x-axis, we do likewise, except that we  place a Not gadget before the chain of Copy gadgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Note that some of the gadgets of Figure 7 need to be rotated or mirrored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Also note that based on the sizes shown in Figure 7, shared mdisks always  appear on grid lines in our construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Given that the total area of the grid  is bounded by O(|C|2 |V |2), and on a segment of unit length, at most four  gadgets can appear, the number of gadgets used in the resulting instance  of Proper MCMD is at most O(|C|2 |V |2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Thus, the size of the resulting  Proper MCMD instance is polynomial in terms of the size of the input  Planar Monotone 3-SAT instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Suppose there is a proper assignment for our Proper MCMD instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  We obtain an assignment A of the variables of our Planar Monotone  11  3-SAT instance as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  We assign one to a variable if the mdisk of  its corresponding Input gadget is merged out, and assign zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Consider any clause c in our Planar Monotone 3-SAT instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Let g  be the Disjunction gadget corresponding to c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  If c is a positive clause, a chain of Copy gadgets connects the Input gadget  of each of the variables that appear in c to g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Therefore, if variable v appears  in the clause and if the shared mdisk of the Input gadget corresponding to  v is merged out, the mdisk of the last Copy gadget of its chain is merged in  inside g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Since, one or more of the shared mdisks of g are merged in, at least  one of the terms in g is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Similarly, if c is a negative clause, because  there is a Not gadget in the chain that connects each variable v of c to its  Disjunction gadget, if the shared mdisk of the Input gadget corresponding  to v is merged out, the mdisk of the last Copy gadget of its chain is also  merged out inside g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Since, one or more of the shared mdisks of g are merged  in, at least one of the variables in g is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Therefore, the Planar Monotone 3-SAT instance is satisfied with  assignment A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  For the reverse direction, suppose there exists an assignment A of the  variables, for which all clauses of G are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We can obtain a proper  assignment in our Proper MCMD instance as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' For each variable  v in V , if v is one, the shared mdisk of the Input gadget corresponding to v  is merged out, and otherwise, it is merged in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Let c be a positive clause in  which variable v with value one appears (since c is satisfied in A, variable  v must exist), and let g be the gadget corresponding to clause c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Since v  is merged out, the mdisk of the last Copy gadget that connects the gadget  corresponding to v to g is merged in with respect to g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' This implies that  one of the shared mdisks of the Disjunction gadget of each positive clause is  merged in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We can similarly show that at least one of the shared mdisks of  the Disjunction gadgets corresponding to negative caluses are also merged  in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' This yields a proper assignment for the Proper MCMD instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  In Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='7 we show that even if all disks have the same radius, the  problem remains NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Proper MCMD remains NP-complete, even if all disks  are required to be of the same radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We fix the radius of mdisks to r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We use the same construction  as Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='6, with the difference that we replace each sdisk with a number  of smaller disks of radius r with the same centre, so that the sum of the radius  of these smaller disks equals the radius of the sdisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Since the disks added  for each sdisk are not centre-disjoint, and their centre cannot be contained  in some other disk, exactly one of them must be selected and after merging  others, it reaches the size of the original sdisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The rest of the proof of  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='6 applies without significant changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='2  Relaxing Merge Order  Due to the first condition of proper assignments (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='3), in a proper  assignment φ of a set of disks D, a disk di can be merged with another disk  dj, only if all closer disks to di than dj are also merged with di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  This  condition, in addition to the second condition of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='3, ensures the  locality of the merges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' By requiring this ordering for merges, however, we  get instances for which there is no solution, such as the one demonstrated  in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' For such instances, a solution can be obtained by relaxing this  condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In this section, we relax the first condition of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In an assignment φ for a set of disks D, let δφ  i denote the  sequence of disks assigned to selected disk di, ordered by their distance to di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Also, let δφ  i (j) denote the j-th disk in this sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' An assignment φ is uproper (short for unordered proper)  if it meets the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' For each pair of possible indices i and j, in which φ(dj) = di, choose k  such that δφ  i (k) = dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The distance between di and dj must be at most  ri + �k−1  x=1 rδφ  i (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In other words, after merging all closer disks in δφ  i ,  di must contain the centre of dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Selected disks must be centre-disjoint with respect to their aggregate  radii;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' none of them can contain the centre of any other selected  disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Given a set of disks, the goal in the Relaxed Maximum  Centre-Disjoint Mergeable Disks Problem ( RMCMD) is to find a uproper  assignment of the maximum possible cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  To show that RMCMD is NP-hard, in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='11 we reduce the Par-  tition problem to RMCMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In Partition, we are given a set of positive  integers and have to decide if there is a subset, whose sum is half of the sum  of all numbers in the input list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Partition is known to be NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' RMCMD is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We reduce Partition to RMCMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Let A = {a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , an} be an  instance of Partition and let s be the sum of the members of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Also let  e be any number smaller than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We create an instance of RMCMD as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Add disk d1 of radius 2s and add d2 with the same radius at distance  3s on the right of d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Add d3 at distance 5s/2 + e above d1 with radius s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Similarly, add d4  at distance 5s/2 + e above d2 with the same radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  13  d1  d2  d3  d4  s  __5  2 s + e  3s  Figure 9: The construction in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Add one disk for each member of A in the midpoint of p1 and p2, such  that the radius of the one corresponding to ai is ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  This is demonstrated in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Let φ be the solution of this RMCMD  instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We show that there is a valid solution to the Partition instance  if and only if the cardinality of φ is four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Suppose X is a subset of A with sum s/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We obtain an assignment  from X as follows: every disk corresponding to a member of X is assigned  to d1 and others are assigned to d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Since the sum of the members of X is  s/2, the aggregate radii of both disks are exactly 5s/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Therefore, the centre  of d3 and d4 are outside these disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' This yields a uproper assignment of  cardinality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  For the reverse direction, suppose the cardinality of φ is four (note that it  cannot be greater).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' If so, all of d1, d2, d3, and d4 are selected, and therefore,  the aggregate radii of d1 and d2 are lower than 5s/2 + e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Given that the  sum of the radii of the disks corresponding to members of A is s, the sum  of the set of disks assigned to d1 and d2 (and therefore the subsets of A  corresponding to them) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  4  Collinear Disk Centres  In this section we present a polynomial-time algorithm for solving MCMD  for a set of disk with collinear centres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Note that even if disk centres are  collinear, there may exist no proper assignments, as demonstrated in Fig-  ure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In the rest of this section, let λ = ⟨d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , dn⟩ be a sequence of  input disks D, ordered by the x-coordinate of their centres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  14  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Let φ be an assignment of {d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , dn} and let φ′ be an  assignment of {d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , dx}, such that x ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' φ is an extension of φ′, if  for every disk di in {d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , dx}, we have φ(di) = φ′(di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In other words,  every selected disk in φ′ is also a selected disk in φ, and every merged disk  in φ′ is also merged with the same disk in φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Equivalently, when φ is limited  to {d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , dx}, φ′ is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' M(x, y, z) denotes the maximum cardinality of a proper  assignment of X = {d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , dx}, such that the following conditions are  met (y ≤ x ≤ z ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' dy is its right-most selected disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' dy+1, dy+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , dx are all merged with dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' dz is the right-most disk in D, where z ≥ x, whose centre is contained  in dy considering its aggregate radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Note that by the third condition of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='2, the centres of dx+1, dx+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , dz  are inside dy, but they are not merged with it, because they are outside X  and not present in the assignment which is limited to set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Also, note that  actually the second condition of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='2 is implied by its first condi-  tion: since dy is the right-most selected disk, all of the disks that appear on  the right of dy in X are surely merged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' On the other hand, none of these  disks can be merged with a selected disk dw on the left of dy, because, in  that case dw would contain the centre of dy and the assignment cannot be  proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' A proper assignment of the maximum cardinality for a set  of n disks D, in which the centres of all disks are collinear, can be computed  in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Let M be defined as in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Obviously, maxn  i=1 M(n, i, n)  is the cardinality of the solution to this MCMD instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  The function M accepts O(n3) different input values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We can compute  and store the value returned by M in a three dimensional table, which we  reference also as M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The values of the entries of M are computed incremen-  tally, as described in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We explain the steps of this algorithm  in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Find a solution to MCMD for a set of collinear disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Compute the sequences δi for 1 ≤ i ≤ n (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Initialize every entry of M to −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' For every possible value of i from 1 to n and for every possible value  of j from 0 to n − 1 perform the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  15  d1  dt  dA  da  di  db  dB  dn  Figure 10: Demonstrating the symbols used in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  (a) Check to see if the first j disks of δi can be merged with di, consid-  ering the conditions of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' If not, skip this iteration  of this loop, and continue with the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  (b) Compute A, a, b, and B: a and b are the left-most and right-  most disks in λ that are merged with di, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Also, A and  B are the left-most and right-most disks of D whose centres are  contained in di, considering its aggregate radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  (c) If a = 1 and M(b, i, B) < 1, assign 1 to M(b, i, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  (d) If a > 1, for every possible value of t from 1 to A − 1 and for  every possible value of k from 0 to n − 2 do as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Check if the first k disks of δt can be merged with dt (based  on Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Compute f and g: df is the right-most disk that is merged  with δt, and dg is the right-most disk of D whose centre is  contained in dt, considering its aggregate radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' If f ≥ a, f ̸= a − 1, or g ≥ i skip this value of k (φL cannot  be extended to obtain φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Replace the value of M(b, i, B) with the maximum of its value  and M(a − 1, t, g) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Compute and return maxn  i=1 M(n, i, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Steps 1 and 2 of the algorithm initialize M and δi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In Step 3, we consider  different cases in which di, for 1 ≤ i ≤ n in order, is selected and update the  value of different entries of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' For every possible value of j from 0 to n−1,  suppose j disks are merged with di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' These disks are the first j disks of δi  by the first condition of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Let S denote the set of such disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  If this is not possible (the centre of one of these disks is not contained in di,  16  after merging its previous disks), we skip this value of j, because it fails the  second condition of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='3 (Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Note that if there exists no  proper assignment in which j disks are merged with di, a greater number of  disks cannot be merged with di in any assignment, and we can safely skip  the remaining values of j and continue the loop of Step 3 by incrementing  the value of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Let a, b, A, and B be defined as Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' If a = 1, selecting di and  merging with it every disk in {d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , db} \\ {di} is a proper assignment  of the first b disks of λ with cardinality one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Therefore, we update the value  of M(b, i, B) to be at least one in step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  If a > 1, let φ be any assignment of {d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , db}, in which i) di is se-  lected, ii) the members of S are merged with di, and iii) the members of  {dA, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , da−1} ∪ {db+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , dB} are contained in di after merging the mem-  bers of S with di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' By the definition of M, the value of M(b, i, B) cannot be  smaller than the cardinality of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' When φ is limited to L = {d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , da−1},  it specifies a proper assignment of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We denote this assignment with φL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  We compute the value of M(b, i, B) by considering all possible assignments  for φL and extending them to obtain φ by selecting di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Let dt be the right-most selected disk of φL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' This is demonstrated in  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The following conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We have t < A, because {dA, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , da−1} are contained in di in φ, and  dt cannot be a selected disk if t ≥ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Therefore, disks {dt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' , da−1}  are merged with dt in φL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Suppose k disks are merged with dt in φL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Let df be the right-most  vertex of D contained in dt after merging disks in φL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We have f < i  (otherwise, df would contain the centre of di, and di cannot be selected  in φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Also let dg be the right-most vertex of D contained in df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We  have g < i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' otherwise, df would contain the centre of di and φ cannot  be an extension of φL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  By trying possible values of t and k that meet these conditions (Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='d),  we find the maximum cardinality of φL, which has been computed in the  previous steps of this algorithm as M(a − 1, t, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Since φ is an extension  of φL by adding exactly one selected disk di, the maximum cardinality of φ  therefore is at least 1 + M(a − 1, t, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Thus, we have  M(b, i, B) ≥ 1 +  max  t and k, as above M(a − 1, t, g)  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='iv updates M(b, i, B) to be at least this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' The time complexity of computing M for a set of n disks,  as described in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='3, is O(n5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  17  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We analyse Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Constructing δi (step 1) can be done in  O(n2 log n) and initializing M (step 2) can be done in O(n3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' For each pair  of values for i and j, steps 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='a-c can be performed in O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' In step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='d,  O(n2) possible cases for t and k are considered, and for each of these cases,  the steps i, ii, iii, and iv can be performed in O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Since the loop of step  3 is repeated O(n2) times, the time complexity of the whole algorithm is  O(n5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  5  Concluding Remarks  We introduced a variant of geometric independent set for a set of disks,  such that the disks that do not appear in the output must be merged with  a nearby disk that does (the problem was stated formally in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We  proved that this problem is NP-hard (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' Also by relaxing one of  the conditions of the problem, we introduced a less restricted variant, which  was proved NP-hard as well (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content=' We presented a polynomial-  time algorithm for the case in which disk centres are collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  Several interesting problems call for further investigation, such as: i)  general approximation algorithms, ii) studying the case in which the number  of disks that can be merged with a selected disk is bounded by some constant,  and iii) solving MCMD when disks are pierced by a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
+page_content='  References  [1] Formann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQfKAnG/content/2301.04350v1.pdf'}
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diff --git a/mNE1T4oBgHgl3EQfhASI/content/tmp_files/2301.03236v1.pdf.txt b/mNE1T4oBgHgl3EQfhASI/content/tmp_files/2301.03236v1.pdf.txt
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,1543 @@
+Optimistic Meta-Gradients
+Sebastian Flennerhag
+DeepMind
+flennerhag@google.com
+Tom Zahavy
+DeepMind
+Brendan O’Donoghue
+DeepMind
+Hado van Hasselt
+DeepMind
+András György
+DeepMind
+Satinder Singh
+DeepMind
+Abstract
+We study the connection between gradient-based meta-learning and convex op-
+timisation. We observe that gradient descent with momentum is a special case
+of meta-gradients, and building on recent results in optimisation, we prove con-
+vergence rates for meta-learning in the single task setting. While a meta-learned
+update rule can yield faster convergence up to constant factor, it is not sufficient
+for acceleration. Instead, some form of optimism is required. We show that op-
+timism in meta-learning can be captured through Bootstrapped Meta-Gradients
+[Flennerhag et al., 2022], providing deeper insight into its underlying mechanics.
+1
+Introduction
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Training Steps
+1e6
+50
+55
+60
+65
+70
+75
+Top-1 Test Accuracy (%)
+SGD
+Standard meta-learning
+Optimistic meta-learning
+Figure 1: ImageNet. We compare training a 50-
+layer ResNet using SGD against variants that tune
+an element-wise learning rate online using standard
+meta-learning or optimistic meta-learning. Shad-
+ing depicts 95% confidence intervals over 3 seeds.
+In meta-learning, a learner is using a param-
+eterised algorithm to adapt to a given task.
+The parameters of the algorithm are then
+meta-learned by evaluating the learner’s result-
+ing performance [Schmidhuber, 1987, Hinton
+and Plaut, 1987, Bengio et al., 1991].
+This
+paradigm has garnered wide empirical success
+[Hospedales et al., 2020]. For instance, it has
+been used to meta-learn how to explore in re-
+inforcement learning (RL) [Xu et al., 2018a,
+Alet et al., 2020], online hyper-parameter tun-
+ing of non-convex loss functions [Bengio, 2000,
+Maclaurin et al., 2015, Xu et al., 2018b, Zahavy
+et al., 2020], discovering black-box loss func-
+tions [Chen et al., 2016, Kirsch et al., 2019, Xu
+et al., 2020, Oh et al., 2020], black-box learning
+algorithms [Hochreiter et al., 2001, Wang et al.,
+2016], or entire training protocols [Real et al.,
+2020]. Yet, very little is known in terms of the
+theoretical properties of meta-learning.
+The reason for this is the complex interac-
+tion between the learner and the meta-learner.
+learner’s problem is to minimize the expected
+loss f of a stochastic objective by adapting its parameters x ∈ Rn. The learner has an update rule ϕ at
+its disposal that generates new parameters xt = xt−1 + ϕ(xt−1, wt); we suppress data dependence to
+simplify notation. A simple example is when ϕ represents gradient descent with wt = η its step size,
+that is ϕ(xt−1, η) = −η∇f(xt−1) [Mahmood et al., 2012, van Erven and Koolen, 2016]; several
+works have explored meta-learning other aspects of a gradient-based update rule [Finn et al., 2017,
+Nichol et al., 2018, Flennerhag et al., 2019, Xu et al., 2018b, Zahavy et al., 2020, Flennerhag et al.,
+arXiv:2301.03236v1  [cs.LG]  9 Jan 2023
+
+2022, Kirsch et al., 2019, Oh et al., 2020]. More generally, ϕ need not be limited to the gradient of
+any function, for instance, it can represent some algorithm implemented within a Recurrent Neural
+Network [Schmidhuber, 1987, Hochreiter et al., 2001, Andrychowicz et al., 2016, Wang et al., 2016].
+The meta-learner’s problem is to optimise the meta-parameters wt to yield effective updates. In
+a typical (gradient-based) meta-learning setting, it does so by treating xt as a function of w. Let
+ht, defined by ht(w) = f(xt−1 + ϕ(xt−1, w)), denote the learner’s post-update performance as a
+function of w. The learner and the meta-learner co-evolve according to
+xt = xt−1 + ϕ(xt−1, wt),
+and
+wt+1 = wt − ∇ht(wt)
+= wt − Dϕ(xt−1, wt)T ∇f(xt),
+where Dϕ(x, w) denotes the Jacobian of ϕ with respect to w. The nested structure between these
+two updates makes it challenging to analyse meta-learning, in particular it depends heavily on the
+properties of the Jacobian. In practice, ϕ is highly complex and so Dϕ is almost always intractable.
+For instance, in Xu et al. [2018a], the meta-parameters define the data-distribution under which
+a stochastic gradient is computed. In Zahavy et al. [2020], the meta-parameters define auxiliary
+objectives that are meant to help with representation learning; in Vinyals et al. [2016] they learn
+an embedding space for nearest-neighbour predictions.
+For this reason, the only theoretical results we are aware of specialise to the multi-task setting and
+assume ϕ represents adaptation by gradient descent. In this setting, at each iteration t, the learner
+must adapt to a new task ft. The learner adapts by taking a (or several) gradient step(s) on ft using
+either a meta-learned initialisation [Flennerhag et al., 2019, Finn et al., 2019, Fallah et al., 2020,
+Wang et al., 2022] or using a meta-learned regulariser [Khodak et al., 2019, Denevi et al., 2019].
+Because the update rule has this form, it is possible to treat the meta-optimisation problem as an
+online learning problem and derive convergence guarantees. Acceleration in this setup is driven by
+the tasks similarity. That is, if all tasks are sufficiently similar, a meta-learned update can accelerate
+convergence [Khodak et al., 2019]. However, these results do not yield acceleration in the absence
+of a task distribution to the best of our knowledge.
+This paper provides an alternative view. We study the classical convex optimisation setting of
+approximating the minimiser minx f(x). We observe that setting the update rule equal to the gradient,
+i.e. ϕ : (x, w) �→ w∇f(x), recovers gradient descent. Similarly, we show in Section 3 that ϕ can
+be chosen to recover gradient descent with momentum. This offers another view of meta-learning
+as a non-linear transformation of classical optimisation. A direct implication of this is that a task
+similarity is not necessary condition for improving the rate of convergence via meta-learning. While
+there is ample empirical evidence to that effect [Xu et al., 2018b, Zahavy et al., 2020, Flennerhag
+et al., 2022, Luketina et al., 2022], we are only aware of theoretical results in the special case of
+meta-learned step sizes [Mahmood et al., 2012, van Erven and Koolen, 2016].
+In particular, we analyse meta-learning using recent techniques developed for convex optimisation
+[Cutkosky, 2019, Joulani et al., 2020, Wang et al., 2021]. Given a function f that is convex with
+Lipschitz smooth gradients, meta-learning improves the rate of convergence by a multiplicative factor
+λ to O(λ/T), via the smoothness of the update rule. Importantly, these works show that to achieve
+accelerated convergence, O(1/T 2), some form of optimism is required. This optimism essentially
+provides a prediction of the next gradient, and hence represents a model of the geometry. We consider
+optimism with meta-learning in the convex setting and prove accelerated rates of convergence,
+O(λ/T 2). Again, meta-learning affects these bounds by a multiplicative factor. We further show
+that optimism in meta-learning can be expressed through the recently proposed Bootstrapped Meta-
+Gradient method [BMG; Flennerhag et al., 2022]. Our analysis provides a first proof of convergence
+for BMG and highlights the underlying mechanics that enable faster learning with BMG. Our main
+contributions are as follows:
+1. We show that meta-learning contains gradient descent with momentum (Heavy Ball [Polyak,
+1964]; Section 3) and Nesterov Acceleration [Nesterov, 1983] as special cases (Section 6).
+2. We show that gradient-based meta-learning can be understood as a non-linear transformation
+of an underlying optimisation method (Section 3).
+3. We establish rates of convergence for meta-learning in the convex setting (Sections 5 and 6).
+4. We show that optimism can be expressed through [Flennerhag et al., 2022]. Our analysis
+(Section 6) provides a first proof of convergence for BMG.
+2
+
+Algorithm 1: Meta-learning in practice.
+input :Weights {βt}T
+t=1
+input :Update rule ϕ
+input :Initialisation (x0, w1)
+for t = 1, 2, . . . , T:
+xt = xt−1 + ϕ(xt−1, wt)
+ht(·) = f(xt−1 + ρtϕ(xt−1, ·))
+wt+1 = wt − βt∇ht(wt)
+return xT
+Algorithm 2: Meta-learning in the convex setting.
+input :Weights {αt}T
+t=1, {βt}T
+t=1
+input :Update rule ϕ
+input :Initialisation (¯x0, w1)
+for t = 1, 2, . . . , T:
+xt = ϕ(¯xt−1, wt)
+¯xt = (1 − αt/α1:t)¯xt−1 + (αt/α1:t)xt
+gt = Dϕ(¯xt−1, wt)T ∇f(¯xt)
+wt+1=arg minw∈W
+�t
+s=1αs⟨gs, w⟩+ 1
+2βt ∥w∥2
+return ¯xT
+2
+Meta-learning meets convex optimisation
+Problem definition.
+This section defines the problem studied in this paper and introduces our
+notation. Let f : X → R be a proper and convex function. The problem of interest is to approximate
+the global minimum minx∈X f(x). We assume a global minimiser exists and is unique, defined by
+x∗ = arg min
+x∈X
+f(x).
+(1)
+We assume that X ⊆ Rn is a closed, convex and non-empty set. f is differentiable and has Lipschitz
+smooth gradients with respect to a norm ∥ · ∥, meaning that there exists L ∈ (0, ∞) such that
+∥∇f(x) − ∇f(y)∥∗ ≤ L∥x − y∥ for all x, y ∈ X, where ∥ · ∥∗ is the dual norm of ∥ · ∥. We consider
+the noiseless setting for simplicity; our results carry over to the stochastic setting by replacing the key
+online-to-batch bound used in our analysis by its stochastic counterpart [Joulani et al., 2020].
+Algorithm.
+Algorithm 1 describes a typical meta-learning algorithm. Unfortunately, at this level
+of generality, little can be said about the its convergence properties. Instead, we consider a stylized
+variant of meta-learning, described in Algorithm 2. This model differs in three regards: (a) it relies
+on moving averages (b) we use a different online learning algorithm for the meta-update, and (c) we
+make stricter assumptions on the update rule. We describe each component in turn.
+Let [T] = {1, 2, . . . , T}. We are given weights {αt}T
+t=1, each αt > 0, and an initialisation (¯x0, w1) ∈
+X × W. At each time t ∈ [T], an update rule ϕ : X × W → X generates the update xt =
+ϕ(¯xt−1, wt), where W ⊆ Rm is closed, convex, and non-empty. We discuss ϕ momentarily. The
+algorithm maintains the online average
+¯xt = x1:t
+α1:t
+= (1 − ρt)¯xt−1 + ρtxt,
+(2)
+where x1:t = �t
+s=1 αsxs, α1:t = �t
+s=1 αs, and ρt = αt/α1:t. Our goal is to establish conditions
+under which {¯xt}T
+t=1 converges to the minimiser x∗. While this moving average is not always used
+in practical applications, it is required for accelerated rates in online-to-batch conversion [Wang and
+Abernethy, 2018, Cutkosky, 2019, Joulani et al., 2020].
+Convergence depends on how each wt is chosen. In Algorithm 1, the meta-learner faces a sequence
+of losses ht : W → R defined by the composition ht(w) = f((1 − ρt)¯xt−1 + ρtϕ(¯xt−1, w)).
+This makes meta-learning a form of online optimisation [McMahan, 2017]. The meta-updates in
+Algorithm 1 is an instance of online gradient descent, which we can model as Follow-The-Regularized-
+Leader (FTRL; reviewed in Section 4). Given some norm ∥ · ∥, an initialization w0 and β > 0,
+FTRL sets each wt according to
+wt+1 = arg min
+w∈W
+�
+t
+�
+s=1
+αs⟨∇hs(ws), w⟩ + 1
+2β ∥w∥2
+�
+.
+(3)
+If ∥ · ∥ is the Euclidean norm, the interior solution to Eq. 3 is given by wt+1 = wt − αtβ∇ht(wt),
+the meta-update in Algorithm 1. It is straightforward to extend Eq. 3 to account for meta-updates that
+use AdaGrad-like [Duchi et al., 2011] acceleration by altering the norms [Joulani et al., 2017].
+3
+
+0
+20
+40
+60
+80
+100
+Iterations
+10
+28
+10
+16
+10
+4
+Loss
+0
+20
+40
+60
+80
+100
+Iterations
+0
+20
+40
+60
+80
+100
+Iterations
+0
+20
+40
+60
+80
+100
+Iterations
+0
+20
+40
+60
+80
+100
+Iterations
+Momentum
+Meta-Momentum
+AdaGrad
+Meta-AdaGrad
+0.1
+0.3
+0.5
+0.7
+0.9
+0.990.9999 3.0
+5.0
+Learning rate
+10
+22
+10
+12
+10
+2
+Loss
+0.1
+0.3
+0.5
+0.7
+0.9
+0.990.9999 3.0
+5.0
+Learning rate
+0.1
+0.3
+0.5
+0.7
+0.9
+0.990.9999 3.0
+5.0
+Learning rate
+0.1
+0.3
+0.5
+0.7
+0.9
+0.990.9999 3.0
+5.0
+Learning rate
+0.1
+0.3
+0.5
+0.7
+0.9
+0.990.9999 3.0
+5.0
+Learning rate
+Momentum
+Meta-Momentum
+AdaGrad
+Meta-AdaGrad
+Figure 2: Convex Quadratic. We generate convex quadratic loss functions with ill-conditioning and
+compare gradient descent with momentum and AdaGrad to meta-learning variants. Meta-Momentum
+uses ϕ : (x, w) �→ w ⊙ ∇f(x) while Meta-AdaGrad uses ϕ : (x, w) �→ ∇f(x)/√w, where division
+is element-wise. Top: loss per iteration for randomly sampled loss functions. Bottom: cumulative
+loss (regret) at the end of learning as a function of learning rate; details in Appendix B.
+Update rule.
+It is not possible to prove convergence outside of the convex setting, since ϕ may
+reach a local minimum where it cannot yield better updates, but the updates are not sufficient to
+converge. Convexity means that each ht must be convex, which requires that ϕ is affine in w (but
+may vary non-linearly in x). We also assume that ϕ is smooth with respect to ∥ · ∥, in the sense that it
+has bounded norm; for all x ∈ X and all w ∈ W we assume that there exists λ ∈ (0, ∞) for which
+∥Dϕ(x, w)T ∇f(x)∥2
+∗ ≤ λ∥∇f(x)∥2
+∗.
+These assumptions hold for any smooth update rule up to first-order Taylor approximation error.
+3
+Meta-Gradients in the Convex Setting - An Overview
+In this section, we provide an informal discussion of our main results (full analysis; Sections 5 and 6).
+Meta-Gradients without Optimism.
+The main difference between classical optimisation and
+meta-learning is the introduction of the update rule ϕ. To see how this acts on optimisation, consider
+two special cases. If the update rule just return the gradient, ϕ = ∇f, Algorithm 2 is reduced
+to gradient descent (with averaging). The inductive bias is fixed and does not change with past
+experience, and so acceleration is not possible—the rate of convergence is O(1/
+√
+T) [Wang et al.,
+2021]. The other extreme is an update rule that only depends on the meta-parameters, ϕ(x, w) = w.
+Here, the meta-learner has ultimate control and selects the next update without constraints. The
+only relevant inductive bias is contained in w. To see how this inductive bias is formed, suppose
+∥ · ∥ = ∥ · ∥2 so that Eq. 3 yields wt+1 = wt − αtρtβ∇f(¯xt) (assuming an interior solution).
+Combining this with the moving average in Eq. 2, we may write the learner’s iterates as
+¯xt = ¯xt−1 + ˜ρt (¯xt−1 − ¯xt−2) − ˜βt∇f(¯xt−1),
+where each ˜ρt = ρt
+1−ρt−1
+ρt−1
+and ˜βt = αtρtβ; setting β = 1/(2L) and each αt = t yields ˜ρt = t−2
+t+1
+and ˜βt = t/(4(t + 1)L). Hence, the canonical momentum algorithm, Polyak’s Heavy-Ball method
+[Polyak, 1964], is obtained as the special case of meta-learning under the update rule ϕ : (x, w) �→ w.
+Because Heavy Ball carries momentum from past updates, it can encode a model of the learning
+dynamics that leads to faster convergence, on the order O(1/T). The implication of this is that the
+dynamics of meta-learning are fundamentally momentum-based and thus learns an update rule in the
+same cumulative manner. This manifests theoretically through its convergence guarantees.
+Theorem 1 (Informal). Set αt = 1 and β =
+1
+λL. If each xt is generated under Algorithm 2, then for
+any viable ϕ, f(¯xT ) − f(x∗) ≤ λL diam(W)
+T
+.
+We refer the reader to Theorem 3 for a formal statement. Compared to Heavy Ball, meta-learning
+introduces a constant λ that captures the smoothness of the update rule. Hence, while meta-learning
+does not achieve better scaling in T through ϕ, it can improve upon classical optimisation by a constant
+factor if λ < 1. That meta-learning can improve upon momentum is borne out experimentally. In
+Figure 2, we consider the problem of minimizing a convex quadratic f : x �→ ⟨x, Qx⟩, where
+Q ∈ Rn×n is PSD but ill-conditioned. We compare momentum to a meta-learned step-size, i.e.
+ϕ : (x, w) �→ w ⊙ ∇f(x), where ⊙ is the Hadamard product. Across randomly sampled Q matrices
+4
+
+(details: Appendix B), we find that introducing a non-linearity ϕ leads to a sizeable improvement in the
+rate of convergence. We also compare AdaGrad to a meta-learned version, ϕ : (x, w) �→ ∇f(x)/√w,
+where division is element-wise. While AdaGrad is a stronger baseline on account of being parameter-
+free, we find that meta-learning the scale vector consistently leads to faster convergence.
+Meta-Gradients with Optimism.
+It is well known that minimizing a smooth convex function
+admits convergence rates of O(1/T 2). Our analysis of standard meta-gradients does not achieve such
+acceleration. Previous work indicate that we should not expect to either; to achieve the theoretical
+lower-limit of O(1/T 2), some form of optimism (reviewed in Section 4) is required. A typical form
+of optimism is to predict the next gradient. This is how Nesterov Acceleration operates [Nesterov,
+1983] and is the reason for its O(1/T 2) convergence guarantee.
+From our perspective, meta-learning is a non-linear transformation of the iterate x. Hence, we should
+expect optimism to play a similarly crucial role. Formally, optimism comes in the form of hint
+functions {˜gt}T
+t=1, each ˜gt ∈ Rm, that are revealed to the meta-learner prior to selecting wt+1. These
+hints give rise to Optimistic Meta-Learning (OML) via meta-updates
+wt+1 = arg min
+w∈W
+�
+αt+1˜gt+1 +
+t
+�
+s=1
+αs⟨∇hs(ws), w⟩ +
+1
+2βt
+∥w∥2
+�
+.
+(4)
+If the hints are accurate, meta-learning with optimism can achieve an accelerated rate of O(˜λ/T 2),
+where ˜λ is a constant that characterises the smoothness of ϕ, akin to λ. Again, we find that meta-
+learning behaves as a non-linear transformation of classical optimism and its rate of convergence is
+governed by the geometry it induces. We summarise this result in the following result.
+Theorem 2 (Informal). Let each hint be given by ˜gt+1 = Dϕ(¯xt−1, wt)T ∇f(¯xt). Assume that ϕ is
+sufficiently smooth. Set αt = t and βt =
+t−1
+2t˜λL, then f(¯xT ) − f(x∗) ≤ 4˜λL diam(W)
+T 2−1
+.
+For a formal statement, see Theorem 4. These predictions hold empirically in a non-convex setting.
+We train a 50-layer ResNet using either SGD with a fixed learning rate, or an update rule that adapts a
+per-parameter learning rate online, ϕ : (x, w) �→ w⊙∇f(x). We compare the standard meta-learning
+approach without optimism to optimistic meta-learning. Figure 1 shows that optimism is critical for
+meta-learning to achieve acceleration, as predicted by theory (experiment details in Appendix C).
+4
+Analysis preliminaries: Online Convex Optimisation
+In this section, we present analytical tools from the optimisation literature that we build upon. In a
+standard optimisation setting, there is no update rule ϕ; instead, the iterates xt are generated by a
+gradient-based algorithm, akin to Eq. 3. In particular, our setting reduces to standard optimisation if
+ϕ is defined by ϕ : (x, w) �→ w, in which case xt = wt. A common approach to analysis is to treat
+the iterates x1, x2, . . . as generated by an online learning algorithm over online losses, obtain a regret
+guarantee for the sequence, and use online-to-batch conversion to obtain a rate of convergence.
+Online Optimisation.
+In online convex optimisation [Zinkevich, 2003], a learner is given a convex
+decision set U and faces a sequence of convex loss functions {αtft}T
+t=1. At each time t ∈ [T], it
+must make a prediction ut prior to observing αtft, after which it incurs a loss αtft(ut) and receives
+a signal—either αtft itself or a (sub-)gradient of αtft(ut). The learner’s goal is to minimise regret,
+R(T) := �T
+t=1 αt(ft(ut) − ft(u)), against a comparator u ∈ U. An important property of a convex
+function f is f(u′) − f(u) ≤ ⟨∇f(u′), u′ − u⟩. Hence, the regret is largest under linear losses:
+�T
+t=1 αt(ft(ut) − ft(u)) ≤ �T
+t=1 αt⟨∇ft(ut), ut − u⟩. For this reason, it is sufficient to consider
+regret under linear loss functions. An algorithm has sublinear regret if limT →∞ R(T)/T = 0.
+FTRL & AO-FTRL.
+The meta-update in Eq. 3 is an instance of Follow-The-Regularised-Leader
+(FTRL) under linear losses. In Section 6, we show that BMG is an instance of the Adaptive-Optimistic
+FTRL (AO-FTRL), which is an extension due to [Rakhlin and Sridharan, 2013, Mohri and Yang,
+2016, Joulani et al., 2020, Wang et al., 2021]. In AO-FTRL, we have a strongly convex regulariser
+∥·∥2. FTRL and AO-FTRL sets the first prediction u1 to minimise ∥·∥2. Given linear losses {gs}t−1
+s=1
+and learning rates {βt}T
+t=1, each βt > 0, the algorithm proceeds according to
+ut = arg min
+u∈U
+�
+αt⟨˜gt, u⟩ +
+t−1
+�
+s=1
+αs⟨gs, u⟩ +
+1
+2βt
+∥u∥2
+�
+,
+(5)
+5
+
+where each ˜gt is a “hint” that enables optimistic learning [Rakhlin and Sridharan, 2013, Mohri and
+Yang, 2016]; setting ˜gt = 0 recovers the original FTRL algorithm. The goal of a hint is to predict
+the next loss vector gt; if the predictions are accurate AO-FTRL can achieve lower regret than its
+non-optimistic counter-part. Since ∥ · ∥2 is strongly convex, FTRL is well defined in the sense that
+the minimiser exists, is unique and finite [McMahan, 2017]. The regret of FTRL and AO-FTRL
+against any comparator u ∈ U can be upper-bounded by
+R(T) =
+T
+�
+t=1
+αt⟨gt, ut − u⟩ ≤ ∥u∥2
+2βT
++ 1
+2
+T
+�
+t=1
+α2
+t βt ∥gt − ˜gt∥2
+∗ .
+(6)
+Hence, hints that predict gt well can reduce the regret substantially. Without hints, FTRL can
+guarantee O(
+√
+T) regret (for non strongly convex loss functions). However, Dekel et al. [2017] show
+that under linear losses, if hints are weakly positively correlated—defined as ⟨gt, ˜gt⟩ ≥ ϵ∥gt∥2 for
+some ϵ > 0—then the regret guarantee improves to O(log T), even for non strongly-convex loss
+functions. We believe optimism provides an exciting opportunity for novel forms of meta-learning.
+Finally, we note that these regret bounds (and hence our analysis) can be extended to stochastic
+optimisation [Mohri and Yang, 2016, Joulani et al., 2017].
+Online-to-batch conversion.
+The main idea behind online to batch conversion is that, for f convex,
+Jensen’s inequality gives f(¯xT )−f(x∗) ≤ �T
+t=1 αt⟨∇f(xt), xt−x∗⟩/α1:T . Hence, one can provide
+a convergence rate by first establishing the regret of the algorithm that generates xt, from which
+one obtains the convergence rate of the moving average of iterates. Applying this naively yields
+O(1/T) rate of convergence. In recent work, Cutkosky [2019] shows that one can upper-bound the
+sub-optimality gap by instead querying the gradient gradient at the average iterate, f(¯xT ) − f(x∗) ≤
+�T
+t=1 αt⟨∇f(¯xt), xt − x∗⟩/α1:T , which can yield faster rates of convergence. Recently, Joulani
+et al. [2020] tightened the analysis and proved that the sub-optimality gap can be bounded by
+f(¯xT ) − f(x∗) ≤
+1
+α1:T
+�
+Rx(T) − αt
+2L∥∇f(¯xt) − ∇f(x∗)∥2
+∗ − α1:t−1
+2L
+∥∇f(¯xt−1) − ∇f(¯xt)∥2
+∗
+�
+,
+(7)
+were we define Rx(T) := �T
+t=1 αt⟨∇f(¯xt), xt − x∗⟩ as the regret of the sequence {xt}T
+t=1 against
+the comparator x∗. With this machinery in place, we now turn to deriving our main results.
+5
+Analysis
+Our analytical goal is to apply the online-to-batch conversion bound in Eq. 7 to the iterates
+x1, x2, . . . , xT that Algorithm 2 generates. Our main challenge is that the update rule ϕ prevents
+a straightforward application of this bound. Instead, we must upper bound the learner’s regret Rx
+by the meta-learner’s regret, which is defined in terms of the iterates w1, w2, . . . , wT . To this end,
+we may decompose Rx as follows:
+Rx(T) =
+T
+�
+t=1
+αt⟨∇f(¯xt), xt − x∗⟩ =
+T
+�
+t=1
+αt⟨∇f(¯xt), ϕ(¯xt−1, wt) − x∗⟩
+=
+T
+�
+t=1
+αt⟨∇f(¯xt), ϕ(¯xt−1, wt) − ϕ(¯xt−1, w∗)⟩ +
+T
+�
+t=1
+αt⟨∇f(¯xt), ϕ(¯xt−1, w∗) − x∗⟩.
+The first term in the final expression can be understood as the regret under convex losses ℓt(·) =
+αt⟨∇f(¯xt), ϕ(¯xt−1, ·)⟩. Since ϕ(¯xt−1, ·) is affine, ℓt is convex and can be upper bounded by its
+linearisation. The linearisation reads ⟨Dϕ(¯xt−1, wt)T ∇f(¯xt), ·⟩, which is identical the linear losses
+⟨∇ht(wt), ·⟩ faced by the meta-learner in Eq. 3. Hence, we may upper bound Rx(T) by
+Rx(T) ≤
+T
+�
+t=1
+αt⟨Dϕ(¯xt−1, wt)T ∇f(¯xt), wt − w∗⟩ +
+T
+�
+t=1
+αt⟨∇f(¯xt), ϕ(¯xt−1, w∗) − x∗⟩
+=
+T
+�
+t=1
+αt⟨∇ht(wt), wt − w∗⟩ +
+T
+�
+t=1
+αt⟨∇f(¯xt), ϕ(¯xt−1, w∗) − x∗⟩
+= Rw(T) +
+T
+�
+t=1
+αt⟨∇f(¯xt), ϕ(¯xt−1, w∗) − x∗⟩,
+(8)
+6
+
+where the last identity follows by definition: Rw(T) := �T
+t=1 αt⟨∇ht(wt), wt − w∗⟩. For the last
+term in Eq. 8 to be negative, so that Rw(T) ≥ Rx(T), we need the relative power of the comparator
+w∗ to be greater than that of the comparator x∗. Intuitively, the comparator x∗ is non-adaptive.
+It must make one choice x∗ and suffer the average loss. In contrast, the comparator w∗ becomes
+adaptive under the update rule; it can only choose one w∗, but on each round it plays ϕ(¯xt−1, w∗).
+If ϕ is sufficiently flexible, this gives the comparator w∗ more power than x∗, and hence it can
+force the meta-learner to suffer greater regret. When this is the case, we say that regret is preserved
+when moving from x∗ to w∗.
+Definition 1. Given f, {αt}T
+t=1, and {xt}T
+t=1, an update rule ϕ : X × W → X preserves regret if
+there exists a comparator w ∈ W that satisfies
+T
+�
+t=1
+αt⟨ϕ(¯xt−1, w), ∇f(¯xt)⟩ ≤
+T
+�
+t=1
+αt⟨x∗, ∇f(¯xt)⟩.
+(9)
+If such w exists, let w∗ denote the comparator with smallest norm ∥w∥.
+By inspecting Eq. 9, we see that if ϕ(¯xt−1, ·) can be made to negatively align with the gradient
+∇f(¯xt), the update rule preserves regret. Hence, any update rule that is gradient-like in its behaviour
+can be made to preserve regret. However, this must not hold on every step, only sufficiently often; nor
+does it imply that the update rule must explicitly invoke ∇f; for instance, update rules that are affine in
+w preserve regret if the diameter of W is sufficiently large, provided the update rule is not degenerate.
+Lemma 1. Given f, {αt}T
+t=1, and {xt}T
+t=1, if ϕ preserves regret, then
+Rx(T) =
+T
+�
+t=1
+αt⟨∇f(¯xt), xt − x∗⟩ ≤
+T
+�
+t=1
+αt⟨∇f(¯xt), ϕ(¯xt−1, wt) − ϕ(¯xt−1, w∗)⟩ = Rw(T).
+Proof: Appendix D. With Lemma 1, we can provide a convergence guarantee for meta-gradients in
+the convex setting. The mechanics of the proof is to use online-to-batch conversion to upper bound
+f(¯xT )−f(x∗) ≤ Rx(T)/α1:T and then appeal to Lemma 1 to obtain f(¯xT )−f(x∗) ≤ Rw(T)/α1:T ,
+from which point we can plug in the FTRL regret bound.
+Theorem 3. Let ϕ preserve regret and assume Algorithm 2 satisfies the assumptions in Section 2.
+Then
+f(¯xT ) − f(x∗) ≤
+1
+α1:T
+�
+∥w∗∥2
+β
++
+T
+�
+t=1
+λβα2
+t
+2
+∥∇f(¯xt)∥2
+∗
+− αt
+2L∥∇f(¯xt) − ∇f(x∗)∥2
+∗ − α1:t−1
+2L
+∥∇f(¯xt−1) − ∇f(¯xt)∥2
+∗
+�
+.
+Moreover, if x∗ is a global minimiser of f, setting αt = 1 and β =
+1
+λL yields
+f(¯xT ) − f(x∗) ≤ λL diam(W)
+T
+.
+Proof: Appendix D.
+6
+Meta-Learning meets Optimism
+The reason Theorem 3 fails to achieve acceleration is because the negative terms, −∥∇f(¯xt−1) −
+∇f(¯xt)∥2
+∗, do not come into play. This is because the positive term in the bound involves the norm of
+the gradient, rather than the norm of the difference of two gradients. The former is typically a larger
+quantity and hence we cannot guarantee that they vanish. To obtain acceleration, we need some form
+of optimism. In this section, we consider an alteration to Algorithm 2 that uses AO-FTRL for the
+meta-updates. Given some sequence of hints {˜gt}T
+t=1, each ˜gt ∈ Rm, each wt+1 is given by
+wt+1 = arg min
+w∈W
+�
+αt+1˜gt+1 +
+t
+�
+s=1
+αs⟨∇hs(ws), w⟩ +
+1
+2βt
+∥w∥2
+�
+.
+(10)
+7
+
+Algorithm 3: BMG in practice.
+input :Weights {βt}T
+t=1
+input :Update rule ϕ
+input :Target oracle
+input :Initialisation (x0, w1)
+for t = 1, 2, . . . , T:
+xt = xt−1 + ϕ(xt−1, wt)
+Query zt from target oracle
+dt(·) = ∥zt − xt + ϕ(xt, ·)∥2
+wt+1 = wt − βt∇dt(wt)
+return xT
+Algorithm 4: Convex optimistic meta-learning.
+input :Weights {αt}T
+t=1, {βt}T
+t=1
+input :Update rule ϕ
+input :Hints {˜gt}T
+t=1
+input :Initialisation (¯x0, w1)
+for t = 1, 2, . . . , T:
+xt = ϕ(¯xt−1, wt)
+¯xt = (1 − αt/α1:t)¯xt−1 + (αt/α1:t)xt
+gt = Dϕ(¯xt−1, wt)T ∇f(¯xt)
+vt = αt+1˜gt+1 + �t
+s=1 αsgs
+wt+1 = arg minw∈W⟨vt, w⟩ +
+1
+2βt ∥w∥2
+return ¯xT
+Otherwise, we proceed as in Algorithm 2; for a complete description, see Algorithm 4. The AO-FTRL
+updates do not correspond to a standard meta-update. However, we show momentarily that optimism
+can be instantiated via the BMG method, detailed in Algorithm 3. The proof for optimistic meta-
+gradients proceed largely as in Theorem 3, it only differs in that we apply the AO-FTRL regret bound.
+Theorem 4. Let ϕ preserve regret and assume Algorithm 4 satisfy the assumptions in Section 2. Then
+f(¯xT ) − f(x∗) ≤
+1
+α1:T
+�
+∥w∗∥2
+βT
++
+T
+�
+t=1
+α2
+t βt
+2
+∥Dϕ(¯xt−1, wt)T ∇f(¯xt) − ˜gt∥2
+∗
+− αt
+2L∥∇f(¯xt) − ∇f(x∗)∥2
+∗ − α1:t−1
+2L
+∥∇f(¯xt−1) − ∇f(¯xt)∥2
+∗
+�
+.
+Moreover, assume each ˜gt is such that ∥Dϕ(¯xt−1, wt)T ∇f(¯xt) − ˜gt∥2
+∗ ≤ q∥∇f(¯xt−1) − ∇f(¯xt)∥2
+∗
+for some q > 0. If each αt = t and βt = t−1
+2tqL, then
+f(¯xt) − f(x∗) ≤ 4qL diam(W)
+T 2 − 1
+.
+Proof. The proof follows the same lines as that of Theorem 3. The only difference is that the regret
+of the {wt}T
+t=1 sequence can be upper bounded by ∥w∗∥2
+βT
++ 1
+2
+�T
+t=1 α2
+t βt∥∇ht(wt) − ˜gt∥2
+∗ instead
+of ∥w∗∥2
+βT
++ 1
+2
+�T
+t=1 α2
+t βt∥∇ht(wt)∥2
+∗, as per the AO-FTRL regret bound in Eq. 6. The final part
+follows immediately by replacing the norms and plugging in the values for α and β.
+■
+From Theorem 4, it is clear that if ˜gt is a good predictor of Dϕ(¯xt−1, wt)T ∇f(¯xt), then the positive
+term in the summation can be cancelled by the negative term. In a classical optimisation setting,
+Dϕ = In, and hence it is easy to see that simply choosing ˜gt to be the previous gradient is sufficient
+to achieve the cancellation [Joulani et al., 2020]. Indeed, this choice gives us Nesterov’s Accelerated
+rate [Wang et al., 2021]. The upshot of this is that we can specialise Algorithm 4 to capture Nesterov’s
+Accelerated method by choosing ϕ : (x, w) �→ w—as in the reduction to Heavy Ball—and setting the
+hints to ˜gt = ∇f(¯xt−1). Hence, while the standard meta-update without optimism contains Heavy
+Ball as a special case, the optimistic meta-update contains Nesterov Acceleration as a special case.
+In the meta-learning setting, Dϕ is not an identity matrix, and hence the best targets for meta-learning
+are different. Naively, choosing ˜gt = Dϕ(¯xt−1, wt)T ∇f(¯xt−1) would lead to a similar cancellation,
+but this is not allowed. At iteration t, we have not computed wt when ˜gt is chosen, and hence
+Dϕ(¯xt−1, wt) is not available. The nearest term that is accessible is Dϕ(¯xt−2, wt−1).
+Corollary 1. Let each ˜gt+1 = Dϕ(¯xt−1, wt)T ∇f(¯xt). Assume that ϕ satisfies
+��Dϕ(x′, w)T ∇f(x) − Dϕ(x′′, w′)T ∇f(x′)
+��2
+∗ ≤ ˜λ ∥∇f(x′) − ∇f(x)∥2
+∗
+for all x′′, x′, x ∈ X and w, w′ ∈ W, for some ˜λ > 0. If each αt = t and βt =
+t−1
+2t˜λL, then
+f(¯xT ) − f(x∗) ≤ 4˜λL diam(W)
+T 2−1
+.
+Proof: Appendix D.
+8
+
+6.1
+Bootstrapped Meta-Gradients
+In this section, we present a simplified version of BMG for clarity, with Appendix E providing
+a fuller comparison. Essentially, BMG alters the meta-update in Algorithm 1; instead of directly
+minimising the loss f, it introduces a sequence of targets z1, z2, . . . and the meta-learner’s goal
+is select w so that the updated parameters minimise the distance these targets. Concretely, given
+an update xt = xt−1 + ϕ(xt−1, wt), targets are bootstrapped from xt, meaning that a vector yt
+is computed to produce the target zt = xt − yt. Assuming the distance to the target is measured
+under 1
+2∥ · ∥2
+2, the BMG meta-update takes the form
+wt+1 = wt − Dϕ(xt−1, wt)T yt.
+Depending on how yt is computed, it can encode optimism. For instance, the authors rely on
+the update rule itself to compute a tangent yt = ϕ(xt, wt) − ∇f(xt + ϕ(xt, wt)). This encodes
+optimism via ϕ because it encourages the meta-learner to build up momentum (i.e. to accumulate
+past updates). We can contrast this with the types of updates produced by AO-FTRL in Eq. 10. If
+we have hints ˜gt+1 = Dϕ(¯xt−1, wt)T ˜yt+1 for some ˜yt+1 ∈ Rn and set ∥ · ∥ = ∥ · ∥2; assuming
+an interior solution, Eq. 10 yields
+wt+1 = wt − Dϕ(¯xt−1, wt)T (αt+1˜yt+1 + αt∇f(¯xt))
+�
+��
+�
+BMG update
++ αtDϕ(¯xt−2, wt−1)T ˜yt
+�
+��
+�
+FTRL error correction
+.
+(11)
+Hence, BMG encodes very similar dynamics to those of AO-FTRL in Eq. 10. Under this choice of
+hints, the main qualitative difference is that AO-FTRL includes a correction term. The effect of this
+term is to “undo” previous hints to avoid feedback loops. Notably, BMG can suffer from divergence
+due to feedback if the gradient in yt is not carefully scaled [Flennerhag et al., 2022]. Our theoretical
+analysis suggests a simple correction method that may stabilize BMG in practice.
+More generally, targets in BMG are isomorphic to the hint function in AO-FTRL if the measure
+of distance in BMG is a Bregman divergence under a strongly convex function (Appendix E). An
+immediate implication of this is that the hints in Corollary 1 can be expressed as targets in BMG, and
+hence if BMG satisfies the assumptions involved, it converges at a rate O(˜λ/T 2). More generally,
+Theorem 4 provides a sufficient condition for any target bootstrap in BMG to achieve acceleration.
+Corollary 2. Let each ˜gt+1 = Dϕ(¯xt−1, wt)T ˜yt+1, for some ˜yt+1 ∈ Rn. If each ˜yt+1 is a better
+predictor of the next gradient than ∇f(¯xt−1), in the sense that
+∥Dϕ(¯xt−2, wt−1)T ˜yt − Dϕ(¯xt−1, wt)T ∇f(¯xt)∥∗ ≤ ˜λ∥∇f(¯xt) − ∇f(¯xt−1)∥∗,
+then Algorithm 4 guarantees convergence at a rate O(˜λ/T 2).
+7
+Conclusion
+This paper explores a connection between convex optimisation and meta-learning. We construct an
+algorithm for convex optimisation that aligns as closely as possible with how meta-learning is done
+in practice. Meta-learning introduces a transformation and we study the effect this transformation has
+on the rate of convergence. We find that, while a meta-learned update rule cannot generate a better
+dependence on the horizon T, it can improve upon classical optimisation up to a constant factor.
+An implication of our analysis is that for meta-learning to achieve acceleration, it is important to
+introduce some form of optimism. From a classical optimisation point of view, such optimism
+arises naturally by providing the meta-learner with hints. If hints are predictive of the learning
+dynamics these can lead to significant acceleration. We show that the recently proposed BMG
+method provides a natural avenue to incorporate optimism in practical application of meta-learning.
+Because targets in BMG and hints in optimistic online learning commute, our results provide first
+rigorous proof of convergence for BMG, while providing a general condition under which optimism
+in BMG yields accelerated learning.
+9
+
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+
+Appendix
+A
+Notation
+Table 1: Notation
+Indices
+t
+Iteration index: t ∈ {1, ..., T}.
+T
+Total number of iterations.
+[T]
+The set {1, 2, . . . , T}.
+i
+Component index: xi is the ith component of x = (x1, . . . , xn).
+αa:b
+Sum of weights: αa:b = �b
+s=a αs
+xa:b
+Weighted sum: xa:b = �b
+s=a αsxs
+¯xa:b
+Weighted average: ¯xa:b = xa:b/αa:b
+Parameters
+x∗ ∈ X
+Minimiser of f.
+xt ∈ X
+Parameter at time t
+¯xt ∈ X
+Moving average of {xs}t
+s=1 under weights {αs}t
+s=1.
+ρt ∈ (0, ∞)
+Moving average coefficient αt/α1:t.
+wt ∈ W
+Meta parameters
+w∗ ∈ X
+w ∈ W that retains regret with smallest norm ∥w∥.
+αt ∈ (0, ∞)
+Weight coefficients
+βt ∈ (0, ∞)
+Meta-learning rate
+Maps
+f : X → R
+Objective function
+∥ · ∥ : X → R
+Norm on X.
+∥ · ∥∗ : X ∗ → R
+Dual norm of ∥ · ∥.
+ht : W → R
+Online loss faced by the meta learner
+Rx(T)
+Regret of {xt}T
+t=1 against x∗: Rx(T) := �T
+t=1 αt⟨∇f(¯xt), xt −x∗⟩.
+Rw(T)
+Rw(T) := �T
+t=1 αt⟨∇f(¯xt), ϕ(¯xt−1, wt) − ϕ(¯xt−1, w∗)⟩.
+ϕ : Rn × Rm → Rn
+Generic update rule used in practice
+Dϕ(x, ·) : Rm →Rn×m
+Jacobian of ϕ w.r.t. its second argument, evaluated at x ∈ Rn.
+ϕ : X × W → X
+Update rule in convex setting
+Dϕ(x, ·) : W → Rn×m
+Jacobian of ϕ w.r.t. its second argument, evaluated at x ∈ X.
+Bµ : Rn×Rn →[0, ∞)
+Bregman divergence under µ : Rn → R.
+µ : Rn → R
+Convex distance generating function.
+13
+
+Table 2: Hyper-parameter sweep on Convex Quadratics. All algorithms are tuned for learning rate
+and initialisation of w. Baselines are tuned for decay rate; meta-learned variant are tuned for the
+meta-learning rate.
+Learning rate
+[.1, .3, .7, .9, 3., 5.]
+w init scale
+[0., 0.3, 1., 3., 10., 30.]
+Decay rate / Meta-learning rate
+[0.001, 0.003, 0.01, .03, .1, .3, 1., 3., 10., 30.]
+B
+Convex Quadratic Experiments
+Loss function.
+We consider the problem of minimising a convex quadratic loss functions f :
+R2 → R of the form f(x) = xT Qx, where Q is randomly sampled as follows. We sample a
+random orthogonal matrix U from the Haar distribution scipy.stats.ortho_group. We con-
+struct a diagonal matrix of eigenvalues, ranked smallest to largest, with λi = i2. Hence, the first
+dimension has an eigenvalue 1 and the second dimension has eigenvalue 4. The matrix Q is given
+by U T diag(λ1, . . . , λn)U.
+Protocol.
+Given that the solution is always (0, 0), this experiment revolves around understanding
+how different algorithms deal with curvature. Given symmetry in the solution and ill-conditioning,
+we fix the initialisation to x0 = (4, 4) for all sampled Qs and all algorithms and train for 100
+iterations. For each Q and each algorithm, we sweep over the learning rate, decay rate, and the
+initialization of w see Table 2. For each method, we then report the results for the combination
+of hyper parameters that performed the best.
+Results.
+We report the learning curves for the best hyper-parameter choice for 5 randomly sampled
+problems in the top row of Figure 2 (columns correspond to different Q). We also study the sensitivity
+of each algorithm to the learning rate in the bottom row Figure 2. For each learning rate, we report
+the cumulative loss during training. While baselines are relatively insensitive to hyper-parameter
+choice, meta-learned improve for certain choices, but are never worse than baselines.
+C
+Imagenet Experiments
+Protocol.
+We train a 50-layer ResNet following the Haiku example, available at https://github.
+com/deepmind/dm-haiku/blob/main/examples/imagenet. We modify the default setting to
+run with SGD. We compare default SGD to variants that meta-learn an element-wise learning rate
+online, i.e. (x, w) �→ w ⊙ ∇f(x). For each variant, we sweep over the learning rate (for SGD) or
+meta-learning rate. We report results for the best hyper-parameter over three independent runs.
+Standard meta-learning.
+In the standard meta-learning setting, we apply the update rule once
+before differentiating w.r.t. the meta-parameters. That is, the meta-update takes the form wt+1 =
+wt − β∇ht(wt), where ht = f(xt + wt ⊙ ∇f(xt)). Because the update rule is linear in w, we
+can compute the meta-gradient analytically:
+∇ht(wt) = ∇wf(x + ϕ(x, w)) = Dϕ(x, w)T ∇f(x′) = ∇f(x) ⊙ ∇f(x′),
+where x′ = x + ϕ(x, w). Hence, we can compute the meta-updates in Algorithm 1 manually as
+wt+1 = max{wt − β∇f(xt) ⊙ ∇f(xt+1), 0.}, where we introduce the max operator on an element-
+wise basis to avoid negative learning rates. Empirically, this was important to stabilize training.
+Optimistic meta-learning.
+For optimistic meta-learning, we proceed much in the same way, but
+include a gradient prediction ˜gt+1. For our prediction, we use the previous gradient, ∇f(xt+1), as
+our prediction. Following Eq. 11, this yields meta-updates of the form
+wt+1 = max
+�
+wt − β∇f(xt+1) ⊙ (∇f(xt+1) + ∇f(xt)) − ∇f(xt) ⊙ ∇f(xt), 0.
+�
+.
+Results.
+We report Top-1 accuracy on the held-out test set as a function of training steps in Figure 1.
+Tuning the learning rate does not yield any statistically significant improvements under standard
+meta-learning. However, with optimistic meta-learning, we obtain a significant acceleration as well
+as improved final performance, increasing the mean final top-1 accuracy from 72% to 75%.
+14
+
+Table 3: Hyper-parameter sweep on Imagenet.
+(Meta-)learning rate
+[0.001, 0.01, 0.02, 0.05, 0.1]
+D
+Proofs
+This section provides complete proofs. We restate the results for convenience.
+Lemma 1. Given f, {αt}T
+t=1, and {xt}T
+t=1, if ϕ preserves regret, then
+Rx(T) =
+T
+�
+t=1
+αt⟨∇f(¯xt), xt − x∗⟩ ≤
+T
+�
+t=1
+αt⟨∇f(¯xt), ϕ(¯xt−1, wt) − ϕ(¯xt−1, w∗)⟩ = Rw(T).
+Proof. Starting from Rx in Eq. 8, if the update rule preserves regret, there exists w∗ ∈ W for which
+Rx(T) =
+T
+�
+t=1
+αt⟨∇f(¯xT ), ϕ(¯xt−1, wt) − x∗⟩
+=
+T
+�
+t=1
+αt⟨∇f(¯xT ), ϕ(¯xt−1, wt) − ϕ(¯xt−1, w∗)⟩ +
+T
+�
+t=1
+αt⟨∇f(¯xT ), ϕ(¯xt−1, w∗) − x∗⟩
+≤
+T
+�
+t=1
+αt⟨∇f(¯xT ), ϕ(¯xt−1, wt) − ϕ(¯xt−1, w∗)⟩ = Rw(T),
+since w∗ is such that �T
+t=1 αt⟨∇f(¯xT ), ϕ(¯xt−1, w∗) − x∗⟩ ≤ 0.
+■
+Theorem 3. Let ϕ preserve regret and assume Algorithm 2 satisfy the assumptions in Section 2. Then
+f(¯xT ) − f(x∗) ≤
+1
+α1:T
+�
+∥w∗∥2
+β
++
+T
+�
+t=1
+λβα2
+t
+2
+∥∇f(¯xt)∥2
+∗
+− αt
+2L∥∇f(¯xt) − ∇f(x∗)∥2
+∗ − α1:t−1
+2L
+∥∇f(¯xt−1) − ∇f(¯xt)∥2
+∗
+�
+.
+If x∗ is a global minimiser of f, setting αt = 1 and β =
+1
+λL yields f(¯xT ) − f(x∗) ≤ λL diam(W)
+T
+.
+Proof. Since ϕ preserves regret, by Lemma 1, the regret term Rx(T) in Eq. 7 is upper bounded by
+Rw(T). We therefore have
+f(¯xT ) − f(x∗) ≤
+1
+α1:T
+�
+Rw(T) − αt
+2L∥∇f(¯xt) − ∇f(x∗)∥2
+∗ − α1:t−1
+2L
+∥∇f(¯xt−1) − ∇f(¯xt)∥2
+∗
+�
+.
+(12)
+Next, we need to upper-bound Rw(T).
+Since, Rw(T) = �T
+t=1 αt⟨∇f(¯xT ), ϕ(¯xt−1, wt) −
+ϕ(¯xt−1, w∗)⟩, the regret of {wt}T
+t=1 is defined under loss functions ht : W → R given by
+ht = αt⟨∇f(¯xT ), ϕ(¯xt−1, w))⟩.
+By assumption of convexity in ϕ, each ht is convex in w.
+Hence, the regret under {αtht}T
+t=1 can be upper bounded by the regret under the linear losses
+{αt⟨∇ht(wt), ·⟩}T
+t=1. These linear losses correspond to the losses used in the meta-update in Eq. 3.
+Since the meta-update is an instance of FTRL, we may upper-bound Rw(T) by Eq. 6 with each
+15
+
+˜gt = 0. Putting this together along with smoothness of ϕ,
+Rx(T) ≤ Rw(T)
+=
+T
+�
+t=1
+αt⟨∇f(¯xT ), ϕ(¯xt−1, wt) − ϕ(¯xt−1, w∗)⟩
+≤
+T
+�
+t=1
+αt⟨∇ht(wt), wt − w∗⟩
+≤ ∥w∗∥2
+β
++ β
+2
+T
+�
+t=1
+α2
+t ∥∇ht(wt)∥2
+∗
+= ∥w∗∥2
+β
++ β
+2
+T
+�
+t=1
+α2
+t ∥Dϕ(¯xt−1, wt)T ∇f(¯xt)∥2
+∗
+≤ ∥w∗∥2
+β
++ λβ
+2
+T
+�
+t=1
+α2
+t ∥∇f(¯xt)∥2
+∗.
+(13)
+Putting Eq. 12 and Eq. 13 together gives the stated bound. Next, if x∗ is the global optimiser,
+∇f(x∗) = 0 by first-order condition. Setting β = 1/(Lλ) and αt = 1 means the first two norm
+terms in the summation cancel. The final norm term in the summation is negative and can be ignored.
+We are left with f(¯xT ) − f(x∗) ≤ λL∥w∗∥2
+T
+≤ λL diam(W)
+T
+.
+■
+Corollary 1. Let each ˜gt+1 = Dϕ(¯xt−1, wt)T ∇f(¯xt). Assume that ϕ satisfies
+��Dϕ(x′, w)T ∇f(x) − Dϕ(x′′, w′)T ∇f(x′)
+��2
+∗ ≤ ˜λ ∥∇f(x′) − ∇f(x)∥2
+∗
+for all x′′, x′, x ∈ X and w, w′ ∈ W, for some ˜λ > 0. If each αt = t and βt =
+t−1
+2t˜λL, then
+f(¯xT ) − f(x∗) ≤
+4˜λL diam(W)
+T 2−1
+.
+Proof. Plugging in the choice of ˜gt and using that
+��Dϕ(¯xt−1, wt)T ∇f(¯xt) − Dϕ(xt−2, wt−1)T ∇f(¯xt−1)
+��2
+∗ ≤ ˜λ ∥∇f(¯xt−1) − ∇f(¯xt)∥2
+∗ ,
+the bound in Theorem 4 becomes
+f(¯xT ) − f(x∗) ≤
+1
+α1:T
+�
+∥w∗∥2
+βT
++ 1
+2
+T
+�
+t=1
+�
+˜λα2
+t βt − α1:t−1
+L
+�
+∥∇f(¯xt) − ∇f(¯xt−1)∥2
+∗
+�
+,
+where we drop the negative terms ∥∇f(¯xt) − ∇f(x∗)∥2
+∗. Setting αt = t yields α1:t−1 = (t−1)t
+2
+,
+while setting βt =
+t−1
+2t˜λL means ˜λα2
+t βt = (t−1)t
+2L . Hence, ˜λα2
+t βt − α1:t−1/L cancels and we get
+f(¯xT ) − f(x∗) ≤ ∥w∗∥2
+βT α1:T
+=
+4∥w∗∥2˜λL
+(T − 1)(T + 1) ≤ 4˜λL diam(W)
+(T − 1)(T + 1) = 4˜λL diam(W)
+T 2 − 1
+.
+■
+Corollary 2. Let each ˜gt+1 = Dϕ(¯xt−1, wt)T ˜yt+1, for some ˜yt+1 ∈ Rn. If each ˜yt+1 is a better
+predictor of the next gradient than ∇f(¯xt−1), in the sense that
+∥Dϕ(¯xt−2, wt−1)T ˜yt − Dϕ(¯xt−1, wt)T ∇f(¯xt)∥∗ ≤ ˜λ∥∇f(¯xt) − ∇f(¯xt−1)∥∗,
+then Algorithm 4 guarantees convergence at a rate O(˜λ/T 2).
+Proof. The proof follows the same argument as Corollary 1.
+■
+16
+
+Algorithm 5: BMG in practice (general version).
+input :Weights {ρt}T
+t=1, {βt}T
+t=1
+input :Update rule ϕ
+input :Matching function Bµ
+input :Target oracle
+input :Initialisation (x0, w1)
+for t = 1, 2, . . . , T:
+xt = xt−1 + ϕ(xt−1, wt)
+Query zt from target oracle
+dt : w �→ Bµ
+zt(xt−1 + ϕ(xt−1, w))
+wt+1 = wt − βt∇dt(wt)
+return xT
+E
+BMG
+Errata: this was incorrectly referred to as Appendix F in our original submission.
+In this section, we provide a more comprehensive reduction of BMG to AO-FTRL. First, we provide
+a more general definition of BMG. Let µ : X → R be a convex distance generating function and
+define the Bregman Divergence Bµ : Rn × Rn → R by
+Bµ
+z (x) = µ(x) − µ(z) − ⟨∇µ(z), x − z⟩.
+Given initial condition (x0, w1), the BMG updates proceed according to
+xt = xt−1 + ϕ(xt−1, wt)
+wt+1 = wt − βt∇dt(wt),
+(14)
+where dt : Rn → R is defined by dt(w) = Bµ
+zt(xt−1 + ϕ(xt−1, wt)), where each zt ∈ Rn is
+referred to as a target. See Algorithm 5 for an algorithmic summary. A bootstrapped target uses
+the meta-learner’s most recent update, xt, to compute the target, zt = xt + yt for some tangent
+vector yt ∈ Rn. This tangent vector represents a form of optimism, and provides a signal to the
+meta-learner as to what would have been a more efficient update. In particular, the author’s consider
+using the meta-learned update rule to construct yt; yt = ϕ(xt, wt) − ∇f(xtϕ(xt, w − t)). Note
+that xt = xt−1 + ϕ(xt−1, wt), and hence this tangent vector is obtained by applying the update rule
+again, but now to xt. For this tangent to represent an improvement, it must be assumed that wt is
+a good parameterisation. Hence, bootstrapping represents a form of optimism. To see how BMG
+relates to Algorithm 4, and in particular, Eq. 10, expand Eq. 14 to get
+wt+1 = wt − βtDϕ(xt−1, wt)T (∇µ(xt) − ∇µ(zt)) .
+(15)
+In contrast, AO-FTRL reduces to a slightly different type of update.
+Lemma 2. Consider Algorithm 4.
+Given online losses ht
+:
+W
+→
+R defined by
+{⟨Dϕ(¯xt−1, wt)T ∇f(¯xt), ·⟩}T
+t=1 and hint functions {⟨˜gt, ·, }⟩T
+t=1, with each ˜gt ∈ Rm. If ∥ · ∥ =
+(1/2)∥ · ∥2, an interior solution to Eq. 10 is given by
+wt+1 =
+βt
+βt−1
+wt − βt
+�
+αt+1˜gt+1 + αt(Dϕ(¯xt−1, wt)T ∇f(¯xt) − ˜gt)
+�
+.
+17
+
+Proof. By direct computation:
+wt+1 = arg min
+w∈W
+�
+αt+1⟨˜gt+1, w⟩ +
+t
+�
+s=1
+αs⟨Dϕ(¯xs−1, ws)T ∇f(¯xs), w⟩ +
+1
+2βt
+∥w∥2
+2
+�
+= −βt
+�
+αt+1˜gt+1 +
+t
+�
+s=1
+αtDϕ(¯xs−1, ws)T ∇f(¯xs))
+�
+= −βt
+�
+αt+1˜gt+1 + αtDϕ(¯xt−1, wt)T ∇f(¯xt) +
+�t−1
+�
+s=1
+αtDϕ(¯xs−1, ws)T ∇f(¯xs))
+��
+= −βt
+�
+αt+1˜gt+1 + αt(Dϕ(¯xt−1, wt)T ∇f(¯xt) − ˜gt)
+�
+− βt
+�
+αt˜gt +
+t−1
+�
+s=1
+αtDϕ(¯xs−1, ws)T ∇f(¯xs))
+�
+=
+βt
+βt−1
+wt − βt
+�
+αt+1˜gt+1 + αt(Dϕ(¯xt−1, wt)T ∇f(¯xt) − ˜gt)
+�
+.
+■
+AO-FTRL includes a decay rate βt/βt−1; this decay rate can be removed by instead using optimistic
+online mirror descent [Rakhlin and Sridharan, 2013, Joulani et al., 2017]—to simplify the exposition
+we consider only FTRL-based algorithms in this paper. An immediate implication of Lemma 2
+is the error-corrected version of BMG.
+Corollary 3. Setting ˜gt+1 = Dϕ(¯xt−1, wt)T ˜gt+1 for some ˜yt+1 ∈ Rn yields an error-corrected
+version of the BMG meta-update in Eq. 14. Specifically, the meta-updates in Lemma 2 becomes
+wt+1 =
+βt
+βt−1
+wt − βtDϕ(¯xt−1, wt)T (αt+1˜yt+1 + αt∇f(¯xt))
+�
+��
+�
+BML update
++ βtαtDϕ(¯xt−2, wt−1)T ˜yt
+�
+��
+�
+FTRL error correction
+.
+Proof. Follows immediately by substituting for each ˜gt+1 in Lemma 2.
+■
+To illustrate this connection, Let µ = f. In this case, the BMG update reads wt+1 = wt −
+βtDϕ(xt−1, wt)T (∇f(zt) − ∇f(xt)). The equivalent update in the convex optimisation setting (i.e.
+Algorithm 4) is obtained by setting ˜yt+1 = ∇f(zt), in which case Corollary 3 yields
+wt+1 = βt+1
+βt
+wt − βtDϕ(¯xt−1, wt)T (αt+1∇f(zt) − αt∇f(¯xt)) + ξt,
+where ξt = βtαtDϕ(¯xt−2, wt−1)T ∇f(¯xt − 1) denotes the error correction term we pick up through
+AO-FTRL. Since Algorithm 5 does not average its iterates—while Algorithm 4 does—we see that
+these updates (ignoring ξt) are identical up to scalar coefficients (that can be controlled for by scaling
+each βt and each ˜gt+1 accordingly).
+More generally, the mapping from targets in BMG and hints in AO-FTRL takes on a more complicated
+pattern. Our next results show that we can always map one into the other. To show this, we need
+to assume a certain recursion. It is important to notice however that at each iteration introduces
+an unconstrained variable and hence the assumption on the recursion is without loss of generality
+(as the free variable can override it).
+Theorem 5. Targets in Algorithm 5 and hints in algorithm 4 commute in the following sense. BMG
+→ AO-FTRL. Let BMG targets {zt}T
+t=1 by given. A sequence of hints {˜g}T
+t=1 can be constructed
+recursively by
+αt+1˜gt+1 = Dϕ(¯xt−1, wt)T (∇µ(¯xt) − ∇µ(zt) − αt∇f(¯xt)) + αt˜gt,
+t ∈ [T],
+(16)
+so that interior updates for Algorithm 4 are given by
+wt+1 =
+βt
+βt−1
+wt − βt (∇µ(zt) − ∇µ(¯xt)) .
+18
+
+AO-FTRL → BMG. Conversely, assume a sequence {˜yt}T
+t=1 are given, each ˜yt ∈ Rn. If µ strictly
+convex, a sequence of BMG targets {zt}T
+t=1 can be constructed recursively by
+zt = ∇µ−1 (∇µ(xt) − (αt+1˜yt+1 + αt∇f(xt)))
+t ∈ [T],
+so that BMG updates in Eq. 14 are given by
+wt+1 = wt − βt
+�
+αt+1˜gt+1 + αt(Dϕ(¯xt−1, wt)T ∇f(¯xt) − ˜gt)
+�
+,
+where each ˜gt+1 is the BMG-induced hint function, given by
+αt+1˜gt+1 = αt+1Dϕ(xt−1, wt)T ˜yt+1 + αt˜gt.
+Proof. First, consider BMG → AO-FTRL. First note that ˜g1 is never used and can thus be chosen
+arbitrarily—here, we set ˜g1 = 0. For w2, Lemma 2 therefore gives the interior update
+w2 = β2
+β1
+w1 − β1(α2˜g2 + α1Dϕ(¯x0, w1)T ∇f(¯x1)).
+Since the formulate for ˜g2 in Eq. 16 only depends on quantities with iteration index t = 0, 1, we may
+set α2˜gt = Dϕ(¯x0, w1)T (∇µ(¯x1) − ∇µ(zt) − αt∇f(¯x1)). This gives the update
+w2 = β2
+β1
+w1 − β1Dϕ(¯x0, w1)T (∇µ(¯x1) − ∇µ(z1)).
+Now assume the recursion holds up to time t. As before, we may choose αt+1˜gt+1 according to
+the formula in Eq. 16 since all quantities on the right-hand side depend on quantities computed at
+iteration t or t − 1. Subtituting this into Lemma 2, we have
+wt+1 =
+βt
+βt−1
+wt − βt
+�
+αt+1˜gt+1 + αt(Dϕ(¯xt−1, wt)T ∇f(¯xt) − ˜gt)
+�
+=
+βt
+βt−1
+wt − βt
+�
+Dϕ(¯xt−1, wt)T (∇µ(¯xt) − ∇µ(zt) − αt∇f(¯xt)) + αt˜gt
++αt(Dϕ(¯xt−1, wt)T ∇f(¯xt) − ˜gt)
+�
+=
+βt
+βt−1
+wt − βtDϕ(¯xt−1, wt)T (∇µ(¯xt) − ∇µ(zt)).
+AO-FTRL → BMG. The proof in the other direction follows similarly. First, note that for µ strictly
+convex, ∇µ is invertible. Then, z1 = ∇µ−1(∇µ(x1) − (α2˜y2 + α1∇f(x1))). This target is
+permissible since x1 is already computed and {˜yt}T
+t=1 is given. Substituting this into the BMG
+meta-update in Eq. 14, we find
+w2 = w1 − β1Dϕ(x0, w1)T (∇µ(x1) − ∇µ(∇µ−1(∇µ(x1) − (α2˜y2 + α1∇f(x1)))))
+= w1 − β1Dϕ(x0, w1)T (α2˜y2 + α1∇f(x1))
+= w1 − β1
+�
+α2˜g2 + α1(Dϕ(¯x0, w1)T ∇f(¯x1) − ˜g1)
+�
+,
+where the last line uses that ˜g2 is defined by α2˜g2 − α1˜g1 = Dϕ(¯x0, w1)T ˜y2 and ˜g1 is arbitrary.
+Again, assume the recursion holds to time t. We then have
+wt+1 = wt − βtDϕ(xt−1, wt)T (∇µ(xt) − ∇µ(zt))
+= wt − βtDϕ(xt−1, wt)T (∇µ(xt)
+− ∇µ(∇µ−1(∇µ(xt) − (αt+1˜yt+1 + αt∇f(xt)))))
+= wt − βtDϕ(xt−1, wt)T (αt+1˜yt+1 + αt∇f(xt))
+= wt − βt(αt+1˜gt+1 + αt(Dϕ(xt−1, wt)T ∇f(xt) − ˜gt)).
+■
+19
+
diff --git a/mNE1T4oBgHgl3EQfhASI/content/tmp_files/load_file.txt b/mNE1T4oBgHgl3EQfhASI/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..799d0690b7ca2ba0497a9f1322c3a3682342fece
--- /dev/null
+++ b/mNE1T4oBgHgl3EQfhASI/content/tmp_files/load_file.txt
@@ -0,0 +1,1022 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf,len=1021
+page_content='Optimistic Meta-Gradients  Sebastian Flennerhag  DeepMind  flennerhag@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='com  Tom Zahavy  DeepMind  Brendan O’Donoghue  DeepMind  Hado van Hasselt  DeepMind  András György  DeepMind  Satinder Singh  DeepMind  Abstract  We study the connection between gradient-based meta-learning and convex op-  timisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We observe that gradient descent with momentum is a special case  of meta-gradients, and building on recent results in optimisation, we prove con-  vergence rates for meta-learning in the single task setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' While a meta-learned  update rule can yield faster convergence up to constant factor, it is not sufficient  for acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Instead, some form of optimism is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We show that op-  timism in meta-learning can be captured through Bootstrapped Meta-Gradients  [Flennerhag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2022], providing deeper insight into its underlying mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  1  Introduction  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='0  Training Steps  1e6  50  55  60  65  70  75  Top-1 Test Accuracy (%)  SGD  Standard meta-learning  Optimistic meta-learning  Figure 1: ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We compare training a 50-  layer ResNet using SGD against variants that tune  an element-wise learning rate online using standard  meta-learning or optimistic meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Shad-  ing depicts 95% confidence intervals over 3 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  In meta-learning, a learner is using a param-  eterised algorithm to adapt to a given task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  The parameters of the algorithm are then  meta-learned by evaluating the learner’s result-  ing performance [Schmidhuber, 1987, Hinton  and Plaut, 1987, Bengio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 1991].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  This  paradigm has garnered wide empirical success  [Hospedales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' For instance, it has  been used to meta-learn how to explore in re-  inforcement learning (RL) [Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2018a,  Alet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2020], online hyper-parameter tun-  ing of non-convex loss functions [Bengio, 2000,  Maclaurin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2015, Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2018b, Zahavy  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2020], discovering black-box loss func-  tions [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2016, Kirsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2019, Xu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2020, Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2020], black-box learning  algorithms [Hochreiter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2001, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=',  2016], or entire training protocols [Real et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=',  2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Yet, very little is known in terms of the  theoretical properties of meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  The reason for this is the complex interac-  tion between the learner and the meta-learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  learner’s problem is to minimize the expected  loss f of a stochastic objective by adapting its parameters x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The learner has an update rule ϕ at  its disposal that generates new parameters xt = xt−1 + ϕ(xt−1, wt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' we suppress data dependence to  simplify notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' A simple example is when ϕ represents gradient descent with wt = η its step size,  that is ϕ(xt−1, η) = −η∇f(xt−1) [Mahmood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2012, van Erven and Koolen, 2016];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' several  works have explored meta-learning other aspects of a gradient-based update rule [Finn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2017,  Nichol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2018, Flennerhag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2019, Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2018b, Zahavy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2020, Flennerhag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=',  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='03236v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='LG]  9 Jan 2023  2022, Kirsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2019, Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' More generally, ϕ need not be limited to the gradient of  any function, for instance, it can represent some algorithm implemented within a Recurrent Neural  Network [Schmidhuber, 1987, Hochreiter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2001, Andrychowicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2016, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  The meta-learner’s problem is to optimise the meta-parameters wt to yield effective updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In  a typical (gradient-based) meta-learning setting, it does so by treating xt as a function of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Let  ht, defined by ht(w) = f(xt−1 + ϕ(xt−1, w)), denote the learner’s post-update performance as a  function of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The learner and the meta-learner co-evolve according to  xt = xt−1 + ϕ(xt−1, wt),  and  wt+1 = wt − ∇ht(wt)  = wt − Dϕ(xt−1, wt)T ∇f(xt),  where Dϕ(x, w) denotes the Jacobian of ϕ with respect to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The nested structure between these  two updates makes it challenging to analyse meta-learning, in particular it depends heavily on the  properties of the Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In practice, ϕ is highly complex and so Dϕ is almost always intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  For instance, in Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' [2018a], the meta-parameters define the data-distribution under which  a stochastic gradient is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In Zahavy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' [2020], the meta-parameters define auxiliary  objectives that are meant to help with representation learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' in Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' [2016] they learn  an embedding space for nearest-neighbour predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  For this reason, the only theoretical results we are aware of specialise to the multi-task setting and  assume ϕ represents adaptation by gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In this setting, at each iteration t, the learner  must adapt to a new task ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The learner adapts by taking a (or several) gradient step(s) on ft using  either a meta-learned initialisation [Flennerhag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2019, Finn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2019, Fallah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2020,  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2022] or using a meta-learned regulariser [Khodak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2019, Denevi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Because the update rule has this form, it is possible to treat the meta-optimisation problem as an  online learning problem and derive convergence guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Acceleration in this setup is driven by  the tasks similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' That is, if all tasks are sufficiently similar, a meta-learned update can accelerate  convergence [Khodak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' However, these results do not yield acceleration in the absence  of a task distribution to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  This paper provides an alternative view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We study the classical convex optimisation setting of  approximating the minimiser minx f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We observe that setting the update rule equal to the gradient,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' ϕ : (x, w) �→ w∇f(x), recovers gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Similarly, we show in Section 3 that ϕ can  be chosen to recover gradient descent with momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' This offers another view of meta-learning  as a non-linear transformation of classical optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' A direct implication of this is that a task  similarity is not necessary condition for improving the rate of convergence via meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' While  there is ample empirical evidence to that effect [Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2018b, Zahavy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2020, Flennerhag  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2022, Luketina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2022], we are only aware of theoretical results in the special case of  meta-learned step sizes [Mahmood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2012, van Erven and Koolen, 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  In particular, we analyse meta-learning using recent techniques developed for convex optimisation  [Cutkosky, 2019, Joulani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2020, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Given a function f that is convex with  Lipschitz smooth gradients, meta-learning improves the rate of convergence by a multiplicative factor  λ to O(λ/T), via the smoothness of the update rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Importantly, these works show that to achieve  accelerated convergence, O(1/T 2), some form of optimism is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' This optimism essentially  provides a prediction of the next gradient, and hence represents a model of the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We consider  optimism with meta-learning in the convex setting and prove accelerated rates of convergence,  O(λ/T 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Again, meta-learning affects these bounds by a multiplicative factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We further show  that optimism in meta-learning can be expressed through the recently proposed Bootstrapped Meta-  Gradient method [BMG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Flennerhag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Our analysis provides a first proof of convergence  for BMG and highlights the underlying mechanics that enable faster learning with BMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Our main  contributions are as follows:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We show that meta-learning contains gradient descent with momentum (Heavy Ball [Polyak,  1964];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Section 3) and Nesterov Acceleration [Nesterov, 1983] as special cases (Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We show that gradient-based meta-learning can be understood as a non-linear transformation  of an underlying optimisation method (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We establish rates of convergence for meta-learning in the convex setting (Sections 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We show that optimism can be expressed through [Flennerhag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Our analysis  (Section 6) provides a first proof of convergence for BMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  2  Algorithm 1: Meta-learning in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  input :Weights {βt}T  t=1  input :Update rule ϕ  input :Initialisation (x0, w1)  for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' , T:  xt = xt−1 + ϕ(xt−1, wt)  ht(·) = f(xt−1 + ρtϕ(xt−1, ·))  wt+1 = wt − βt∇ht(wt)  return xT  Algorithm 2: Meta-learning in the convex setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  input :Weights {αt}T  t=1, {βt}T  t=1  input :Update rule ϕ  input :Initialisation (¯x0, w1)  for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' , T:  xt = ϕ(¯xt−1, wt)  ¯xt = (1 − αt/α1:t)¯xt−1 + (αt/α1:t)xt  gt = Dϕ(¯xt−1, wt)T ∇f(¯xt)  wt+1=arg minw∈W  �t  s=1αs⟨gs, w⟩+ 1  2βt ∥w∥2  return ¯xT  2  Meta-learning meets convex optimisation  Problem definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  This section defines the problem studied in this paper and introduces our  notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Let f : X → R be a proper and convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The problem of interest is to approximate  the global minimum minx∈X f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We assume a global minimiser exists and is unique, defined by  x∗ = arg min  x∈X  f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  (1)  We assume that X ⊆ Rn is a closed, convex and non-empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' f is differentiable and has Lipschitz  smooth gradients with respect to a norm ∥ · ∥, meaning that there exists L ∈ (0, ∞) such that  ∥∇f(x) − ∇f(y)∥∗ ≤ L∥x − y∥ for all x, y ∈ X, where ∥ · ∥∗ is the dual norm of ∥ · ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We consider  the noiseless setting for simplicity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' our results carry over to the stochastic setting by replacing the key  online-to-batch bound used in our analysis by its stochastic counterpart [Joulani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Algorithm 1 describes a typical meta-learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Unfortunately, at this level  of generality, little can be said about the its convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Instead, we consider a stylized  variant of meta-learning, described in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' This model differs in three regards: (a) it relies  on moving averages (b) we use a different online learning algorithm for the meta-update, and (c) we  make stricter assumptions on the update rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We describe each component in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Let [T] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' , T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We are given weights {αt}T  t=1, each αt > 0, and an initialisation (¯x0, w1) ∈  X × W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' At each time t ∈ [T], an update rule ϕ : X × W → X generates the update xt =  ϕ(¯xt−1, wt), where W ⊆ Rm is closed, convex, and non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We discuss ϕ momentarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The  algorithm maintains the online average  ¯xt = x1:t  α1:t  = (1 − ρt)¯xt−1 + ρtxt,  (2)  where x1:t = �t  s=1 αsxs, α1:t = �t  s=1 αs, and ρt = αt/α1:t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Our goal is to establish conditions  under which {¯xt}T  t=1 converges to the minimiser x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' While this moving average is not always used  in practical applications, it is required for accelerated rates in online-to-batch conversion [Wang and  Abernethy, 2018, Cutkosky, 2019, Joulani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Convergence depends on how each wt is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In Algorithm 1, the meta-learner faces a sequence  of losses ht : W → R defined by the composition ht(w) = f((1 − ρt)¯xt−1 + ρtϕ(¯xt−1, w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  This makes meta-learning a form of online optimisation [McMahan, 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The meta-updates in  Algorithm 1 is an instance of online gradient descent, which we can model as Follow-The-Regularized-  Leader (FTRL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' reviewed in Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Given some norm ∥ · ∥, an initialization w0 and β > 0,  FTRL sets each wt according to  wt+1 = arg min  w∈W  �  t  �  s=1  αs⟨∇hs(ws), w⟩ + 1  2β ∥w∥2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  (3)  If ∥ · ∥ is the Euclidean norm, the interior solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 3 is given by wt+1 = wt − αtβ∇ht(wt),  the meta-update in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' It is straightforward to extend Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 3 to account for meta-updates that  use AdaGrad-like [Duchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2011] acceleration by altering the norms [Joulani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  3  0  20  40  60  80  100  Iterations  10  28  10  16  10  4  Loss  0  20  40  60  80  100  Iterations  0  20  40  60  80  100  Iterations  0  20  40  60  80  100  Iterations  0  20  40  60  80  100  Iterations  Momentum  Meta-Momentum  AdaGrad  Meta-AdaGrad  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='9999 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='0  Learning rate  10  22  10  12  10  2  Loss  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
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+page_content='990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='9999 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='0  Learning rate  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
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+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='9999 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='0  Learning rate  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='9999 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
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+page_content='0  Learning rate  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='9999 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='0  Learning rate  Momentum  Meta-Momentum  AdaGrad  Meta-AdaGrad  Figure 2: Convex Quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We generate convex quadratic loss functions with ill-conditioning and  compare gradient descent with momentum and AdaGrad to meta-learning variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Meta-Momentum  uses ϕ : (x, w) �→ w ⊙ ∇f(x) while Meta-AdaGrad uses ϕ : (x, w) �→ ∇f(x)/√w, where division  is element-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Top: loss per iteration for randomly sampled loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Bottom: cumulative  loss (regret) at the end of learning as a function of learning rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' details in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Update rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  It is not possible to prove convergence outside of the convex setting, since ϕ may  reach a local minimum where it cannot yield better updates, but the updates are not sufficient to  converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Convexity means that each ht must be convex, which requires that ϕ is affine in w (but  may vary non-linearly in x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We also assume that ϕ is smooth with respect to ∥ · ∥, in the sense that it  has bounded norm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' for all x ∈ X and all w ∈ W we assume that there exists λ ∈ (0, ∞) for which  ∥Dϕ(x, w)T ∇f(x)∥2  ∗ ≤ λ∥∇f(x)∥2  ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  These assumptions hold for any smooth update rule up to first-order Taylor approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  3  Meta-Gradients in the Convex Setting - An Overview  In this section, we provide an informal discussion of our main results (full analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Sections 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Meta-Gradients without Optimism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  The main difference between classical optimisation and  meta-learning is the introduction of the update rule ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' To see how this acts on optimisation, consider  two special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' If the update rule just return the gradient, ϕ = ∇f, Algorithm 2 is reduced  to gradient descent (with averaging).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The inductive bias is fixed and does not change with past  experience, and so acceleration is not possible—the rate of convergence is O(1/  √  T) [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=',  2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The other extreme is an update rule that only depends on the meta-parameters, ϕ(x, w) = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Here, the meta-learner has ultimate control and selects the next update without constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The  only relevant inductive bias is contained in w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' To see how this inductive bias is formed, suppose  ∥ · ∥ = ∥ · ∥2 so that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 3 yields wt+1 = wt − αtρtβ∇f(¯xt) (assuming an interior solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Combining this with the moving average in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 2, we may write the learner’s iterates as  ¯xt = ¯xt−1 + ˜ρt (¯xt−1 − ¯xt−2) − ˜βt∇f(¯xt−1),  where each ˜ρt = ρt  1−ρt−1  ρt−1  and ˜βt = αtρtβ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' setting β = 1/(2L) and each αt = t yields ˜ρt = t−2  t+1  and ˜βt = t/(4(t + 1)L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Hence, the canonical momentum algorithm, Polyak’s Heavy-Ball method  [Polyak, 1964], is obtained as the special case of meta-learning under the update rule ϕ : (x, w) �→ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Because Heavy Ball carries momentum from past updates, it can encode a model of the learning  dynamics that leads to faster convergence, on the order O(1/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The implication of this is that the  dynamics of meta-learning are fundamentally momentum-based and thus learns an update rule in the  same cumulative manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' This manifests theoretically through its convergence guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Theorem 1 (Informal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Set αt = 1 and β =  1  λL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' If each xt is generated under Algorithm 2, then for  any viable ϕ, f(¯xT ) − f(x∗) ≤ λL diam(W)  T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  We refer the reader to Theorem 3 for a formal statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Compared to Heavy Ball, meta-learning  introduces a constant λ that captures the smoothness of the update rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Hence, while meta-learning  does not achieve better scaling in T through ϕ, it can improve upon classical optimisation by a constant  factor if λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' That meta-learning can improve upon momentum is borne out experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In  Figure 2, we consider the problem of minimizing a convex quadratic f : x �→ ⟨x, Qx⟩, where  Q ∈ Rn×n is PSD but ill-conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We compare momentum to a meta-learned step-size, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  ϕ : (x, w) �→ w ⊙ ∇f(x), where ⊙ is the Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Across randomly sampled Q matrices  4  (details: Appendix B), we find that introducing a non-linearity ϕ leads to a sizeable improvement in the  rate of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We also compare AdaGrad to a meta-learned version, ϕ : (x, w) �→ ∇f(x)/√w,  where division is element-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' While AdaGrad is a stronger baseline on account of being parameter-  free, we find that meta-learning the scale vector consistently leads to faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Meta-Gradients with Optimism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  It is well known that minimizing a smooth convex function  admits convergence rates of O(1/T 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Our analysis of standard meta-gradients does not achieve such  acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Previous work indicate that we should not expect to either;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' to achieve the theoretical  lower-limit of O(1/T 2), some form of optimism (reviewed in Section 4) is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' A typical form  of optimism is to predict the next gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' This is how Nesterov Acceleration operates [Nesterov,  1983] and is the reason for its O(1/T 2) convergence guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  From our perspective, meta-learning is a non-linear transformation of the iterate x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Hence, we should  expect optimism to play a similarly crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Formally, optimism comes in the form of hint  functions {˜gt}T  t=1, each ˜gt ∈ Rm, that are revealed to the meta-learner prior to selecting wt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' These  hints give rise to Optimistic Meta-Learning (OML) via meta-updates  wt+1 = arg min  w∈W  �  αt+1˜gt+1 +  t  �  s=1  αs⟨∇hs(ws), w⟩ +  1  2βt  ∥w∥2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  (4)  If the hints are accurate, meta-learning with optimism can achieve an accelerated rate of O(˜λ/T 2),  where ˜λ is a constant that characterises the smoothness of ϕ, akin to λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Again, we find that meta-  learning behaves as a non-linear transformation of classical optimism and its rate of convergence is  governed by the geometry it induces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We summarise this result in the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Theorem 2 (Informal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Let each hint be given by ˜gt+1 = Dϕ(¯xt−1, wt)T ∇f(¯xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Assume that ϕ is  sufficiently smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Set αt = t and βt =  t−1  2t˜λL, then f(¯xT ) − f(x∗) ≤ 4˜λL diam(W)  T 2−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  For a formal statement, see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' These predictions hold empirically in a non-convex setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  We train a 50-layer ResNet using either SGD with a fixed learning rate, or an update rule that adapts a  per-parameter learning rate online, ϕ : (x, w) �→ w⊙∇f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We compare the standard meta-learning  approach without optimism to optimistic meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Figure 1 shows that optimism is critical for  meta-learning to achieve acceleration, as predicted by theory (experiment details in Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  4  Analysis preliminaries: Online Convex Optimisation  In this section, we present analytical tools from the optimisation literature that we build upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In a  standard optimisation setting, there is no update rule ϕ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' instead, the iterates xt are generated by a  gradient-based algorithm, akin to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In particular, our setting reduces to standard optimisation if  ϕ is defined by ϕ : (x, w) �→ w, in which case xt = wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' A common approach to analysis is to treat  the iterates x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' as generated by an online learning algorithm over online losses, obtain a regret  guarantee for the sequence, and use online-to-batch conversion to obtain a rate of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Online Optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  In online convex optimisation [Zinkevich, 2003], a learner is given a convex  decision set U and faces a sequence of convex loss functions {αtft}T  t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' At each time t ∈ [T], it  must make a prediction ut prior to observing αtft, after which it incurs a loss αtft(ut) and receives  a signal—either αtft itself or a (sub-)gradient of αtft(ut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The learner’s goal is to minimise regret,  R(T) := �T  t=1 αt(ft(ut) − ft(u)), against a comparator u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' An important property of a convex  function f is f(u′) − f(u) ≤ ⟨∇f(u′), u′ − u⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Hence, the regret is largest under linear losses:  �T  t=1 αt(ft(ut) − ft(u)) ≤ �T  t=1 αt⟨∇ft(ut), ut − u⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' For this reason, it is sufficient to consider  regret under linear loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' An algorithm has sublinear regret if limT →∞ R(T)/T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  FTRL & AO-FTRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  The meta-update in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 3 is an instance of Follow-The-Regularised-Leader  (FTRL) under linear losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In Section 6, we show that BMG is an instance of the Adaptive-Optimistic  FTRL (AO-FTRL), which is an extension due to [Rakhlin and Sridharan, 2013, Mohri and Yang,  2016, Joulani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2020, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In AO-FTRL, we have a strongly convex regulariser  ∥·∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' FTRL and AO-FTRL sets the first prediction u1 to minimise ∥·∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Given linear losses {gs}t−1  s=1  and learning rates {βt}T  t=1, each βt > 0, the algorithm proceeds according to  ut = arg min  u∈U  �  αt⟨˜gt, u⟩ +  t−1  �  s=1  αs⟨gs, u⟩ +  1  2βt  ∥u∥2  �  ,  (5)  5  where each ˜gt is a “hint” that enables optimistic learning [Rakhlin and Sridharan, 2013, Mohri and  Yang, 2016];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' setting ˜gt = 0 recovers the original FTRL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The goal of a hint is to predict  the next loss vector gt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' if the predictions are accurate AO-FTRL can achieve lower regret than its  non-optimistic counter-part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Since ∥ · ∥2 is strongly convex, FTRL is well defined in the sense that  the minimiser exists, is unique and finite [McMahan, 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The regret of FTRL and AO-FTRL  against any comparator u ∈ U can be upper-bounded by  R(T) =  T  �  t=1  αt⟨gt, ut − u⟩ ≤ ∥u∥2  2βT  + 1  2  T  �  t=1  α2  t βt ∥gt − ˜gt∥2  ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  (6)  Hence, hints that predict gt well can reduce the regret substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Without hints, FTRL can  guarantee O(  √  T) regret (for non strongly convex loss functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' However, Dekel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' [2017] show  that under linear losses, if hints are weakly positively correlated—defined as ⟨gt, ˜gt⟩ ≥ ϵ∥gt∥2 for  some ϵ > 0—then the regret guarantee improves to O(log T), even for non strongly-convex loss  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We believe optimism provides an exciting opportunity for novel forms of meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Finally, we note that these regret bounds (and hence our analysis) can be extended to stochastic  optimisation [Mohri and Yang, 2016, Joulani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Online-to-batch conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  The main idea behind online to batch conversion is that, for f convex,  Jensen’s inequality gives f(¯xT )−f(x∗) ≤ �T  t=1 αt⟨∇f(xt), xt−x∗⟩/α1:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Hence, one can provide  a convergence rate by first establishing the regret of the algorithm that generates xt, from which  one obtains the convergence rate of the moving average of iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Applying this naively yields  O(1/T) rate of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In recent work, Cutkosky [2019] shows that one can upper-bound the  sub-optimality gap by instead querying the gradient gradient at the average iterate, f(¯xT ) − f(x∗) ≤  �T  t=1 αt⟨∇f(¯xt), xt − x∗⟩/α1:T , which can yield faster rates of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Recently, Joulani  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' [2020] tightened the analysis and proved that the sub-optimality gap can be bounded by  f(¯xT ) − f(x∗) ≤  1  α1:T  �  Rx(T) − αt  2L∥∇f(¯xt) − ∇f(x∗)∥2  ∗ − α1:t−1  2L  ∥∇f(¯xt−1) − ∇f(¯xt)∥2  ∗  �  ,  (7)  were we define Rx(T) := �T  t=1 αt⟨∇f(¯xt), xt − x∗⟩ as the regret of the sequence {xt}T  t=1 against  the comparator x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' With this machinery in place, we now turn to deriving our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  5  Analysis  Our analytical goal is to apply the online-to-batch conversion bound in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 7 to the iterates  x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' , xT that Algorithm 2 generates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Our main challenge is that the update rule ϕ prevents  a straightforward application of this bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Instead, we must upper bound the learner’s regret Rx  by the meta-learner’s regret, which is defined in terms of the iterates w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' , wT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' To this end,  we may decompose Rx as follows:  Rx(T) =  T  �  t=1  αt⟨∇f(¯xt), xt − x∗⟩ =  T  �  t=1  αt⟨∇f(¯xt), ϕ(¯xt−1, wt) − x∗⟩  =  T  �  t=1  αt⟨∇f(¯xt), ϕ(¯xt−1, wt) − ϕ(¯xt−1, w∗)⟩ +  T  �  t=1  αt⟨∇f(¯xt), ϕ(¯xt−1, w∗) − x∗⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  The first term in the final expression can be understood as the regret under convex losses ℓt(·) =  αt⟨∇f(¯xt), ϕ(¯xt−1, ·)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Since ϕ(¯xt−1, ·) is affine, ℓt is convex and can be upper bounded by its  linearisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The linearisation reads ⟨Dϕ(¯xt−1, wt)T ∇f(¯xt), ·⟩, which is identical the linear losses  ⟨∇ht(wt), ·⟩ faced by the meta-learner in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Hence, we may upper bound Rx(T) by  Rx(T) ≤  T  �  t=1  αt⟨Dϕ(¯xt−1, wt)T ∇f(¯xt), wt − w∗⟩ +  T  �  t=1  αt⟨∇f(¯xt), ϕ(¯xt−1, w∗) − x∗⟩  =  T  �  t=1  αt⟨∇ht(wt), wt − w∗⟩ +  T  �  t=1  αt⟨∇f(¯xt), ϕ(¯xt−1, w∗) − x∗⟩  = Rw(T) +  T  �  t=1  αt⟨∇f(¯xt), ϕ(¯xt−1, w∗) − x∗⟩,  (8)  6  where the last identity follows by definition: Rw(T) := �T  t=1 αt⟨∇ht(wt), wt − w∗⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' For the last  term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 8 to be negative, so that Rw(T) ≥ Rx(T), we need the relative power of the comparator  w∗ to be greater than that of the comparator x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Intuitively, the comparator x∗ is non-adaptive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  It must make one choice x∗ and suffer the average loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In contrast, the comparator w∗ becomes  adaptive under the update rule;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' it can only choose one w∗, but on each round it plays ϕ(¯xt−1, w∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  If ϕ is sufficiently flexible, this gives the comparator w∗ more power than x∗, and hence it can  force the meta-learner to suffer greater regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' When this is the case, we say that regret is preserved  when moving from x∗ to w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Given f, {αt}T  t=1, and {xt}T  t=1, an update rule ϕ : X × W → X preserves regret if  there exists a comparator w ∈ W that satisfies  T  �  t=1  αt⟨ϕ(¯xt−1, w), ∇f(¯xt)⟩ ≤  T  �  t=1  αt⟨x∗, ∇f(¯xt)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  (9)  If such w exists, let w∗ denote the comparator with smallest norm ∥w∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  By inspecting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 9, we see that if ϕ(¯xt−1, ·) can be made to negatively align with the gradient  ∇f(¯xt), the update rule preserves regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Hence, any update rule that is gradient-like in its behaviour  can be made to preserve regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' However, this must not hold on every step, only sufficiently often;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' nor  does it imply that the update rule must explicitly invoke ∇f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' for instance, update rules that are affine in  w preserve regret if the diameter of W is sufficiently large, provided the update rule is not degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Given f, {αt}T  t=1, and {xt}T  t=1, if ϕ preserves regret, then  Rx(T) =  T  �  t=1  αt⟨∇f(¯xt), xt − x∗⟩ ≤  T  �  t=1  αt⟨∇f(¯xt), ϕ(¯xt−1, wt) − ϕ(¯xt−1, w∗)⟩ = Rw(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Proof: Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' With Lemma 1, we can provide a convergence guarantee for meta-gradients in  the convex setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The mechanics of the proof is to use online-to-batch conversion to upper bound  f(¯xT )−f(x∗) ≤ Rx(T)/α1:T and then appeal to Lemma 1 to obtain f(¯xT )−f(x∗) ≤ Rw(T)/α1:T ,  from which point we can plug in the FTRL regret bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Let ϕ preserve regret and assume Algorithm 2 satisfies the assumptions in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Then  f(¯xT ) − f(x∗) ≤  1  α1:T  �  ∥w∗∥2  β  +  T  �  t=1  λβα2  t  2  ∥∇f(¯xt)∥2  ∗  − αt  2L∥∇f(¯xt) − ∇f(x∗)∥2  ∗ − α1:t−1  2L  ∥∇f(¯xt−1) − ∇f(¯xt)∥2  ∗  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Moreover, if x∗ is a global minimiser of f, setting αt = 1 and β =  1  λL yields  f(¯xT ) − f(x∗) ≤ λL diam(W)  T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Proof: Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  6  Meta-Learning meets Optimism  The reason Theorem 3 fails to achieve acceleration is because the negative terms, −∥∇f(¯xt−1) −  ∇f(¯xt)∥2  ∗, do not come into play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' This is because the positive term in the bound involves the norm of  the gradient, rather than the norm of the difference of two gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The former is typically a larger  quantity and hence we cannot guarantee that they vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' To obtain acceleration, we need some form  of optimism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In this section, we consider an alteration to Algorithm 2 that uses AO-FTRL for the  meta-updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Given some sequence of hints {˜gt}T  t=1, each ˜gt ∈ Rm, each wt+1 is given by  wt+1 = arg min  w∈W  �  αt+1˜gt+1 +  t  �  s=1  αs⟨∇hs(ws), w⟩ +  1  2βt  ∥w∥2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  (10)  7  Algorithm 3: BMG in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  input :Weights {βt}T  t=1  input :Update rule ϕ  input :Target oracle  input :Initialisation (x0, w1)  for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' , T:  xt = xt−1 + ϕ(xt−1, wt)  Query zt from target oracle  dt(·) = ∥zt − xt + ϕ(xt, ·)∥2  wt+1 = wt − βt∇dt(wt)  return xT  Algorithm 4: Convex optimistic meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  input :Weights {αt}T  t=1, {βt}T  t=1  input :Update rule ϕ  input :Hints {˜gt}T  t=1  input :Initialisation (¯x0, w1)  for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' , T:  xt = ϕ(¯xt−1, wt)  ¯xt = (1 − αt/α1:t)¯xt−1 + (αt/α1:t)xt  gt = Dϕ(¯xt−1, wt)T ∇f(¯xt)  vt = αt+1˜gt+1 + �t  s=1 αsgs  wt+1 = arg minw∈W⟨vt, w⟩ +  1  2βt ∥w∥2  return ¯xT  Otherwise, we proceed as in Algorithm 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' for a complete description, see Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The AO-FTRL  updates do not correspond to a standard meta-update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' However, we show momentarily that optimism  can be instantiated via the BMG method, detailed in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The proof for optimistic meta-  gradients proceed largely as in Theorem 3, it only differs in that we apply the AO-FTRL regret bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Let ϕ preserve regret and assume Algorithm 4 satisfy the assumptions in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Then  f(¯xT ) − f(x∗) ≤  1  α1:T  �  ∥w∗∥2  βT  +  T  �  t=1  α2  t βt  2  ∥Dϕ(¯xt−1, wt)T ∇f(¯xt) − ˜gt∥2  ∗  − αt  2L∥∇f(¯xt) − ∇f(x∗)∥2  ∗ − α1:t−1  2L  ∥∇f(¯xt−1) − ∇f(¯xt)∥2  ∗  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Moreover, assume each ˜gt is such that ∥Dϕ(¯xt−1, wt)T ∇f(¯xt) − ˜gt∥2  ∗ ≤ q∥∇f(¯xt−1) − ∇f(¯xt)∥2  ∗  for some q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' If each αt = t and βt = t−1  2tqL, then  f(¯xt) − f(x∗) ≤ 4qL diam(W)  T 2 − 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The proof follows the same lines as that of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The only difference is that the regret  of the {wt}T  t=1 sequence can be upper bounded by ∥w∗∥2  βT  + 1  2  �T  t=1 α2  t βt∥∇ht(wt) − ˜gt∥2  ∗ instead  of ∥w∗∥2  βT  + 1  2  �T  t=1 α2  t βt∥∇ht(wt)∥2  ∗, as per the AO-FTRL regret bound in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The final part  follows immediately by replacing the norms and plugging in the values for α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  ■  From Theorem 4, it is clear that if ˜gt is a good predictor of Dϕ(¯xt−1, wt)T ∇f(¯xt), then the positive  term in the summation can be cancelled by the negative term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In a classical optimisation setting,  Dϕ = In, and hence it is easy to see that simply choosing ˜gt to be the previous gradient is sufficient  to achieve the cancellation [Joulani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Indeed, this choice gives us Nesterov’s Accelerated  rate [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The upshot of this is that we can specialise Algorithm 4 to capture Nesterov’s  Accelerated method by choosing ϕ : (x, w) �→ w—as in the reduction to Heavy Ball—and setting the  hints to ˜gt = ∇f(¯xt−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Hence, while the standard meta-update without optimism contains Heavy  Ball as a special case, the optimistic meta-update contains Nesterov Acceleration as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  In the meta-learning setting, Dϕ is not an identity matrix, and hence the best targets for meta-learning  are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Naively, choosing ˜gt = Dϕ(¯xt−1, wt)T ∇f(¯xt−1) would lead to a similar cancellation,  but this is not allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' At iteration t, we have not computed wt when ˜gt is chosen, and hence  Dϕ(¯xt−1, wt) is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The nearest term that is accessible is Dϕ(¯xt−2, wt−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Let each ˜gt+1 = Dϕ(¯xt−1, wt)T ∇f(¯xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Assume that ϕ satisfies  ��Dϕ(x′, w)T ∇f(x) − Dϕ(x′′, w′)T ∇f(x′)  ��2  ∗ ≤ ˜λ ∥∇f(x′) − ∇f(x)∥2  ∗  for all x′′, x′, x ∈ X and w, w′ ∈ W, for some ˜λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' If each αt = t and βt =  t−1  2t˜λL, then  f(¯xT ) − f(x∗) ≤ 4˜λL diam(W)  T 2−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Proof: Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  8  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='1  Bootstrapped Meta-Gradients  In this section, we present a simplified version of BMG for clarity, with Appendix E providing  a fuller comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Essentially, BMG alters the meta-update in Algorithm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' instead of directly  minimising the loss f, it introduces a sequence of targets z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' and the meta-learner’s goal  is select w so that the updated parameters minimise the distance these targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Concretely, given  an update xt = xt−1 + ϕ(xt−1, wt), targets are bootstrapped from xt, meaning that a vector yt  is computed to produce the target zt = xt − yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Assuming the distance to the target is measured  under 1  2∥ · ∥2  2, the BMG meta-update takes the form  wt+1 = wt − Dϕ(xt−1, wt)T yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Depending on how yt is computed, it can encode optimism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' For instance, the authors rely on  the update rule itself to compute a tangent yt = ϕ(xt, wt) − ∇f(xt + ϕ(xt, wt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' This encodes  optimism via ϕ because it encourages the meta-learner to build up momentum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' to accumulate  past updates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We can contrast this with the types of updates produced by AO-FTRL in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' If  we have hints ˜gt+1 = Dϕ(¯xt−1, wt)T ˜yt+1 for some ˜yt+1 ∈ Rn and set ∥ · ∥ = ∥ · ∥2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' assuming  an interior solution, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 10 yields  wt+1 = wt − Dϕ(¯xt−1, wt)T (αt+1˜yt+1 + αt∇f(¯xt))  �  ��  �  BMG update  + αtDϕ(¯xt−2, wt−1)T ˜yt  �  ��  �  FTRL error correction  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  (11)  Hence, BMG encodes very similar dynamics to those of AO-FTRL in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Under this choice of  hints, the main qualitative difference is that AO-FTRL includes a correction term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The effect of this  term is to “undo” previous hints to avoid feedback loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Notably, BMG can suffer from divergence  due to feedback if the gradient in yt is not carefully scaled [Flennerhag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Our theoretical  analysis suggests a simple correction method that may stabilize BMG in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  More generally, targets in BMG are isomorphic to the hint function in AO-FTRL if the measure  of distance in BMG is a Bregman divergence under a strongly convex function (Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' An  immediate implication of this is that the hints in Corollary 1 can be expressed as targets in BMG, and  hence if BMG satisfies the assumptions involved, it converges at a rate O(˜λ/T 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' More generally,  Theorem 4 provides a sufficient condition for any target bootstrap in BMG to achieve acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Let each ˜gt+1 = Dϕ(¯xt−1, wt)T ˜yt+1, for some ˜yt+1 ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' If each ˜yt+1 is a better  predictor of the next gradient than ∇f(¯xt−1), in the sense that  ∥Dϕ(¯xt−2, wt−1)T ˜yt − Dϕ(¯xt−1, wt)T ∇f(¯xt)∥∗ ≤ ˜λ∥∇f(¯xt) − ∇f(¯xt−1)∥∗,  then Algorithm 4 guarantees convergence at a rate O(˜λ/T 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  7  Conclusion  This paper explores a connection between convex optimisation and meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We construct an  algorithm for convex optimisation that aligns as closely as possible with how meta-learning is done  in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Meta-learning introduces a transformation and we study the effect this transformation has  on the rate of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We find that, while a meta-learned update rule cannot generate a better  dependence on the horizon T, it can improve upon classical optimisation up to a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  An implication of our analysis is that for meta-learning to achieve acceleration, it is important to  introduce some form of optimism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' From a classical optimisation point of view, such optimism  arises naturally by providing the meta-learner with hints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' If hints are predictive of the learning  dynamics these can lead to significant acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We show that the recently proposed BMG  method provides a natural avenue to incorporate optimism in practical application of meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Because targets in BMG and hints in optimistic online learning commute, our results provide first  rigorous proof of convergence for BMG, while providing a general condition under which optimism  in BMG yields accelerated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  9  References  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
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+page_content=' Cloutier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Learning a Synaptic Learning Rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Université de Montréal,  Département d’informatique et de recherche opérationnelle, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
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+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Kurth-Nelson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Tirumala, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Soyer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
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+page_content=' Leibo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Munos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Blundell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Kumaran,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Botvinick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Learning to Reinforcement Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In Annual Meeting of the Cognitive Science  Society, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Xu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Zhao, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Peng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Learning to Explore with Meta-Policy Gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In International  Conference on Machine Learning, 2018a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' van Hasselt, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Silver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Meta-Gradient Reinforcement Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In Advances in  Neural Information Processing Systems, 2018b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  11  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' van Hasselt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Hessel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Oh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Singh, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Silver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Meta-gradient reinforcement  learning with an objective discovered online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Advances in Neural Information Processing Systems,  33:15254–15264, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Zahavy, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Xu, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Veeriah, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Hessel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Oh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' van Hasselt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Silver, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' A  Self-Tuning Actor-Critic Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In Advances in Neural Information Processing Systems,  volume 33, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Zinkevich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Online Convex Programming and Generalized Infinitesimal Gradient Ascent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In  International Conference on Machine Learning, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  12  Appendix  A  Notation  Table 1: Notation  Indices  t  Iteration index: t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  T  Total number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  [T]  The set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' , T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  i  Component index: xi is the ith component of x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' , xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  αa:b  Sum of weights: αa:b = �b  s=a αs  xa:b  Weighted sum: xa:b = �b  s=a αsxs  ¯xa:b  Weighted average: ¯xa:b = xa:b/αa:b  Parameters  x∗ ∈ X  Minimiser of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  xt ∈ X  Parameter at time t  ¯xt ∈ X  Moving average of {xs}t  s=1 under weights {αs}t  s=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  ρt ∈ (0, ∞)  Moving average coefficient αt/α1:t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  wt ∈ W  Meta parameters  w∗ ∈ X  w ∈ W that retains regret with smallest norm ∥w∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  αt ∈ (0, ∞)  Weight coefficients  βt ∈ (0, ∞)  Meta-learning rate  Maps  f : X → R  Objective function  ∥ · ∥ : X → R  Norm on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  ∥ · ∥∗ : X ∗ → R  Dual norm of ∥ · ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  ht : W → R  Online loss faced by the meta learner  Rx(T)  Regret of {xt}T  t=1 against x∗: Rx(T) := �T  t=1 αt⟨∇f(¯xt), xt −x∗⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Rw(T)  Rw(T) := �T  t=1 αt⟨∇f(¯xt), ϕ(¯xt−1, wt) − ϕ(¯xt−1, w∗)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  ϕ : Rn × Rm → Rn  Generic update rule used in practice  Dϕ(x, ·) : Rm →Rn×m  Jacobian of ϕ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' its second argument, evaluated at x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  ϕ : X × W → X  Update rule in convex setting  Dϕ(x, ·) : W → Rn×m  Jacobian of ϕ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' its second argument, evaluated at x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Bµ : Rn×Rn →[0, ∞)  Bregman divergence under µ : Rn → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  µ : Rn → R  Convex distance generating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  13  Table 2: Hyper-parameter sweep on Convex Quadratics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' All algorithms are tuned for learning rate  and initialisation of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Baselines are tuned for decay rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' meta-learned variant are tuned for the  meta-learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Learning rate  [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='7, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='9, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=']  w init scale  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=']  Decay rate / Meta-learning rate  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='003, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='01, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='03, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=']  B  Convex Quadratic Experiments  Loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  We consider the problem of minimising a convex quadratic loss functions f :  R2 → R of the form f(x) = xT Qx, where Q is randomly sampled as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We sample a  random orthogonal matrix U from the Haar distribution scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='stats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='ortho_group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We con-  struct a diagonal matrix of eigenvalues, ranked smallest to largest, with λi = i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Hence, the first  dimension has an eigenvalue 1 and the second dimension has eigenvalue 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The matrix Q is given  by U T diag(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' , λn)U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Given that the solution is always (0, 0), this experiment revolves around understanding  how different algorithms deal with curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Given symmetry in the solution and ill-conditioning,  we fix the initialisation to x0 = (4, 4) for all sampled Qs and all algorithms and train for 100  iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' For each Q and each algorithm, we sweep over the learning rate, decay rate, and the  initialization of w see Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' For each method, we then report the results for the combination  of hyper parameters that performed the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  We report the learning curves for the best hyper-parameter choice for 5 randomly sampled  problems in the top row of Figure 2 (columns correspond to different Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We also study the sensitivity  of each algorithm to the learning rate in the bottom row Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' For each learning rate, we report  the cumulative loss during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' While baselines are relatively insensitive to hyper-parameter  choice, meta-learned improve for certain choices, but are never worse than baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  C  Imagenet Experiments  Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  We train a 50-layer ResNet following the Haiku example, available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  com/deepmind/dm-haiku/blob/main/examples/imagenet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We modify the default setting to  run with SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We compare default SGD to variants that meta-learn an element-wise learning rate  online, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' (x, w) �→ w ⊙ ∇f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' For each variant, we sweep over the learning rate (for SGD) or  meta-learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We report results for the best hyper-parameter over three independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Standard meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  In the standard meta-learning setting, we apply the update rule once  before differentiating w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' the meta-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' That is, the meta-update takes the form wt+1 =  wt − β∇ht(wt), where ht = f(xt + wt ⊙ ∇f(xt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Because the update rule is linear in w, we  can compute the meta-gradient analytically:  ∇ht(wt) = ∇wf(x + ϕ(x, w)) = Dϕ(x, w)T ∇f(x′) = ∇f(x) ⊙ ∇f(x′),  where x′ = x + ϕ(x, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Hence, we can compute the meta-updates in Algorithm 1 manually as  wt+1 = max{wt − β∇f(xt) ⊙ ∇f(xt+1), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' }, where we introduce the max operator on an element-  wise basis to avoid negative learning rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Empirically, this was important to stabilize training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Optimistic meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  For optimistic meta-learning, we proceed much in the same way, but  include a gradient prediction ˜gt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' For our prediction, we use the previous gradient, ∇f(xt+1), as  our prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 11, this yields meta-updates of the form  wt+1 = max  �  wt − β∇f(xt+1) ⊙ (∇f(xt+1) + ∇f(xt)) − ∇f(xt) ⊙ ∇f(xt), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  We report Top-1 accuracy on the held-out test set as a function of training steps in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Tuning the learning rate does not yield any statistically significant improvements under standard  meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' However, with optimistic meta-learning, we obtain a significant acceleration as well  as improved final performance, increasing the mean final top-1 accuracy from 72% to 75%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  14  Table 3: Hyper-parameter sweep on Imagenet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  (Meta-)learning rate  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='1]  D  Proofs  This section provides complete proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We restate the results for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Given f, {αt}T  t=1, and {xt}T  t=1, if ϕ preserves regret, then  Rx(T) =  T  �  t=1  αt⟨∇f(¯xt), xt − x∗⟩ ≤  T  �  t=1  αt⟨∇f(¯xt), ϕ(¯xt−1, wt) − ϕ(¯xt−1, w∗)⟩ = Rw(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Starting from Rx in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 8, if the update rule preserves regret, there exists w∗ ∈ W for which  Rx(T) =  T  �  t=1  αt⟨∇f(¯xT ), ϕ(¯xt−1, wt) − x∗⟩  =  T  �  t=1  αt⟨∇f(¯xT ), ϕ(¯xt−1, wt) − ϕ(¯xt−1, w∗)⟩ +  T  �  t=1  αt⟨∇f(¯xT ), ϕ(¯xt−1, w∗) − x∗⟩  ≤  T  �  t=1  αt⟨∇f(¯xT ), ϕ(¯xt−1, wt) − ϕ(¯xt−1, w∗)⟩ = Rw(T),  since w∗ is such that �T  t=1 αt⟨∇f(¯xT ), ϕ(¯xt−1, w∗) − x∗⟩ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  ■  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Let ϕ preserve regret and assume Algorithm 2 satisfy the assumptions in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Then  f(¯xT ) − f(x∗) ≤  1  α1:T  �  ∥w∗∥2  β  +  T  �  t=1  λβα2  t  2  ∥∇f(¯xt)∥2  ∗  − αt  2L∥∇f(¯xt) − ∇f(x∗)∥2  ∗ − α1:t−1  2L  ∥∇f(¯xt−1) − ∇f(¯xt)∥2  ∗  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  If x∗ is a global minimiser of f, setting αt = 1 and β =  1  λL yields f(¯xT ) − f(x∗) ≤ λL diam(W)  T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Since ϕ preserves regret, by Lemma 1, the regret term Rx(T) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 7 is upper bounded by  Rw(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We therefore have  f(¯xT ) − f(x∗) ≤  1  α1:T  �  Rw(T) − αt  2L∥∇f(¯xt) − ∇f(x∗)∥2  ∗ − α1:t−1  2L  ∥∇f(¯xt−1) − ∇f(¯xt)∥2  ∗  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  (12)  Next, we need to upper-bound Rw(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Since, Rw(T) = �T  t=1 αt⟨∇f(¯xT ), ϕ(¯xt−1, wt) −  ϕ(¯xt−1, w∗)⟩, the regret of {wt}T  t=1 is defined under loss functions ht : W → R given by  ht = αt⟨∇f(¯xT ), ϕ(¯xt−1, w))⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  By assumption of convexity in ϕ, each ht is convex in w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Hence, the regret under {αtht}T  t=1 can be upper bounded by the regret under the linear losses  {αt⟨∇ht(wt), ·⟩}T  t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' These linear losses correspond to the losses used in the meta-update in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Since the meta-update is an instance of FTRL, we may upper-bound Rw(T) by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 6 with each  15  ˜gt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Putting this together along with smoothness of ϕ,  Rx(T) ≤ Rw(T)  =  T  �  t=1  αt⟨∇f(¯xT ), ϕ(¯xt−1, wt) − ϕ(¯xt−1, w∗)⟩  ≤  T  �  t=1  αt⟨∇ht(wt), wt − w∗⟩  ≤ ∥w∗∥2  β  + β  2  T  �  t=1  α2  t ∥∇ht(wt)∥2  ∗  = ∥w∗∥2  β  + β  2  T  �  t=1  α2  t ∥Dϕ(¯xt−1, wt)T ∇f(¯xt)∥2  ∗  ≤ ∥w∗∥2  β  + λβ  2  T  �  t=1  α2  t ∥∇f(¯xt)∥2  ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  (13)  Putting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 12 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 13 together gives the stated bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Next, if x∗ is the global optimiser,  ∇f(x∗) = 0 by first-order condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Setting β = 1/(Lλ) and αt = 1 means the first two norm  terms in the summation cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The final norm term in the summation is negative and can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  We are left with f(¯xT ) − f(x∗) ≤ λL∥w∗∥2  T  ≤ λL diam(W)  T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  ■  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Let each ˜gt+1 = Dϕ(¯xt−1, wt)T ∇f(¯xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Assume that ϕ satisfies  ��Dϕ(x′, w)T ∇f(x) − Dϕ(x′′, w′)T ∇f(x′)  ��2  ∗ ≤ ˜λ ∥∇f(x′) − ∇f(x)∥2  ∗  for all x′′, x′, x ∈ X and w, w′ ∈ W, for some ˜λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' If each αt = t and βt =  t−1  2t˜λL, then  f(¯xT ) − f(x∗) ≤  4˜λL diam(W)  T 2−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Plugging in the choice of ˜gt and using that  ��Dϕ(¯xt−1, wt)T ∇f(¯xt) − Dϕ(xt−2, wt−1)T ∇f(¯xt−1)  ��2  ∗ ≤ ˜λ ∥∇f(¯xt−1) − ∇f(¯xt)∥2  ∗ ,  the bound in Theorem 4 becomes  f(¯xT ) − f(x∗) ≤  1  α1:T  �  ∥w∗∥2  βT  + 1  2  T  �  t=1  �  ˜λα2  t βt − α1:t−1  L  �  ∥∇f(¯xt) − ∇f(¯xt−1)∥2  ∗  �  ,  where we drop the negative terms ∥∇f(¯xt) − ∇f(x∗)∥2  ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Setting αt = t yields α1:t−1 = (t−1)t  2  ,  while setting βt =  t−1  2t˜λL means ˜λα2  t βt = (t−1)t  2L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Hence, ˜λα2  t βt − α1:t−1/L cancels and we get  f(¯xT ) − f(x∗) ≤ ∥w∗∥2  βT α1:T  =  4∥w∗∥2˜λL  (T − 1)(T + 1) ≤ 4˜λL diam(W)  (T − 1)(T + 1) = 4˜λL diam(W)  T 2 − 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  ■  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Let each ˜gt+1 = Dϕ(¯xt−1, wt)T ˜yt+1, for some ˜yt+1 ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' If each ˜yt+1 is a better  predictor of the next gradient than ∇f(¯xt−1), in the sense that  ∥Dϕ(¯xt−2, wt−1)T ˜yt − Dϕ(¯xt−1, wt)T ∇f(¯xt)∥∗ ≤ ˜λ∥∇f(¯xt) − ∇f(¯xt−1)∥∗,  then Algorithm 4 guarantees convergence at a rate O(˜λ/T 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The proof follows the same argument as Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  ■  16  Algorithm 5: BMG in practice (general version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  input :Weights {ρt}T  t=1, {βt}T  t=1  input :Update rule ϕ  input :Matching function Bµ  input :Target oracle  input :Initialisation (x0, w1)  for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' , T:  xt = xt−1 + ϕ(xt−1, wt)  Query zt from target oracle  dt : w �→ Bµ  zt(xt−1 + ϕ(xt−1, w))  wt+1 = wt − βt∇dt(wt)  return xT  E  BMG  Errata: this was incorrectly referred to as Appendix F in our original submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  In this section, we provide a more comprehensive reduction of BMG to AO-FTRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' First, we provide  a more general definition of BMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Let µ : X → R be a convex distance generating function and  define the Bregman Divergence Bµ : Rn × Rn → R by  Bµ  z (x) = µ(x) − µ(z) − ⟨∇µ(z), x − z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Given initial condition (x0, w1), the BMG updates proceed according to  xt = xt−1 + ϕ(xt−1, wt)  wt+1 = wt − βt∇dt(wt),  (14)  where dt : Rn → R is defined by dt(w) = Bµ  zt(xt−1 + ϕ(xt−1, wt)), where each zt ∈ Rn is  referred to as a target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' See Algorithm 5 for an algorithmic summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' A bootstrapped target uses  the meta-learner’s most recent update, xt, to compute the target, zt = xt + yt for some tangent  vector yt ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' This tangent vector represents a form of optimism, and provides a signal to the  meta-learner as to what would have been a more efficient update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In particular, the author’s consider  using the meta-learned update rule to construct yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' yt = ϕ(xt, wt) − ∇f(xtϕ(xt, w − t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Note  that xt = xt−1 + ϕ(xt−1, wt), and hence this tangent vector is obtained by applying the update rule  again, but now to xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' For this tangent to represent an improvement, it must be assumed that wt is  a good parameterisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Hence, bootstrapping represents a form of optimism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' To see how BMG  relates to Algorithm 4, and in particular, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 10, expand Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 14 to get  wt+1 = wt − βtDϕ(xt−1, wt)T (∇µ(xt) − ∇µ(zt)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  (15)  In contrast, AO-FTRL reduces to a slightly different type of update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Consider Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Given online losses ht  :  W  →  R defined by  {⟨Dϕ(¯xt−1, wt)T ∇f(¯xt), ·⟩}T  t=1 and hint functions {⟨˜gt, ·, }⟩T  t=1, with each ˜gt ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' If ∥ · ∥ =  (1/2)∥ · ∥2, an interior solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 10 is given by  wt+1 =  βt  βt−1  wt − βt  �  αt+1˜gt+1 + αt(Dϕ(¯xt−1, wt)T ∇f(¯xt) − ˜gt)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  17  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' By direct computation:  wt+1 = arg min  w∈W  �  αt+1⟨˜gt+1, w⟩ +  t  �  s=1  αs⟨Dϕ(¯xs−1, ws)T ∇f(¯xs), w⟩ +  1  2βt  ∥w∥2  2  �  = −βt  �  αt+1˜gt+1 +  t  �  s=1  αtDϕ(¯xs−1, ws)T ∇f(¯xs))  �  = −βt  �  αt+1˜gt+1 + αtDϕ(¯xt−1, wt)T ∇f(¯xt) +  �t−1  �  s=1  αtDϕ(¯xs−1, ws)T ∇f(¯xs))  ��  = −βt  �  αt+1˜gt+1 + αt(Dϕ(¯xt−1, wt)T ∇f(¯xt) − ˜gt)  �  − βt  �  αt˜gt +  t−1  �  s=1  αtDϕ(¯xs−1, ws)T ∇f(¯xs))  �  =  βt  βt−1  wt − βt  �  αt+1˜gt+1 + αt(Dϕ(¯xt−1, wt)T ∇f(¯xt) − ˜gt)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  ■  AO-FTRL includes a decay rate βt/βt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' this decay rate can be removed by instead using optimistic  online mirror descent [Rakhlin and Sridharan, 2013, Joulani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=', 2017]—to simplify the exposition  we consider only FTRL-based algorithms in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' An immediate implication of Lemma 2  is the error-corrected version of BMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Setting ˜gt+1 = Dϕ(¯xt−1, wt)T ˜gt+1 for some ˜yt+1 ∈ Rn yields an error-corrected  version of the BMG meta-update in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Specifically, the meta-updates in Lemma 2 becomes  wt+1 =  βt  βt−1  wt − βtDϕ(¯xt−1, wt)T (αt+1˜yt+1 + αt∇f(¯xt))  �  ��  �  BML update  + βtαtDϕ(¯xt−2, wt−1)T ˜yt  �  ��  �  FTRL error correction  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Follows immediately by substituting for each ˜gt+1 in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  ■  To illustrate this connection, Let µ = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' In this case, the BMG update reads wt+1 = wt −  βtDϕ(xt−1, wt)T (∇f(zt) − ∇f(xt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The equivalent update in the convex optimisation setting (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Algorithm 4) is obtained by setting ˜yt+1 = ∇f(zt), in which case Corollary 3 yields  wt+1 = βt+1  βt  wt − βtDϕ(¯xt−1, wt)T (αt+1∇f(zt) − αt∇f(¯xt)) + ξt,  where ξt = βtαtDϕ(¯xt−2, wt−1)T ∇f(¯xt − 1) denotes the error correction term we pick up through  AO-FTRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Since Algorithm 5 does not average its iterates—while Algorithm 4 does—we see that  these updates (ignoring ξt) are identical up to scalar coefficients (that can be controlled for by scaling  each βt and each ˜gt+1 accordingly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  More generally, the mapping from targets in BMG and hints in AO-FTRL takes on a more complicated  pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Our next results show that we can always map one into the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' To show this, we need  to assume a certain recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' It is important to notice however that at each iteration introduces  an unconstrained variable and hence the assumption on the recursion is without loss of generality  (as the free variable can override it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Targets in Algorithm 5 and hints in algorithm 4 commute in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' BMG  → AO-FTRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Let BMG targets {zt}T  t=1 by given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' A sequence of hints {˜g}T  t=1 can be constructed  recursively by  αt+1˜gt+1 = Dϕ(¯xt−1, wt)T (∇µ(¯xt) − ∇µ(zt) − αt∇f(¯xt)) + αt˜gt,  t ∈ [T],  (16)  so that interior updates for Algorithm 4 are given by  wt+1 =  βt  βt−1  wt − βt (∇µ(zt) − ∇µ(¯xt)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  18  AO-FTRL → BMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Conversely, assume a sequence {˜yt}T  t=1 are given, each ˜yt ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' If µ strictly  convex, a sequence of BMG targets {zt}T  t=1 can be constructed recursively by  zt = ∇µ−1 (∇µ(xt) − (αt+1˜yt+1 + αt∇f(xt)))  t ∈ [T],  so that BMG updates in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 14 are given by  wt+1 = wt − βt  �  αt+1˜gt+1 + αt(Dϕ(¯xt−1, wt)T ∇f(¯xt) − ˜gt)  �  ,  where each ˜gt+1 is the BMG-induced hint function, given by  αt+1˜gt+1 = αt+1Dϕ(xt−1, wt)T ˜yt+1 + αt˜gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' First, consider BMG → AO-FTRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' First note that ˜g1 is never used and can thus be chosen  arbitrarily—here, we set ˜g1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' For w2, Lemma 2 therefore gives the interior update  w2 = β2  β1  w1 − β1(α2˜g2 + α1Dϕ(¯x0, w1)T ∇f(¯x1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Since the formulate for ˜g2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 16 only depends on quantities with iteration index t = 0, 1, we may  set α2˜gt = Dϕ(¯x0, w1)T (∇µ(¯x1) − ∇µ(zt) − αt∇f(¯x1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' This gives the update  w2 = β2  β1  w1 − β1Dϕ(¯x0, w1)T (∇µ(¯x1) − ∇µ(z1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Now assume the recursion holds up to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' As before, we may choose αt+1˜gt+1 according to  the formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 16 since all quantities on the right-hand side depend on quantities computed at  iteration t or t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Subtituting this into Lemma 2, we have  wt+1 =  βt  βt−1  wt − βt  �  αt+1˜gt+1 + αt(Dϕ(¯xt−1, wt)T ∇f(¯xt) − ˜gt)  �  =  βt  βt−1  wt − βt  �  Dϕ(¯xt−1, wt)T (∇µ(¯xt) − ∇µ(zt) − αt∇f(¯xt)) + αt˜gt  +αt(Dϕ(¯xt−1, wt)T ∇f(¯xt) − ˜gt)  �  =  βt  βt−1  wt − βtDϕ(¯xt−1, wt)T (∇µ(¯xt) − ∇µ(zt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  AO-FTRL → BMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' The proof in the other direction follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' First, note that for µ strictly  convex, ∇µ is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Then, z1 = ∇µ−1(∇µ(x1) − (α2˜y2 + α1∇f(x1))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' This target is  permissible since x1 is already computed and {˜yt}T  t=1 is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' Substituting this into the BMG  meta-update in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' 14, we find  w2 = w1 − β1Dϕ(x0, w1)T (∇µ(x1) − ∇µ(∇µ−1(∇µ(x1) − (α2˜y2 + α1∇f(x1)))))  = w1 − β1Dϕ(x0, w1)T (α2˜y2 + α1∇f(x1))  = w1 − β1  �  α2˜g2 + α1(Dϕ(¯x0, w1)T ∇f(¯x1) − ˜g1)  �  ,  where the last line uses that ˜g2 is defined by α2˜g2 − α1˜g1 = Dϕ(¯x0, w1)T ˜y2 and ˜g1 is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  Again, assume the recursion holds to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content=' We then have  wt+1 = wt − βtDϕ(xt−1, wt)T (∇µ(xt) − ∇µ(zt))  = wt − βtDϕ(xt−1, wt)T (∇µ(xt)  − ∇µ(∇µ−1(∇µ(xt) − (αt+1˜yt+1 + αt∇f(xt)))))  = wt − βtDϕ(xt−1, wt)T (αt+1˜yt+1 + αt∇f(xt))  = wt − βt(αt+1˜gt+1 + αt(Dϕ(xt−1, wt)T ∇f(xt) − ˜gt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
+page_content='  ■  19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE1T4oBgHgl3EQfhASI/content/2301.03236v1.pdf'}
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+arXiv:2301.05441v1  [math.DS]  13 Jan 2023
+On M-dynamics and Li-Yorke chaos of extensions of minimal
+dynamics
+Xiongping Dai
+Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
+Abstract
+Let π: (T, X) → (T, Y) be an extension of minimal compact metric flows with discrete phase
+group T such that Rπ � ∆X. A subflow of Rπ is called an M-flow if it is T.T. and it contains a
+dense set of a.p. points. In this paper we mainly prove the following:
+(1) π is PI iff every M-flow containing ∆X in Rπ is just equal to ∆X.
+(2) If π is not PI, then there exists a canonical Li-Yorke chaotic M-flow in Rπ. In particular, an
+Ellis weak-mixing non-proximal extension is non-PI and so Li-Yorke chaotic.
+(3) If (T, X) is a unbounded M-flow or a locally compact noncompact M-flow, then it is sensitive
+on initial conditions.
+In addition, we show that every syndetically distal flow is pointwise Bohr a.p.
+Keywords: Minimal flow · PI-extension · M-flow · Li-Yorke chaos · Syndetic distability
+2010 MSC: 37B05
+1. Introduction
+We begin by reviewing briefly the basic notions needed. Let T be a Hausdorff topological
+group with identity e and X a Hausdorff space. Unless specified otherwise, by (T, X) we mean a
+flow [28, 23, 44, 8, 2, 15] with phase group T and with phase space X; that is to say, there is a
+continuous phase mapping T × X → X, denoted (t, x) �→ tx, such that ex = x and (st)x = s(tx)
+for all x ∈ X and s, t ∈ T. If T is only a topological monoid, then (T, X) will be called a semiflow.
+If X is compact (resp. metrizable, . . . ), then (T, X) will be called a compact (resp. metric, . . . )
+dynamic where T may be a group or monoid.
+The dynamics (T, X), (T, Y), (T, Z), . . . will be sometimes written as X , Y , Z , . . . , respec-
+tively. Write X × X for (T, X × X) defined by (t, (x1, x2)) �→ (tx1, tx2).
+Let X be a dynamic. A point x ∈ X is said to be almost periodic (a.p.) under X if T x is a
+minimal subset of X, i.e., Ty = T x ∀y ∈ T x (see Def. I. in §2.2.1 for the precise definition). X
+is called topologically transitive (T.T.) if for all non-empty open sets U, V in X,
+(1.1) NT(V, U) := {t ∈ T | U ∩ tV � ∅} is nonempty.
+Email address: xpdai@nju.edu.cn (Xiongping Dai)
+Preprint submitted to JDE (28 Jan. 2022)
+January 16, 2023
+
+We say X is weakly mixing if X × X is a T.T. dynamic. Following Glasner-Weiss [26]:
+(1.2) X is referred to as an M-dynamic if X is T.T. and X has a dense set of a.p. points.
+See Theorems 2.1 and 2.2 for a sufficient condition of M-dynamics. Moreover, we will prove
+that every syndetically distal flow is pointwise Bohr a.p. (see Def. III. and v. in §2.2.1 and Theo-
+rem 2.10).
+Let X and Y be two compact dynamics. A continuous map π: X → Y is called an ex-
+tension of dynamics, denoted by π: X → Y , if πX = Y and πtx = tπx for all t ∈ T and
+x ∈ X. Extensions are important elements in the structure theory of minimal topological dynam-
+ics (cf. [23, 44, 8, 2, 15, 9]). For π: X → Y , we write
+(1.3) Rπ = {(x, x′) | x, x′ ∈ X s.t. πx = πx′},
+which is an invariant closed equivalence relation on X. So Rπ = (T, Rπ) is a dynamic. Usually
+one is only interested to the dynamics of Rπ that is driven by Y , for example, in skew-product
+flows and random dynamical systems.
+Let π: X → Y be an extension of minimal compact dynamics. As usual we say that:
+(1.4) π is proximal if Tz ∩ ∆X � ∅ ∀z ∈ Rπ.
+(1.5) π is almost periodic (a.p.) if there is no (x, x′) ∈ Rπ \ ∆X such that T(U × V ∩ Rπ) ∩ ∆X � ∅
+for all U ∈ Nx and V ∈ Nx′. Here Nx stands for the neighborhood system of x in X.
+(1.6) π is a PI-extension [18, 23, 44, 8, 2, 22], provided that there exists a minimal proximal
+extension ρ: X ′ → X such that π′ = π ◦ ρ: X ′ → Y can be built by successive
+proximal and a.p. extensions; that is, there exists a “PI-tower”:
+X ′
+Y
+π′
+∨
+< ψ1
+0
+Y1 < ψ2
+1
+π′
+1
+>
+Y2 <
+...
+π′
+2
+>
+· · · <
+Yγ <
+ψγ+1
+γ
+π′
+γ
+>
+Yγ+1 <
+...
+π′
+γ+1
+>
+· · · <
+Yϑ
+π′
+ϑ
+>
+such that:
+a. π′
+ϑ is an isomorphism,
+b. ψγ+1
+γ
+is proximal or a.p. for all ordinal γ < ϑ, and
+c. if γ with γ ≤ ϑ is a limit ordinal then Yγ = lim
+←−−λ<γYλ.
+In the special case that ρ is an isomorphism, we say that π is strictly PI.
+In particular, if X is a nontrivial minimal weakly mixing compact dynamics with T nilpotent
+([9, Def. 2.3.8]), then X → {pt} is not PI. Here {pt} stands for the one-point dynamic.
+A non-minimal compact M-flow is sensitive on initial conditions (cf. [26, Thm. 1.3] and [14,
+Prop. 2.5]). This implies that every non-minimal T.T. compact flow with a dense set of periodic
+points is Devaney chaotic (cf., e.g., [6, 11, 38]). See [46, 45] on sensitivity and its variations of
+extensions of minimal metric flows.
+In this paper, we shall study sensitivity of noncompact M-flows; moreover, we will define and
+study the Li-Yorke chaos of extensions of minimal compact metric dynamics from the viewpoint
+of M-dynamics (see §§4 and 5); and prove that every non-PI extension of minimal compact
+dynamics can canonically induce M-dynamics (cf. Theorem 3.5). We shall improve Bronstein’s
+intrinsic characterization of PI-extensions; see Theorem 3.10. Moreover we shall consider three
+examples in §6.
+We now conclude our Introduction with a remark on an important open problem of Robert
+Ellis (1970s), which is relative closely to the definition of M-dynamics.
+2
+
+1.7 Remark. Let X be a T.T. pointwise a.p. compact dynamic, where the phase space X is
+non-metrizable.
+Question (R. Ellis; cf. [15, p. 263]). Is X a minimal dynamic?
+Using metric approaches there are some confirmative answers ((1.7a), (1.7b), and (1.7c)) to Ques-
+tion above.
+(1.7a) If T is σ-compact, then X is minimal.
+Proof. See [9, Thm. B.2.1] for T a semigroup; cf. [15, Prop. 4.23] for T a countable
+group.
+(1.7b) If X is distal (see Def. 2.0f), then X is minimal.
+Proof. See [4, Cor. 1.31] for T a semigroup; cf. [16, Prop. 1.9] and [15, Prop. 4.24] for T
+a group.
+(1.7c) If θ: X → Z is a distal extension with Z a minimal compact dynamic (see Def. 2.0f),
+then X is minimal.
+Proof. See [9, Thm. B.2.4].
+(1.7d) If X is weakly a.p. (Gottschalk; i.e., x �→ T x is continuous), then X is minimal.
+Proof. Obvious.
+(1.7e) If X is locally a.p. (see Def. II. in §2.2.1), then X is minimal.
+Proof. First we claim that P(X ) = RP(X ) (see (2.0a) and (2.0b)). In fact, if X is a flow,
+this is [17, Lem. 13(4)]. Now assume X is a semiflow. Let Z ⊂ X and x ∈ X such that
+x is distal from Z (i.e., there is an index α ∈ UX with (tx, tz) � α ∀t ∈ T, z ∈ Z). Take
+β ∈ UX with β3 ⊆ α. Pick U ∈ Nx and a syndetic set A in T such that AU ⊆ β[x]. Clearly,
+we have for all a ∈ A, y ∈ U and z ∈ Z that (ay, az) � β. Next, select a compact set
+K in T such that Kt ∩ A � ∅ ∀t ∈ T. Take γ ∈ UX so small that Kγ ⊆ β. Now for all
+y ∈ U, z ∈ Z and t ∈ T, we have that (ty, tz) � γ; for otherwise, we can pick some k ∈ K
+with a = kt ∈ A so that (ay, az) ∈ β. This implies that if (x, x′) � P(X ), then there exist
+U ∈ Nx and V ∈ Nx′ such that U is distal from V under X , and further, (x, x′) � RP(X ).
+Thus, P(X ) = RP(X ). If M1 � M2 are two minimal sets in X, then by T.T. it follows that
+there is a pair (x1, x2) ∈ M1 × M2 such that (x1, x2) ∈ RP(X ). Then (x1, x2) ∈ P(X ) and
+T x1 = T x2, contrary to M1 � M2. Thus, X is minimal.
+(1.7f) If X is a pointwise regularly a.p. flow (see Def. V. in §2.2.1), then X is minimal.
+Proof. By [28, Thm. 5.18], (T, T x) is minimal regularly a.p. for all x ∈ X so that X is
+distal. Then by (1.7b), X is a minimal regularly a.p. flow.
+Standing notation.
+1. In a non-metric space, the convergence “x j → x” is always under the sense of net.
+2. By Nx we will denote the neighborhood system of a point x in the ambit containing it.
+3. If X is a Hausdorff uniform space, UX stands for a compatible symmetric uniformity structure
+of X.
+3
+
+2. AP-Transitive relations and stability
+We shall present a sufficient necessary condition for a subdynamic of Rπ to be an M-dynamic
+in §2.1. Moreover, we shall study in §2.2 the stability of noncompact flows.
+2.1. AP-Transitive relations
+Let π: X → Y be an extension of compact dynamics, where Y is minimal. We first need
+to introduce a relation on X. Let L ⊆ Rπ, L � ∅. We say that
+• L is an AP-transitive relation for π on X, provided that L is a reflexive symmetric relation on
+X such that if (x1, x2), (x2, x3) ∈ L are both a.p. points, then (x1, x3) ∈ L.
+Clearly, an invariant closed AP-transitive relation for π on X need not be an equivalence
+relation. Of course, if X is a π-distal extension of Y , then every AP-transitive relation for π is
+an equivalence relation on X.
+• Let I be a minimal left ideal of the Stone-ˇCech compactification βT and J = {u ∈ I | u2 = u}. If
+L = {z ∈ Rπ | z is a.p.} such that whenever u, v ∈ J with ux = vx for some x ∈ X then ux′ = vx′
+for all x′ ∈ L[x] (= {x′ ∈ X | (x, x′) ∈ L}), then L is an AP-transitive relation for π on X.
+Proof. Let (x1, x2) and (x2, x3) ∈ L be a.p. Then there are u, v ∈ J with u(x1, x2) = (x1, x2) and
+v(x2, x3) = (x2, x3). So ux2 = vx2. Then ux1 = vx1 and (x1, x3) = v(x1, x3) ∈ Rπ is a.p. so that
+(x1, x3) ∈ L.
+Let Pπ and RPπ be the π-relative proximal and regionally proximal relations on X, respec-
+tively; that is,
+(2.0a)
+Pπ = {˜x ∈ Rπ | ∃ t j ∈ T, ˜x′ ∈ ∆X s.t. t j ˜x → ˜x′}.
+and
+(2.0b)
+RPπ = {˜x ∈ Rπ | ∃ ˜x j ∈ Rπ, t j ∈ T, ˜x′ ∈ ∆X s.t. ˜x j → ˜x, t j ˜x j → ˜x′}.
+In the case of Y = {pt}, we will write P(X ) = Pπ and RP(X ) = RPπ. Let L ⊆ Rπ, we then define
+(2.0c)
+Pπ|L = Pπ ∩ L
+and
+(2.0d)
+RPπ|L = {˜x ∈ L | ∃ ˜x j ∈ L, t j ∈ T, ˜x′ ∈ ∆X s.t. ˜x j → ¯x, t j ˜x j → ˜x′}.
+In the case L = Rπ, Pπ|L = Pπ and RPπ|L = RPπ.
+Then:
+2.0e. π is proximal iff Pπ = Rπ;
+2.0f. a point x ∈ X is π-distal iff Pπ[x] ∩ T x = {x}; π is distal iff Pπ = ∆X; in the case P = {pt},
+X is distal iff P(X ) = ∆X.
+2.0g. π is distal-equicontinuous (d.e.; cf. [9]) iff RPπ = ∆X iff π is a.p.
+Here the π-proximal cell of x ∈ X is defined by Pπ[x] = {x′ | x′ ∈ X, (x, x′) ∈ Pπ}. Notice that d.e.
+extension is also called a.p. extension or isometric extension in flows (cf., e.g., [23, 44, 2]).
+4
+
+2.1 Theorem (cf. [33] [44] for L = Rπ). Let π: X → Y be an extension of minimal compact
+flows. Suppose L ⊆ Rπ is a closed invariant AP-transitive relation on X such that
+(a) L has a dense set of a.p. points and
+(b) RPπ|L = L.
+Then (T, L) is T.T.; that is, (T, L) is an M-flow.
+Since L is not necessarily an equivalence relation on X, so T × X/L → X/L, defined by
+(t, L[x]) �→ L[tx], need not be a well-defined phase mapping. In view of this, Theorem 2.1 is not
+a corollary of the classical McMahon-Veech theorem of T.T. In fact, using Ellis’ algebraic theory
+we can obtain the semiflow version of the above theorem as follows:
+2.2 Theorem (cf. [9] for L = Rπ). Let π: X → Y be an extension of minimal compact semi-
+flows. Suppose L ⊆ Rπ is a closed invariant AP-transitive relation on X such that
+(a) L has a dense set of a.p. points and
+(b) RPπ|L = L.
+Then (T, L) is an M-semiflow.
+Before proving Theorem 2.2 we need to introduce some terms for our convenience. In the
+sequel, T is thought of as a discrete monoid; and let I be any fixed minimal left ideal in βT and
+J = {u ∈ I | u2 = u}. By Aut (I ) we denote the set of automorphisms of the universal minimal
+compact dynamic I = (T, I). Put
+Γ[α] = {(m, αm) | m ∈ I}
+∀α ∈ Aut (I ).
+The so-called τ-topology on Aut (I ) is defined as follows: Let αi ∈ Aut (I ) be a net and let
+α ∈ Aut (I ). We say αi →τ α iff for every m ∈ I there exists a net mi → m in I such that
+αimi → αm in I (cf. [9, Appendix A]).
+Under the τ-topology Aut (I ) is a compact T1-space (cf. [23, 2] and [9, Prop. A.2.5]). Let F
+be a τ-closed subgroup of Aut (I ); then its derived subgroup is defined as follows:
+F′ = {α ∈ F | ∃ δi ∈ F s.t. δi →τ α & δi →τ idI}
+(cf. [9, Def. A.4.1]). Here F′ measures clearly the degree to which the τ-topology on F fails to
+be a Hausdorff space.
+Proof of Theorem 2.2. At first we can construct a CD of minimal compact semiflows and ho-
+momorphisms as follows:
+I
+πX > X
+Y
+π
+∨
+πY
+>
+and set
+F = {α ∈ Aut (I ) | πY = πY ◦ α};
+that is, the Ellis group of Y rel. πY. Let
+L = {α ∈ F | (πX × πX)Γ[α] ⊆ L}.
+5
+
+Notice that Γ[α] consists of a.p. points of Rπ for every α ∈ F. Since L is an AP-transitive relation
+on X, (α−1m, m) = (α−1m, α(α−1m)) ∀m ∈ I, and F is a group, hence L is a subgroup of F.
+Moreover, L is τ-closed. Indeed, if α j ∈ L and α j →τ α, then there exists a net m j → m in I
+such that α jm j → αm so (πX × πX)(m, αm) ∈ L and α ∈ L, for L is closed. Thus, L is a τ-closed
+subgroup of F.
+Let (x, x′) ∈ L be an a.p. point. There exists u ∈ J with u(x, x′) = (x, x′). Since (x, x′) ∈ RPπ|L
+and L has a dense set of a.p. points, by a standard argument (cf., e.g., [9, Proof of Thm. 3.1.3])
+we can select (m, m′), (m j, m′
+j), (n, n′) in RπY ⊆ I × I and t j ∈ T with
+(πX × πX)(m, m′) = (x, x′), (πX × πX)(m j, m′
+j) ∈ L, πXn = πXn′,
+and
+(m j, m′
+j) → (m, m′), t j(m j, m′
+j) → (n, n′)
+such that (m, m′) and (m j, m′
+j) are all a.p. under (T, I × I). Then by regularity of I , there exist
+α j, γ ∈ L such that
+m′ = γm, m′
+j = α jm j, (m j, α jm j) → (m, γm), (t jm j, α jt jm j) → (n, n′).
+Now since πXn = πXn′, we can take n′′ ∈ I such that (n′, n′′) is a.p., n is proximal with n′′
+and πXn′′ = πXn′. Thus, there exists some ξ ∈ Aut (I ) with n′′ = ξn′. Then πXξn′ = πXn′ so
+πX ◦ ξ = πX, ξ ∈ L, and (m j, ξα jm j) → (m, ξγm). Furthermore, there is a net sj ∈ T such that
+(sjm j, ξα jsjm j) → (n, n). Thus, ξα j →τ ξγ, ξα j →τ idI, ξα j ∈ L. So ξγ ∈ L′.
+To sum up, we have concluded that if (x, x′) ∈ L is a.p., then there exists an α ∈ L with
+(x, x′) ∈ (πX × πX)Γ[α] such that there is a net δ j ∈ L with δ j →τ α and δ j →τ idI; that is, α ∈ L′.
+Since L′ is a τ-closed subgroup (cf. [9, Lem. A.4.2]), hence if α, γ ∈ L′, then there exists a
+net δ j ∈ L such that δ j →τ α and δ j →τ γ (cf. [9, Lem. A.4.3]).
+Now let ¯x, ¯w be two a.p. points in L, and, let U and V be two neighborhoods of ¯x and ¯w in
+L, respectively. Then we can take α, γ ∈ L′ and m, n ∈ I such that (πX × πX)(m, αm) = ¯x and
+(πX × πX)(n, γn) = ¯w. So there exists a net δ j ∈ L such that δ j →τ α and δ →τ γ. Further, there
+are nets m j → m and n j → n in I such that δ jm j → αm and δn j → γn. Since (πX × πX)(m j, δ jm j)
+and (πX ×πX)(n j, δ jn j) lie in a same minimal subset of L, this implies that NT(U, V) � ∅. As L has
+already a dense set of a.p. points, (T, L) is T.T. and an M-semiflow. The proof is complete.
+As a matter of fact, if (T, L) is an M-semiflow, then conditions (a) and (b) in Theorem 2.2 are
+clearly fulfilled.
+2.2. Lyapunov stability and Bohr/regular almost periodicity
+Let X be an arbitrary flow, not necessarily minimal, with T a topological group not nec-
+essarily discrete and with X a uniform Hausdorff space not necessarily compact. We will now
+consider conditions under which an (regularly) a.p. point is Lyapunov stable.
+2.2.1. Basic definitions
+We first introduce some notions needed.
+• Let S and A be subsets of T. Then S is said to be thick if for all compact subset K of T there
+exists an element t ∈ T such that Kt ⊆ S ; A is called a syndetic subset of T if there exists a
+compact subset K of T such that T = K−1A.
+6
+
+It is easy to check
+• A subset of T is syndetic iff it intersects non-voidly every thick subset of T.
+Recall that:
+I. A point x ∈ X is almost periodic (a.p.) under X iff NT(x, U) is a syndetic subset of T for
+every U ∈ Nx in X.
+II. A point x ∈ X is locally a.p. under X iff for every U ∈ Nx there exists a V ∈ Nx and a
+syndetic subset A of T such that AV ⊆ U. We say X is locally a.p. if it is pointwise locally
+a.p. under X .
+III. X is called a.p. if for every α ∈ UX there exists a syndetic set A in T such that Ax ⊆ α[x]
+for all x ∈ X. A point x ∈ X is Bohr a.p. under X if (T, T x) is an a.p. flow. We say X is
+pointwise Bohr a.p. iff each point of X is a Bohr a.p. point under X .
+IV. X is called equicontinuous if given ǫ ∈ UX and x ∈ X there exists an index δ ∈ UX such
+that t(δ[x]) ⊆ ǫ[tx] for all t ∈ T. X is said to be uniformly equicontinuous if for all ε ∈ UX
+there exists an index δ ∈ UX with Tδ ⊆ ε.
+V. X is regularly a.p. at x ∈ X, or x is a regularly a.p. point under X , iff NT(x, U) contains a
+syndetic normal closed subgroup of T for all U ∈ Nx. We say X is regularly a.p. iff given
+ε ∈ UX there exists a syndetic normal closed subgroup A of T such that Ax ⊆ ε[x] for all
+x ∈ X. X is called point-regularly a.p. iff there exists a regularly a.p. point x such that
+T x = X.
+If X is compact, then x ∈ X is a.p. under X iff T x is minimal under X ; and, X is a.p. iff it
+is uniformly equicontinuous iff it is equicontinuous iff RP(X ) = ∆X (see Lemma 2.3).
+We say that
+i. X is thickly stable iff for every index ε ∈ UX and all point x ∈ X there exists an index
+δ ∈ UX and a thick subset S of T such that s(δ[x]) ⊆ ε[sx] for all s ∈ S . Here δ and S
+depend on the choice of x.
+ii. X is thickly regularly stable iff for every index ε ∈ UX and all point x ∈ X there exists an
+index δ ∈ UX and a thick subsemigroup S of T such that s(δ[x]) ⊆ ε[sx] for all s ∈ S .
+iii. X is uniformly thickly regularly stable iff for every index ε ∈ UX there exists an index
+δ ∈ UX and a thick subsemigroup S of T such that S δ ⊆ ε.
+iv. X is syndetically stable provided that for every index ε ∈ UX and all point x ∈ X there
+exists an index δ ∈ UX and a syndetic subset S of T such that s(δ[x]) ⊆ ε[sx] for all s ∈ S .
+X is uniformly syndetically stable iff for every index ε ∈ UX there exists an index δ ∈ UX
+and a syndetic subset S of T such that S δ ⊆ ε.
+v. X is syndetically distal if for every x ∈ X and ε ∈ UX there exists a syndetic subset S of
+T and an index δ ∈ UX such that if y � ε[x] then ty � δ[tx] for all t ∈ S . X is uniformly
+syndetically distal if for every ε ∈ UX there exists a syndetic subset S of T and an index
+δ ∈ UX such that if y � ε[x] then ty � δ[tx] for all t ∈ S and x ∈ X.
+Let S ⊆ T and X0 ⊆ X be non-empty sets. Following [19, Def. 2] and [8, 2.8.6 and 2.8.11] we
+say that:
+a. X is Lyapunov S -stable iff for every index ε ∈ UX and all x ∈ X there exists an index δ ∈ UX
+such that t(δ[x]) ⊆ ε[tx] for all t ∈ S .
+b. X is uniformly Lyapunov S -stable iff for every index ε ∈ UX there exists an index δ ∈ UX
+such that S δ ⊆ ε.
+7
+
+c. X is Lyapunov S -stable at L ⊆ X w.r.t. X0 iff for every index ε ∈ UX and all x ∈ L there
+exists an index δ ∈ UX such that t(δ[x] ∩ X0) ⊆ ε[tx] for all t ∈ S .
+Clearly the uniformly Lyapunov T-stable is exactly the uniformly equicontinuous as in IV. If
+X is compact, then a. and b. are equivalent.
+Notice that comparing with “Lyapunov S -stable” and “uniformly Lyapunov S -stable” of
+England [19, Def. 2] and Bronstein [8, 2.8.6 and 2.8.11], S may vary with ε in our thick and
+syndetic stability cases i., ii., iii., iv., and v. above. In addition, if T is not abelian, the inverse of
+a syndetic (resp. thick) set need not be syndetic (resp. thick).
+The following important result is due to W.H. Gottschalk. However, we will give an alterna-
+tive simple proof here for reader’s convenience.
+2.3 Lemma (cf. [27]; see also [8, Thm. 1.6.23] and [2, Thm. 2.2]). A compact flow X is equicon-
+tinuous (or equivalently Lyapunov T-stable) iff it is an a.p. flow.
+Proof. Assume X is an a.p. flow. Clearly, it is uniformly syndetically stable and distal so that
+RP(X ) = P(X ) = ∆X. Thus X is equicontinuous for RP(X ) = �
+α∈UX Tα and X compact.
+Conversely, suppose X is equicontinuous; and then, X is distal for P(X ) = ∆X. Let ε ∈ UX
+and β ∈ UX with β3 ⊂ ε. Then there exists an index δ ∈ UX with δ ⊂ β such that Tδ ⊂ β. Since X
+is compact, there exists a finite set {x1, . . ., xn} in X such that X = δ[x1]∪· · ·∪δ[xn]. As (x1, . . ., xn)
+is an a.p. point in Xn, there is a syndetic subset A of T such that A(x1, . . ., xn) ⊆ δ[x1]×· · ·×δ[xn].
+Now for y ∈ X we have for some 1 ≤ i ≤ n that (y, xi) ∈ δ, A(y, xi) ⊂ β and Axi ⊂ δ[xi]. Thus
+Ay ⊂ β3[y]. Hence Ay ⊂ ε[y] for all y ∈ Y. The proof is complete.
+2.2.2. Thick stability
+2.4 Lemma. Let X be a minimal flow and S a thick subset of T. If X is Lyapunov S -stable at
+some point x ∈ X w.r.t. X, then X is thickly stable.
+Proof. Obvious.
+2.5 Theorem. Let X be a thickly stable flow with T an abelian group. If x ∈ X is an a.p. point
+under X , then (T, T x) is an a.p. subflow of X with discrete phase group T, and moreover,
+(T, T x) is an equicontinuous subflow of X .
+Proof. Since X is a regular space so that the orbit closure of an a.p. point is minimal (cf. [8,
+Thm. 1.5.3]), we can assume X = (T, T x) is minimal without loss of generality. Moreover,
+we may assume x is a.p. under X with T a discrete group. Let ε, α ∈ UX with α2 ⊆ ε. Since
+X is thickly stable at x by hypothesis, there exists an index γ ∈ UX and a thick subset S of
+T such that s(γ[x]) ⊆ α[sx] for all s ∈ S . Take an index δ ∈ UX with δ2 ⊆ γ. Since x is a.p.
+under X , there exists a discretely syndetic subset A of T such that Ax ⊆ δ[x]. To show that
+X is a.p., it suffices to prove that Atx ⊆ ε[tx] for all t ∈ T. For this, let t ∈ T, a ∈ A. There
+exists an index σ ∈ UX with σ ⊂ δ such that a(σ[x]) ⊆ δ[ax]. Then there is a syndetic subset
+B of T such that Bx ⊆ σ[x]. Since S is thick, so is S −1 in T. Since t−1B is syndetic in T, thus
+t−1B ∩ S −1 � ∅. Then there are elements b ∈ B and s ∈ S such that t−1b = s−1 so b−1t = s. By
+bx ∈ Bx ⊆ σ[x], it follows that abx ∈ δ[ax] ⊆ γ[x]. By b−1t = s ∈ S and commutativity of T,
+it follows that atx = (b−1t)(ab)x ∈ α[b−1tx]. Moreover, b ∈ B implies bx ∈ δ[x] ⊆ γ[x]. Hence
+tx = (b−1t)bx ∈ α[b−1tx]. Therefore, (atx, tx) ∈ α2 ⊆ ε for all t ∈ T and all a ∈ A.
+Next we shall prove that X is equicontinuous. Let ε ∈ UX there exists an index δ1 ∈ UX and
+a syndetic set A ⊆ T such that Aδ1 ⊆ ε. Select a compact set K ⊆ T such that KA = T = AK
+8
+
+(for T is abelian). Given x0 ∈ X, there exists an δ ∈ UX such that kδ[x0] ⊆ δ1[kx0] for all k ∈ K.
+Thus, tδ[x0] ⊆ ε[tx0] for all t ∈ T and then X is equicontinuous. The proof is complete.
+2.5a Corollary. If X is a locally a.p. thickly stable compact flow with T abelian, then X is an
+a.p. flow.
+Proof. Since X is locally a.p., X is pointwise a.p. and P(X ) = RP(X ). Then RP(X ) = ∆X by
+Theorem 2.5. This together with Lemma 2.3 proves Corollary 2.5a.
+2.5b Corollary (cf. [19, Thm. 2] or [8, Thm. 2.8.8]). Let X be a flow with T abelian. Let S be
+a thick subset of T and x ∈ X an a.p. point under X . If X is Lyapunov S -stable at x w.r.t. T x,
+then x is a Bohr a.p. point under X .
+Proof. By Lemma 2.4 and Theorem 2.5.
+When T is not necessarily an abelian topological group, we can conclude the following result
+with “x regularly a.p.” instead of “a.p.” under X :
+2.6 Theorem. Let X be a flow with X a Baire space. If X is thickly stable and it is regularly
+a.p. at some point x ∈ X, then (T, T x) is a pointwise regularly a.p. and equicontinuous subflow
+of X .
+Proof. We can assume X = T x without loss of generality for T x is a minimal subset of X. Let
+ε, α ∈ UX with α2 ⊆ ε. There exists an index γ ∈ UX and a thick subset S of T such that
+sγ[x] ⊆ α[sx] for all s ∈ S . Take an index δ ∈ UX with δ2 ⊆ γ. Since x is a.p. under X , there is a
+discretely syndetic set A in T with Ax ⊆ δ[x] and int Ax � ∅. To show X is Lyapunov T-stable
+at x w.r.t. X, it suffices to prove that tax ∈ ε[tx] for all t ∈ T and every a ∈ A.
+For that, let t ∈ T and a ∈ A be arbitrarily given. Select an index σ ∈ UX with aσ[x] ⊆ δ[ax].
+Since x is regularly a.p. under X , there exists a syndetic normal closed subgroup B of T such
+that Bx ⊆ σ[x]. Thus aBx ⊆ δ[ax] ⊆ γ[x]. Since S is thick and tB is syndetic in T, tB ∩ S � ∅.
+Thus tb = s for some b ∈ B and s ∈ S . Then
+tax = (tb)(b−1a)x ∈ (tb)(aB)x ⊆ α[tbx]
+and
+tx = (tb)b−1x ∈ (tb)Bx ⊆ (tb)γ[x] ⊆ α[tbx].
+Whence (tax, tx) ∈ α2 ⊆ ε for all t ∈ T and all a ∈ A.
+Further by minimality of X, X is Lyapunov T-stable. On the other hand, X is regularly a.p.
+at each point of T x. Then by Lyapunov stability, each point of X = T x is regularly a.p. under X .
+The proof is completed.
+2.6a Corollary. If X is a point-regularly a.p. thickly stable compact flow, then X is a (uni-
+formly) regularly a.p. flow.
+Proof. Use Theorem 2.6 and the uniform equicontinuity of X .
+2.6b Corollary. Let X be a flow with X a Baire space. Let S be a thick subset of T and x ∈ X
+a regularly a.p. point under X . If X is Lyapunov S -stable at x w.r.t. T x, then X is Lyapunov
+T-stable at x w.r.t. T x.
+Proof. By Lemma 2.4 and Theorem 2.6.
+Note that a point-regularly a.p. compact flow need not be a.p.; for example, [28, Thm. 12.55].
+So condition “X is thickly stable” is critical for Theorem 2.6.
+9
+
+2.2.3. Lyapunov thick-subsemigroup stability
+2.7 (P-limit set). Let x ∈ X, and P ⊆ T. The P-limit set of x, denoted Px, is defined to be
+�
+t∈T Ptx; that is, y ∈ Px iff for every t ∈ T there is a net p j ∈ P such that p jtx → y. Each point of
+Px is called a P-limit point of x under X . See [28, Def. 6.33] or [8, Def. 2.8.9].
+• If x ∈ X is a.p. under X and P a thick set in T, then x ∈ Px.
+Note here that Px is different from the proximal cell P[x] under X .
+2.7a Lemma (cf. [28, Thm. 6.07] for T a generative group). If P is a thick semigroup in T and
+K a compact subset of T with e ∈ K, then �
+k∈K kP is a thick semigroup in T.
+Proof. Write Q = �
+k∈K kP. Since P ∩ kP is a semigroup for all k ∈ K, Q is a semigroup in T.
+Let C be a compact subset of T and set D = C ∪ K−1C. Since D is compact, we can select t ∈ T
+such that Dt ⊆ P. Now we have for k ∈ K that C ⊆ D ∩ kD and Ct ⊆ Dt ∩ kDt ⊆ P ∩ kP. Thus
+Ct ⊆ Q and Q is thick in T. The proof is completed.
+2.7b Lemma (cf. [8, Lem. 2.8.10] for T an abelian group). Let P be a normal thick subsemigroup
+of T. Let x ∈ X. Then based on X :
+(1) Px is closed and T-invariant.
+(2) If Px is compact, then Px � ∅.
+(3) Px = �
+t∈P Ptx.
+Here P ‘normal’ means ‘Pt = tP ∀t ∈ T’.
+Proof.
+(1): Clearly, Px is closed. Let y ∈ Px and s, t ∈ T. Since y ∈ Ps−1tx, so sy ∈ sPs−1tx = Ptx and
+sy ∈ Px. Thus Px is T-invariant.
+(2): Suppose Px is compact. Let t1, . . ., tn ∈ T. By Lemma 2.7a, P ∩ t1P ∩ · · · ∩ tnP is a thick
+semigroup in T. Thus �n
+i=1Ptix = �n
+i=1tiPx ⊇
+��n
+i=1tiP
+�
+x � ∅ and Px � ∅.
+(3): Since P is normal thick in T, P−1 is also thick in T. Then for all t ∈ T, there is some st ∈ P
+with stt ∈ P. So
+Px ⊆
+�
+p∈PPpx ⊆
+�
+t∈T Psttx ⊆
+�
+t∈T Ptx ⊆ Px,
+The proof is complete.
+2.8 Lemma (a special case of [4, Thm. 1.15]). Let X be a compact flow and S a subsemigroup
+of T. If X is Lyapunov S -stable, then it is Lyapunov ⟨S ⟩-stable. Here ⟨S ⟩ is the subgroup of T
+generated by S .
+2.9 Theorem. Let X be a compact flow and S a thick subsemigroup of T. If X is Lyapunov
+S -stable, then X is an a.p. flow.
+Proof. By Lemma 2.8, (⟨S ⟩, X) is an equicontinuous flow. Since S is thick in T, so is ⟨S ⟩. How-
+ever, every thick subgroup of T coincides with T. Thus ⟨S ⟩ = T and X is equicontinuous. Since
+X is compact, X is a.p. by Lemma 2.3. The proof is complete.
+2.9a Corollary (cf. [32] [36, Thm. V.8.12] for T = R and S = R+; [19, Thm. 3] [8, Thm. 2.8.12]
+for T abelian). Let X be a flow and let S be a thick normal subsemigroup of T. Let x ∈ X such
+that S −1x is compact and that (T, T x) is a uniformly Lyapunov S -stable subflow of X . Then
+(T, T x) is a compact minimal a.p. subflow of X .
+10
+
+Proof. Let P = S −1; then P ⊳ T and P is thick in T. By Lemma 2.7b, Px is an T-invariant closed
+non-empty compact subset of X. Then we can take a T-minimal set M with M ⊆ Px ⊆ T x.
+Since (T, T x) is uniformly Lyapunov S -stable, (T, T x) is also uniformly Lyapunov S -stable. Let
+y ∈ M. Since M ⊆ S −1x, there is a net sn ∈ S such that s−1
+n x → y in T x. By uniform S -stability,
+sny → x ∈ M. Thus T x = M is compact. Then by Theorem 2.9, (T, T x) is equicontinuous and so
+an a.p. minimal compact subflow of X . The proof is completed.
+2.9b Corollary (cf. [19, Thm. 4] [8, Thm. 2.8.14] for T abelian). Let X be a flow and let P be
+a thick normal subsemigroup of T. Let x ∈ X. Then following statements are equivalent:
+(1) x is Bohr a.p. under X with T x compact.
+(2) x ∈ Px, Px is compact, and X is uniformly Lyapunov P-stable at T x w.r.t. Px.
+Proof. Assume (1). Clearly, Px = T x is compact minimal by Lemma 2.7b, and (T, T x) is
+equicontinuous. Thus x ∈ Px and T x is uniformly Lyapunov P-stable w.r.t. Px. Then (1) implies
+(2). Conversely assume (2). By Lemma 2.7b, T x ⊆ Px and T x = Px. Now applying Theorem 2.9
+with T x and P instead of X and S respectively, (T, Px) is a.p. Since x ∈ Px, so x is Bohr a.p. and
+(2) implies (1). The proof is complete.
+2.9c Corollary (cf. [19, Thm. 5] for T abelian). Let X be a flow and let P be a thick normal
+subsemigroup of T. Suppose y ∈ X such that:
+(1) The P-limit set of y, Py, is compact non-empty with Px = Py for all x ∈ Py.
+(2) X is Lyapunov P-stable at Ty w.r.t. Py.
+Then Py is the closure of a Bohr a.p. point of X .
+Proof. Condition (1) implies that Py is a minimal set for X . Since Py is compact, condition (2)
+implies that (T, Py) is uniformly Lyapunov P-stable. Then by Theorem 2.9, (T, Py) is (Bohr) a.p.
+so that Py is the closure of a Bohr a.p. orbit of X . The proof is complete.
+2.2.4. Syndetic distality
+It is easy to verify that every uniformly distal flow is a.p. (cf. [4, Lem. 1.7(2)] and [9,
+Thm. 1.1.5]). We now improve this result as follows:
+2.10 Theorem. If X is a syndetically distal compact flow, then it is pointwise Bohr a.p.
+Proof. First we claim that X is distal. Let x ∈ X and y ∈ P[x] with x � y. Then there is an
+index ǫ ∈ UX with (x, y) � ǫ. Since X is syndetically distal, there exists an index γ ∈ UX
+and a syndetic subset A of T such that t−1(γ[tx]) ⊆ ǫ[x] for all t ∈ A. However, by y ∈ P[x],
+B := {t ∈ T | (tx, ty) ∈ γ} is a thick subset of T. By A ∩ B � ∅, take t0 ∈ A ∩ B. Then t0y ∈ γ[t0x]
+and t−1
+0 (γ[t0x]) ⊆ ǫ[x]. So (x, y) ∈ ǫ, a contradiction. Thus P(X ) = ∆X and X is a distal flow.
+So to show X pointwise Bohr a.p., we can assume X is minimal distal, and we shall show
+X is a.p. For this, define the strong regionally proximal relation on X by
+U(X ) = {(x, y) | ∃ yn ∈ X → y and tn ∈ T s.t. (tnx, tnyn) → ∆X}
+and
+U[x] = {y ∈ X | (x, y) ∈ U(X )}
+∀x ∈ X.
+11
+
+Then
+U[x] =
+�
+α∈UX
+�
+t∈Tt−1(α[tx])
+∀x ∈ X.
+Since X is minimal distal, X admits an invariant Borel probability measure (cf. [20]). Then it
+follows from [34] that RP(X ) = U(X ). Therefore,
+RP[x] = U[x] =
+�
+α∈UX
+�
+t∈Tt−1(α[tx])
+for every x ∈ X.
+Let x ∈ X and ε ∈ UX. There exists an index δ ∈ UX and a syndetic subset S of T such that
+s−1(δ[sx]) ⊆ ε[x] for all s ∈ S . Since S is syndetic, there is a compact subset K of T such that
+T = KS . Further, since X is compact and the phase mapping T × X → X is jointly continuous,
+there exists an index α ∈ UX such that K−1α ⊆ δ. Let t ∈ T be arbitrary. Then we can write t = ks
+for some k ∈ K and some s ∈ S . Thus
+t−1(α[tx]) = s−1(k−1(α[ksx])) ⊆ s−1(δ[sx]) ⊆ ε[x].
+This shows that RP[x] ⊆ ε[x]. Since ε is arbitrary, so RP[x] = {x} and RP(X ) = ∆X. Then X is
+equicontinuous and a.p. by Lemma 2.3. The proof is complete.
+2.10a Corollary. If X is a syndetically distal and syndetically stable compact flow, then it is an
+a.p. flow.
+Proof. Since X is syndetically stable, so P(X ) = RP(X ). Then by Theorem 2.10, RP(X ) = ∆X
+so X is an a.p. flow by Lemma 2.3. The proof is complete.
+2.10b Corollary. If X is a syndetically distal T.T. compact flow, then it is an a.p. minimal flow.
+Proof. By Theorem 2.10 and (1.7b).
+Recall that X is said to be uniformly syndetically stable/equicontinuous if for every index
+ε ∈ UX there exists an index δ ∈ UX and a syndetic subset S of T such that S δ ⊆ ε.
+2.10c Corollary ([13]). If X is a uniformly syndetically stable compact flow with T an abelian
+group, then X is an a.p. flow.
+Proof. Since T is abelian, X is syndetically distal. Then X is a.p. by Corollary 2.10a.
+2.10d Question. If X is a uniformly syndetically stable compact flow with T non-abelian, is X
+an a.p. flow?
+We say that
+• X is uniformly syndetically regularly stable if for every index ε ∈ UX there exists an index
+δ ∈ UX and a syndetic subsemigroup A of T such that Aδ ⊂ ε.
+• X is uniformly left-syndetically stable if for every index ε ∈ UX there exists an index δ ∈ UX
+and a “left-syndetic” subset A of T such that Aδ ⊂ ε.
+• X is uniformly left-thickly stable if for every index ε ∈ UX there exists an index δ ∈ UX and
+a “left-thick” subset A of T such that Aδ ⊂ ε.
+12
+
+Here A is left-syndetic iff there exists a compact subset K of T such that A ∩ tK � ∅ for all t ∈ T,
+and, A is left-thick iff for every compact subset K of T there exists an element t ∈ T such that
+tK ⊆ A.
+2.10e Corollary. If X is a uniformly syndetically regularly stable compact flow, then it is an
+a.p. flow.
+Proof. Let ε ∈ UX. We can take an index α ∈ UX such that ¯α ⊆ ε. Further there exists an index
+δ ∈ UX and a syndetic subsemigroup A of T such that Aδ ⊆ α. Let S = A. Then S is a closed syn-
+detic subsemigroup of T such that S δ ⊆ ε. Noting that S is a subgroup of T (cf. [8, Lem. 2.8.17]),
+X is syndetically distal and syndetically stable. Thus X is a.p. by Corollary 2.10a. The proof
+is complete.
+2.10f Theorem. If X is a uniformly left-syndetically stable compact flow, then it is an a.p. flow.
+Proof. Let ε ∈ UX. Then there exists an index δ ∈ UX and a left-syndetic subset A of T such that
+Aδ ⊂ ε. Since A is left-syndetic, there is a compact subset K of T such that T = AK. Further,
+there exists an index α ∈ UX such that Kα ⊆ δ. Thus Tα = AKα ⊆ Aδ ⊂ ε. Hence X is an
+equicontinuous and a.p. flow. The proof is complete.
+2.10g Corollary. If X is a uniformly syndetically distal compact flow, then it is an a.p. flow.
+Proof. Let X be uniformly syndetically distal. Since the inverse of a syndetic set is left-syndetic,
+X is clearly a uniformly left-syndetic stable flow. Thus X is a.p. by Theorem 2.10f. The proof
+is complete.
+Theorem 2.10f is comparable with Clay’s theorem ([12, Thm. 14] [8, Thm. 1.6.21]): X is
+uniformly left-thickly stable iff it is equicontinuous. Moreover, following [4, Thm. 8.3], X is
+equicontinuous iff for every α ∈ UX there exists a syndetic subset A of T such that A = A−1 (so
+A is left-syndetic) and Ax ⊆ α[x] for all x ∈ X.
+3. Canonical M-dynamics of non-PI extensions
+Let π: X → Y be a nontrivial extension of minimal compact dynamics. Recall that π is
+weakly mixing if Rπ is T.T.; and π is RIC if π is open such that
+Rn
+π = {(x1, . . . , xn) ∈ Xn | πx1 = · · · = πxn}
+has a dense set of a.p. points for all n ≥ 2 (cf. [9]). Here we shall define canonical M-dynamics
+in Rπ, which will be useful for describing the π-relative unpredicted dynamics of X . Moreover,
+we will characterize intrinsically PI-extensions of minimal dynamics in this section.
+In the sequel, T is thought of as a discrete space and let I be a minimal left ideal in βT.
+Let I = (T, I) as in §2.1. Let πX : I → X and πY : I → Y be two extensions such that
+πY = π ◦ πX. Associated to πX and πY we can define two groups in Aut (I ):
+3.1. A = {α ∈ Aut (I ) | πX = πX ◦ α} and F = {α ∈ Aut (I ) | πY = πY ◦ α}.
+Then A and F are τ-closed subgroups of Aut (T, I) with A ⊆ F, and, A = F iff π is proximal;
+see [9, Appendix A]. Put F0 = F, F1 = F′, F2 = F′
+1, and for every ordinal o, Fo+1 = F′
+o. If o is a
+limit ordinal, put Fo = �
+γ<o Fγ. Then by transfinite induction there there exists a least ordinal o
+such that F0 ⊋ F1 ⊋ F2 ⊋ · · · ⊋ Fo = Fo+1 = · · · . Write
+13
+
+3.2. F∞ = Fo, which is called the core of F and which is a τ-closed subgroup of F such that
+AF∞ = F∞A.
+Then:
+3.3. π is a PI-extension iff F∞ ⊆ A (cf. [23, Thm. X.4.2] or [2, Thm. 14.23]; it may follow from
+3.6 below).
+Notice that the PI condition of π is independent of the choice of πX and πY. So PI-extension is an
+intrinsic concept. Every PI-extension has “good” structure; see 3.6 below.
+There are many intrinsic characterizations of PI-extensions; see [7, 35, 42]. In particular, “π
+is PI iff every M-dynamic in Rπ is minimal” (due to Bronstein, cf. [7, 42, 22]). In fact, we will
+show that π is PI iff every M-dynamic containing ∆X in Rπ is just equal to ∆X (see Theorem 3.10
+below). So if π is not PI, then Rπ contains a non-minimal M-dynamic.
+However, we shall be only interested to the non-minimal M-dynamics that contains ∆X. This
+type of M-dynamic is useful for us to define chaos of non-PI extensions. For this, we now put
+3.4. M∞ = (πX × πX)Γ[F∞], where Γ[F∞] = �
+α∈F∞Γ[α].
+Clearly, M∞ is a closed invariant symmetric subset of Rπ containing a dense set of a.p. points of
+RPπ (cf. [9, Lem. A.4.4]).
+Although we have not shown that M∞ is an AP-transitive relation on X here, yet we can first
+obtain the following:
+3.5 Theorem. Let π: X → Y be an extension of minimal dynamics. If π is not PI, then (T, M∞)
+is an M-dynamic in Rπ with ∆X ⊊ M∞ and RPπ|M∞ = M∞.
+Proof. Since π is not PI, F∞ ⊈ A by 3.3 and idI ∈ F∞ ∩ A. Thus, ∆X ⊊ M∞. Let U and V be two
+non-empty open sets in M∞. There exist two elements α, γ ∈ F∞ such that U ∩ (πX × πX)Γ[α] � ∅
+and V ∩(πX ×πX)Γ[γ] � ∅. As F∞ = F′
+∞, it follows that (T, M∞) is T.T. (cf. Proof of Theorem 2.2)
+so that (T, M∞) is an M-dynamic and then RPπ|M∞ = M∞. The proof is complete.
+Therefore, in view of Bronstein’s theorem and Theorem 3.5, if Rπ has a non-minimal M-
+subdynamic, then (T, M∞) is an M-dynamic in Rπ with ∆X ⊊ M∞.
+3.6 (Ellis-Glasner-Shapiro tower). Based on the classical Ellis-Glasner-Shapiro structure theo-
+rem (cf. [18, 23, 44, 2] and [22]), for π: X → Y , there exists the following canonical CD of
+minimal compact dynamics and T-extensions:
+I
+X <
+φ
+πX
+>
+X∞
+πX∞
+>
+Y
+π
+∨
+<
+ψ
+πY
+>
+Y∞
+π∞
+∨
+πY∞ − − − −
+>
+s.t.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ψ has a PI-tower,
+φ is proximal,
+π∞ is RIC weakly mixing,
+g(X∞) = A,
+g(Y∞) = AF∞.
+Here A, F are as in Def. 3.1 and
+g(X∞) = {α ∈ Aut (I ) | πX∞ = πX∞ ◦ α},
+g(Y∞) = {α ∈ Aut (I ) | πY∞ = πY∞ ◦ α}.
+Notice that (φ×φ)Rπ∞ ⊊ Rπ in general, for ψ need generally not be proximal and π need not satisfy
+the Bronstein condition. Moreover, π is PI iff π∞ is 1-1 onto iff π∞ is proximal iff A = AF∞.
+14
+
+3.7 Theorem. Let π: X → Y be a non-PI extension of minimal compact dynamics. Then
+M∞ = (φ × φ)Rπ∞ and (πX × πX)Γ[F∞] is the set of all a.p. points of (T, M∞).
+Proof. By Def. 3.4 and F∞ ⊆ g(Y∞), it follows that M∞ ⊆ (φ×φ)Rπ∞. Let (w, w′) ∈ Rπ∞ be an a.p.
+point. Since π∞ satisfies the Bronstein condition and (w, w′) ∈ RPπ∞, so by [9, Theorem 3.1.3]
+we have that (w, w′) ∈ (πX∞ × πX∞)Γ[α] for some α ∈ (AF∞)′ ⊆ AF∞. This implies that (φ ×
+φ)(w, w′) ∈ M∞. Thus M∞ ⊇ (φ × φ)Rπ∞ and M∞ = (φ × φ)Rπ∞.
+Let (x1, x2) ∈ M∞ be an a.p. point. By M∞ = (φ × φ)Rπ∞, we can take an a.p. point (w1, w2)
+in Rπ∞ such that φw1 = x1 and φw2 = x2. Further, there exist elements α ∈ A and γ ∈ F∞ such
+that (w1, w2) = (πX∞ × πX∞)(m, αγm) = (πX∞ × πX∞)(m, γm) for some m ∈ I. So by πX = φ ◦ πX∞,
+(x1, x2) = (πX × πX)(m, γm). The proof is complete.
+3.8 Lemma (cf. [5, Lem. 2.10] for T a group). Let φ: X → Y be a proximal extension of
+compact dynamics with X having a dense set of a.p. points. Then X is an M-dynamic iff so is
+Y .
+Proof. Necessity is obvious. Conversely, suppose Y is an M-dynamic. Let K be an invariant
+closed set in X with int K � ∅. We need to prove K = X. Suppose the contrary that K � X. Then
+X \ K � ∅. Since a.p. points are dense in X, so (X \ K) ∩ t−1int K = ∅ for all t ∈ T. It is clear
+that L := X \ K = X \ int K and t(X \ K) ⊆ L for every t ∈ T. Thus L is an invariant closed set
+in X with int L � ∅ and K ∪ L = X. As φK ∪ φL = φX = Y, φK or φL must have a non-empty
+interior in Y, and so, by T.T. of Y , φK = Y or φL = Y.
+Assume φK = Y. Let x ∈ X \ K be an a.p. point. Then we can find x′ ∈ K with φx = φx′ such
+that (x, x′) is an a.p. point. Since φ is proximal, so x = x′ and K = X. If φL = Y, then similarly it
+follows that L = X, contrary to L ∩ int K = ∅ and int K � ∅. Hence K = X and X is T.T. and
+an M-dynamic. The proof is complete.
+3.9 Lemma. Let φ: Z → Y be an extension of minimal compact dynamics, let (T, N) be an
+M-dynamic in Rφ. Then:
+(1) If φ is proximal, then N = ∆Z.
+(2) If φ is a.p., then N is minimal.
+Proof. (1) is obvious. For (2), we consider θ: N
+→ Y , which is induced canonically by φ.
+Then N is T.T. and θ-distal. By (1.7c), N is minimal.
+3.10 Theorem (cf. [7, 42, 22] for (1) ⇔ (2)). Let π: X → Y be an extension of minimal
+compact dynamics. Then the following are pairwise equivalent:
+(1) π is a PI-extension.
+(2) Every M-dynamic in Rπ is minimal.
+(3) Every M-dynamic containing ∆X in Rπ is minimal.
+Proof.
+(1) ⇒ (2): Clearly by 3.3, X∞ � Y∞ in the CD of 3.6. Set ψ∞ = ψ ◦ π∞. Let (T, M) be an
+M-dynamic in Rπ. There exists an invariant closed subset N of Rψ∞, which has a dense set of
+a.p. points such that M = (φ × φ)N. Then by Lemma 3.8, (T, N) is an M-dynamic in Rψ∞. Since
+ψ∞ is strictly PI, thus N is minimal by using the PI-tower of ψ∞ and Lemma 3.9, and M is also
+minimal. Then (1) implies (2).
+15
+
+(2) ⇒ (3): Obvious.
+(3) ⇒ (1): This follows easily from Theorem 3.5. The proof is complete.
+3.11 (Ellis weak-mixing extension). Recall that for π: X → Y :
+• X is called an π-Ellis-weak-mixing extension of Y or simply π is Ellis weak-mixing [25, 3] if
+F = AF′, where A and F are defined as in Def. 3.1.
+It is well known that if there is no the Bronstein condition and Y � {pt}, an Ellis weak-mixing
+extension of minimal compact flows need not be a weak-mixing extension (see [25]); and, an Ellis
+weak-mixing minimal compact semiflow need not be weakly mixing (see [9, Exa. 4.3.2]).
+Moreover, by using the canonical Ellis-Glasner-Shapiro construction of RIC extensions [18,
+23, 44, 2] we can easily obtain the following:
+3.12 Lemma. If π is Ellis weak-mixing non-proximal, then it is not PI so that there exist non-
+minimal canonical M-flows (T, M∞) in Rπ.
+Notice that the definition of Ellis weak-mixing is independent of the choice of πX and πY.
+Moreover, a proximal minimal flow is Ellis weak-mixing and PI (cf. [23, 9]).
+4. Li-Yorke chaos of extension, sensitivity and asymptotically a.p. motions
+We will introduce the definition of Li-Yorke chaotic extension in §4.1, provide a sufficient
+condition for sensitivity on initial conditions in §4.2, and consider asymptotically a.p. motions
+in §4.3.
+4.1 (Li-Yorke chaos of extension). In this part let π: X → Y be a nontrivial extension of
+minimal compact metric dynamics, unless specified otherwise. Let ρ be a metric on the compact
+metric space X. Then (x, x′) ∈ Pπ iff (x, x′) ∈ Rπ and there exists a sequence {tn}∞
+n=1 in T such that
+limn→∞ ρ(tnx, tnx′) = 0.
+Since π is nontrivial, so Rπ � ∆X. Recall that whenever T = Z+ and (x, x′) ∈ X × X with
+x � x′, we say (x, x′) is a Li-Yorke pair iff
+lim infn→∞ρ(nx, nx′) = 0
+and
+lim supn→∞ρ(nx, nx′) > 0.
+In view of this, the definition of Li-Yorke chaos has been generalized to group actions; see, e.g.,
+[30, 37, 14, 1] and so on. Moreover, Devaney’s chaos implies the Li-Yorke’s chaos (cf., e.g., [29]
+for Z-flow, [14, Prop. 2.21] and [1] for T an abelian group). We now introduce the Li-Yorke
+chaos of extensions of minimal dynamics as follows:
+4.1a. Let L be a closed subset of X × X such that ∆X ⊊ L ⊆ Rπ.
+(a) We say (x, x′) ∈ X × X is a Li-Yorke pair for π rel. L if (x, x′) ∈ Rπ and L ⊆ T(x, x′).
+(b) A set S in X is called Li-Yorke scrambled for π rel. L if every pair (x, x′) ∈ S × S \ ∆X is a
+Li-Yorke pair for π rel. L.
+4.1b Lemma. Let L = (T, L) be an M-dynamic in Rπ, and define Trans(L ) to be the set
+Trans(L ) = {˜x ∈ L | T ˜x = L}.
+If ∆X ⊊ L, then each point of Trans(L ) is a Li-Yorke pair for π rel. L.
+16
+
+Since (T, L) is an M-dynamic such that ∆X ⊊ L ⊆ Rπ, so if (x, x′) ∈ Trans(L ), then the set
+of t ∈ T at which T x and T x′ are sufficiently close and the set of t ∈ T at which T x and T x′
+are far away are both very big (it is at least syndetic). In view of this, the Li-Yorke chaos here is
+much stronger than the classical sense.
+4.2 (Sensitive to initial conditions). Let X be a dynamic with X a uniform Hausdorff space not
+necessarily compact.
+4.2a. X is said to be bounded if given α ∈ UX there exists a point x0 ∈ X and a compact set K
+in T such that α[Kx0] = X. If X is not bounded, then it called unbounded.
+Clearly, if (X, ρ) is a metric space such that there is a point x0 ∈ X such that {ρ(x0, x) | x ∈ X}
+is unbounded in R, then X is unbounded. Moreover, we can easily construct a compact metric
+space that can support a unbounded pointwise a.p. flow. By virtue of these, our “unbounded”
+condition is then much general than the one introduced in [40, (D1)] for only metric spaces.
+In fact,
+(i) If X is bounded having a dense set of a.p. points, then X is T.T. Moreover, if X is
+bounded, then it is totally bounded.
+Proof. Suppose this is not true. Then X would be bounded and thus it would be T.T.
+Indeed, let U, V be non-empty open sets in X. Then there exists a point x0 ∈ X and s, t ∈ T
+such that sx0 ∈ U and tx0 ∈ V. Further, there exists an open W ∈ Nx0 such that sW ⊆ U
+and tW ⊆ V. Since the a.p. points are dense in X, we can pick an a.p. point, say x1, in sW
+and some s1 ∈ T such that s1x1 ∈ W. So ts1x1 ∈ V and NT(U, V) � ∅.
+(ii) Let T be σ-compact. If X is a pointwise a.p. non-minimal compact dynamic, then X is
+unbounded.
+Proof. If this is false, then X is T.T. by (i) and thus, X would be minimal by (1.7a) in §1.
+4.2b. We say that X is sensitive to initial conditions on X0, where X0 ⊆ X, if there exists an
+index ε ∈ UX such that for every x ∈ X0 and all U ∈ Nx(X) there exists some y ∈ U and some
+t ∈ T with (tx, ty) � ε.
+4.2c Lemma. If X0 is dense in X and X is sensitive to initial conditions on X0, then X is
+sensitive to initial conditions on X.
+Proof. Obvious.
+4.2d Theorem (cf. [40, Thm. 1, Thm. 1.1] for T = R+ or Z+ with X a unbounded metric space,
+under a stronger definition of a.p. points). Let X be a unbounded M-flow; then X is sensitive
+to initial conditions on X.
+Proof. Let X0 be the set consist of a.p. points of X . Then X0 is dense in X. By Lemma 4.2c, it
+is enough to show that X is sensitive to initial conditions on X0. Suppose this is false. Let α,
+ε ∈ UX with ε2 ⊆ α. Then there is an index δ ∈ UX and a point x0 ∈ X0 such that for all y ∈ δ[x0],
+(ty, tx0) ∈ ε for all t ∈ T. Moreover, since x0 is a.p. under X , there is a compact set K in T such
+that KNT(x0, δ[x0]) = T. Thus
+Tδ[x0] ⊆ ε[Kδ[x0]] ⊆ ε2[Kx0]
+and
+Tδ[x0] ⊆ α[Kx0].
+Since X is T.T., X = Tδ[x0] and X = α[Kx0]. This is contrary to that X is unbounded. The proof
+is completed.
+17
+
+It should be noted that the unboundedness of X is critical for Theorem 4.2d; for example, a
+minimal equicontinuous compact flow is a nonsensitive M-flow.
+4.2e Corollary (cf. [40, Thm. 1.2] for T = Z+ and R+ and RE(X ) dense). Let X be an M-flow
+with X a complete noncompact space (like X = Rn). Then X is sensitive to initial conditions on
+X.
+Proof. First note that X is unbounded; for otherwise, X would be totally bounded complete so
+that X would be compact. Then by Theorem 4.2d, X is sensitive.
+4.2f. A point x ∈ X is called a Birkhoff recurrent point under X [36, 10] if given ε ∈ UX there
+exists a compact set K in T such that T x ⊆ ε[Ktx] for all t ∈ T. The set of Birkhoff recurrent
+points of X is denoted RE(X ) in [40].
+A Birkhoff recurrent point must be an a.p. point. In fact, in compact dynamics a point is a.p.
+iff it is Birkhoff recurrent (cf. [10, Thm. 4.1]); in locally compact metric flows, a point is a.p. iff
+it is Birkhoff recurrent (cf. [10, Cor. 4.2]). However, in locally compact metric semiflows an a.p.
+point need not be Birkhoff recurrent (see [4, Rem. 3.15(c)] for a counterexample).
+So in Theorem 4.2d, our condition “a.p. points are dense in X” is generally weaker than
+Seifert’s condition “RE(X ) is dense in X” in [40].
+• Let X be a flow. If x ∈ X such that given ε ∈ UX there exists a compact set L ⊆ T with
+ε[tx] ∩ Latx � ∅ for all t, a ∈ T, then x ∈ RE(X ) (cf. [40, p. 1721] for T = R+ and Z+).
+Proof. Let s ∈ T and t ∈ T. Select a ∈ T with s = at or equivalently t = a−1s. Then tx ∈ ε[Lsx].
+This shows that x ∈ RE(X ).
+Thus, x is Birkhoff recurrent under X iff (T, T x) is a weakly a.p. subflow of X (in the sense
+of Gottschalk [4]: Given ε ∈ UX there exists a compact set L in T such that for all y ∈ T x,
+Lt ∩ NT(y, ε[y]) � ∅ ∀t ∈ T).
+4.2g. By Equi(X ) we denote the set of points at which X is equicontinuous/T-stable. That is,
+x ∈ Equi(X ) iff given ε ∈ UX there is a member U ∈ Nx such that (tx, ty) ∈ ε for all t ∈ T and
+all y ∈ U. See Def. IV. and a. in §2.2.1.
+4.2h Corollary. Let X be a unbounded T.T. flow. If Equi(X ) � ∅, then the a.p. points are not
+dense in X (and consequently RE(X ) is not dense in X and X is an M-flow).
+Proof. Suppose this is false. Then by Theorem 4.2d, X would be sensitive to initial conditions
+on X. So there would be an index ε ∈ UX such that for x ∈ Equi(X ) and U ∈ Nx there exists
+a point y ∈ U with (tx, ty) � ε for some t ∈ T. This contradicts equicontinuity of X at x. The
+proof is completed.
+4.2i Remark. If “a.p.” points are replaced by “Birkhoff recurrent” points, then Theorem 4.2d
+and Corollary 4.2h still hold for T a semigroup. By virtue of this, Corollary 4.2h is in conflict
+with an open question in [40, p. 1725].
+4.2j Remark. Using “X locally compact noncompact” instead of “X unbounded”, by the same
+argument as that of Theorem 4.2d, we can conclude the following:
+• Let X be an M-flow with X a locally compact noncompact space. Then X is sensitive to
+initial conditions on X.
+18
+
+4.3 (Asymptotically a.p. motions). In this part, let X be a semiflow with X a uniform Hausdorff
+space. We first define
+(i) the ω-limit set ω[x] = {w ∈ X | ∀ U ∈ Nw and s ∈ T, ∃ t ∈ T s.t. tsx ∈ U}; Clearly, for
+T = R+ or Z+ and X a metric space, w ∈ ω[x] iff ∃ tn → ∞ such that tnx → w. Moreover,
+the P-limit set of x, Px = ω[x], for P = T.
+(ii) AAP(X ) = {x ∈ X | ∃ an a.p. point y s.t. ∀ ε ∈ UX, ∃ s ∈ T s.t. (tsx, tsy) ∈ ε ∀t ∈ T}.
+Clearly, for T = R+ or Z+ and (X, ρ) a metric space, x ∈ AAP(X ) iff ∃ an a.p. point y such
+that ρ(tx, ty) → 0 as t → ∞.
+We call AAP(X ) the set of asymptotically a.p. points of X . By virtue of 4.2f, this is equivalent
+to that defined by Seifert [39, (iv)] under compact ambit.
+4.3a Lemma. Let T be such that �
+s∈F T s � ∅ for every finite set F ⊂ T. If x ∈ X such that T x
+is compact, then ω[x] � ∅ is compact invariant.
+Proof. By ω[x] = �
+s∈T T sx and compactness, it follows easily that ω[x] � ∅ is compact. The
+invariance is evident.
+4.3b Theorem. Let T be such that �
+s∈F T s � ∅ for every finite set F ⊂ T. Let x ∈ X such
+that T x is compact and ω[x] ⊆ Equi(X ). Then (T, ω[x]) is a minimal subsemiflow of X ,
+ω[x] ⊆ RE(X ), and x ∈ AAP(X ). If, in addition, T is an amenable semigroup or T is a group,
+then (T, ω[x]) is an a.p. subdynamic of X .
+Proof. By Lemma 4.3a, ω[x] is an invariant compact nonempty subset of X. Let M be a minimal
+subset of ω[x]. If M � ω[x], then we can take a point w ∈ ω[x]\ M. Further, there exists an index
+ε ∈ UX such that ε[w] ∩ ε[M] = ∅. Let y ∈ M. Select U ∈ Ny with U ⊂ ε[M] such that if sx ∈ U
+then tsx ∈ ε[M] for all t ∈ T. This is contrary to w ∈ ω[x]. Thus ω[x] is a minimal set in X. This
+also shows that ω[x] ⊆ RE(X ).
+Now we will prove that x ∈ AAP(X ). By a theorem of Auslander-Ellis (cf., e.g., [21,
+Thm. 8.7]), we can find a point y ∈ ω[x] such that x is proximal to y. Let ε ∈ UX. We note
+here that since ω[x] ⊆ Equi(X ) is compact, there is an index δ ∈ UX such that (z, w) ∈ δ
+and w ∈ ω[x] implies that (tz, tw) ∈ ε for all t ∈ T. Now, we can take some s ∈ T such that
+(sx, sy) ∈ δ, and further, (tsx, tsy) ∈ ε for all t ∈ T. Thus x ∈ AAP(X ).
+Finally, If T is an amenable semigroup or T is a group, then by [4, Thm. 1.15] it follows that
+(T, ω[x]) is an a.p. subdynamic of X . The proof is completed.
+4.3c Corollary (cf. [39, Thm. 1] for d = 1). Let T = Rd
++ or Zd
++ where d ≥ 1; let T x be
+compact such that ω[x] ⊆ Equi(X ). Then x ∈ AAP(X ), ω[x] ⊂ RE(X ), and (T, ω[x]) is an a.p.
+subsemiflow of X .
+Proof. Clearly, Rd
++ and Zd
++ both satisfy the condition of Lemma 4.3a (by Remark 4.3g below).
+Then Theorem 4.3b follows Corollary 4.3c.
+4.3d Corollary. Let x ∈ X such that
+(1) ω[x] � ∅ is compact with ω[y] = ω[x] for all y ∈ ω[x] and
+(2) ω[x] ∩ Equi(X ) � ∅.
+Then x ∈ AAP(X ).
+19
+
+Proof. By condition (1), ω[x] is a compact minimal set of X . Then from condition (2), it follows
+that ω[x] ⊆ Equi (X ). Now using Theorem 4.3b, we have that x ∈ AAP(X ).
+4.3e Remark (cf. [39, Lem. 1] for T = R+). In Theorem 4.3b, if T is an amenable semigroup
+or T is a group, then for x ∈ AAP(X ), the corresponding shadowing a.p. motion as in 4.3(ii) is
+unique.
+Proof. Suppose this is false. Then there are two distinct points y1, y2 ∈ ω[x] such that for every
+ε ∈ UX there are s1, s2 ∈ T such that (tsix, tsiyi) ∈ ε ∀t ∈ T, for i = 1, 2. By T s1 ∩ T s2 � ∅
+for all s1, s2 ∈ T, it follows that y1 is proximal to y2 under X . However, since y1, y2 ∈ ω[x] and
+(T, ω[x]) is a.p., thus (y1, y2) is a.p. under X × X so that y1 = y2. The proof is complete.
+It should be mentioned that our unicity proof here is much simpler than Seifert’s proof for
+the special case that T = R+ and X a complete metric space [39, Lem. 1]. There Seifert need to
+use y1, y2 ∈ RE(X ) instead of our “proximality” argument.
+4.3f Remark. Recall that a topological semigroup T is called a “C-semigroup” [31] if T \ T s is
+compact for all s ∈ T. For example, (R+, +) and (Z+, +) are C-semigroups. Then:
+• If T is a noncompact C-semigroup, then �n
+i=1 T si � ∅ for all n ≥ 2 and s1, . . . , sn ∈ T.
+Proof. Let n ≥ 2 and s1, . . ., sn ∈ T. If �n
+i=1 T si = ∅, then (T \ T s1) ∪ · · · ∪ (T \ T sn) = T so that
+T is compact, contrary to the hypothesis. Thus �n
+i=1 T si � ∅.
+4.3g Remark. Let T = Rd
++ and Zd
++, where d ≥ 1 is an integer. Then �n
+i=1(T + si) � ∅ for all
+n ≥ 2 and s1, . . . , sn ∈ T.
+Proof. Given n ≥ 2, let K = {o, s1, . . . , sn}, which is a compact subset of G, where G = Rd or Zd
+and o = (0, . . ., 0). Noting that T is a thick semigroup in G. Then by Lemma 2.7a, �n
+i=1(T + si)
+is a thick set in G so that �n
+i=1(T + si) � ∅. The proof is completed.
+5. Li-Yorke chaos of non-PI extensions
+If π: X → Y is a non-PI extension of minimal compact flows, then we can define a canoni-
+cal M-flow (T, M∞) such that ∆X ⊊ M∞ ⊆ Rπ as in Def. 3.4 in §3. Moreover, M∞ has been described
+by Theorem 3.7. We can then state our main result of this section as follows:
+5.1 Theorem. Let π: X → Y be a non-PI extension of minimal compact metric flows. Then
+there exists a residual set YLY in Y such that for every y ∈ YLY and for generic x ∈ π−1y there is a
+residual set S [x] in M∞[x] with the property that (x, x′) is a Li-Yorke pair for π rel. M∞ for each
+x′ ∈ S [x].
+Recall that a mapping is said to be semi-open if the image under this mapping of every open
+non-empty set contains an open non-empty set. If φ: X → Y is an extension of compact flows
+with Y minimal and X having a dense set of a.p. points, then φ is semi-open ([8, Lem. 3.12.15],
+[2, Thm. 1.15]).
+To prove Theorem 5.1, we shall need two auxiliary lemmas. The first is a special case of the
+mentioned semi-openness theorem.
+5.2 Lemma. Let π: X → Y be an extension of compact flows with Y minimal. If (T, L) is an
+M-flow in Rπ, then p: L → Y defined by (x, x′) �→ πx is semi-open onto.
+20
+
+5.3 Lemma (a topological ‘Fubini theorem’; cf. [43, Prop. 3.1], [24, Lem. 5.2] for other ver-
+sions). Let φ: W → Z be a semi-open continuous onto map of Polish spaces. Suppose K is a
+residual subset of W. Let
+ZK = {z ∈ Z | K ∩ φ−1z is a residual subset of φ−1z}.
+Then ZK is residual in Z.
+Proof. Since K is residual in W, there exists a sequence F1, F2, . . . of closed nowhere dense sets
+in W such that W \ K ⊆ �∞
+i=1 Fi. Let Kz = K ∩ φ−1z for all z ∈ Z, and we notice that
+φ−1z \ Kz ⊆
+�∞
+i=1(Fi ∩ φ−1z).
+Now let
+B = {z ∈ Z | ∃ i s.t. intφ−1z(Fi ∩ φ−1z) � ∅}.
+So if z � B, then Kz is of course residual in φ−1z. Then Z \ ZK ⊆ B. It will therefore suffice to
+prove B is of the first category.
+Let U1, U2, . . . be a basis for the topology on W. If z ∈ B, then we have for some integers m
+and i that ∅ � Um ∩ φ−1z ⊆ Fi. Put
+Cmi = {z ∈ B | ∅ � Um ∩ φ−1z ⊆ Fi}
+for 1 ≤ m, i < ∞.
+Then B = �∞
+m,i=1 Cmi and B is a first category set in Z if each set Cmi is nowhere dense in Z.
+Let Dmi = intZclsZCmi, and suppose Dmi � ∅ for some two positive integers m, i. Since
+Cmi ∩ Dmi � ∅, and, Um ∩ φ−1z � ∅ whenever z ∈ Cmi ∩ Dmi, hence the set
+F := Um ∩ φ−1Dmi
+is an open non-empty subset of Um. We shall prove F ⊆ Fi, contrary to that Fi is nowhere dense,
+then Lemma 5.3 will follow.
+Let Wo = {w ∈ W | φ is open at w}. Since φ is semi-open and W is a Polish space, Wo is
+a residual subset of W. Indeed, let Gn = {x | x ∈ W s.t. x ∈ B1/n(x) ∩ φ−1intWφB1/n(x)} for
+n = 1, 2, . . .; then Gn is open dense in W, and, Wo = �
+n Gn.
+Now put Fo = F ∩ Wo. Then Fo is dense in F. If w ∈ Fo and z = φw ∈ Cmi, then by
+definition w ∈ Um ∩ φ−1z ⊆ Fi. Now let w ∈ Fo but z = φw � Cmi. Then at least z ∈ clsZCmi, or,
+z = limn→∞ zn, zn ∈ Cmi. So there exists a sequence wn → w in F such that φwn = zn for all n.
+Eventually, wn ∈ Um, and since zn ∈ Cmi, we have wn ∈ Fi for all n. Therefore w ∈ Fi = Fi and
+Fo ⊆ Fi. Since Fi is closed and Fo is dense in F, so F ⊆ Fi. The proof is complete.
+Notice here that comparing with Veech [43, Prop. 3.1] and Glasner [24, Lem. 5.2], the point
+of Lemma 5.3 is that W need not be a minimal flow. Moreover, as a result of Lemma 5.3, we can
+conclude the following useful result:
+Corollary ([29, Lem. 3.2]). Let E be a Polish space and R a relation on E such that R contains
+a residual subset of E × E. Then there exists a residual subset A of E such that for all x ∈ A there
+exists a residual subset Kx of E with {x} × Kx ⊂ R.
+21
+
+Proof of Theorem 5.1. Since M∞ := (T, M∞) is T.T. by Theorem 3.5 and M∞ is a compact metric
+space, Trans(M∞) is residual in M∞. By Lemma 4.2, each point of Trans(M∞) is a Li-Yorke
+pair for π rel. M∞. Let λ: M∞ → X be defined by (x, x′) �→ x. Clearly λ is an extension of flows.
+Since λ: M∞ → X is semi-open onto by Lemma 5.2, there exists by Lemma 5.3 a residual set
+XLY in X such that for every x ∈ XLY, the set S [x] = {x′ ∈ X | (x, x′) ∈ Trans(M∞)} is relatively
+residual in λ−1x = M∞[x]. Furthermore, since π: X → Y is semi-open onto, by Lemma 5.3 there
+exists a residual set YLY in Y such that for every y ∈ YLY, the set π−1y ∩ XLY is relatively residual
+in π−1y. The proof is complete.
+5.4 Lemma (cf. [29, Lem. 3.1] for E having no isolated point & R ⊂ E × E). Let E be a Polish
+space. Let R ⊂ E × E \ ∆E be a symmetric relation on E with the property that there exists a
+residual subset A of E such that for each a ∈ A, R[a] contains a residual subset of E. Then there
+is a dense uncountable subset B of E such that B ⊆ A and B × B \ ∆E ⊆ R.
+Proof. Let B be the family of dense subsets B of E such that B × B \ ∆E ⊆ R and B ⊆ A. Clearly,
+B � ∅. Indeed, if B ⊂ A is at most countable with B × B \ ∆ ⊆ R and B � E, then there exists an
+element x ∈ A ∩ (E \ B) ∩
+��
+b∈B R[b]
+�
+. So B1 = B ∪ {x} ⊂ A such that B1 × B1 \ ∆E ⊆ R. Then
+by induction we can find a subset of A that belongs to B.
+Now by Zorn’s lemma there exists a maximal element B in (B, ⊆). Then B is uncountable.
+Otherwise, we can choose an element x ∈ A∩
+��
+b∈B R[b]
+�
+such that B∪{x} ∈ B, contrary to that
+B is a maximal element in B. This proves Lemma 5.4.
+Theorem 5.1 is an extension of [14, Prop. 3.21]. As a matter of fact, using Rπ instead of M∞
+in the above arguments, we can obtain the following:
+5.5 Theorem. Let π: X → Y be a nontrivial extension of metric flows with Y minimal. Suppose
+that Rπ has a dense set of a.p. points and that π is weakly mixing. Then there exists a residual
+set YLY in Y such that for every y ∈ YLY, π−1y has no isolated points and it contains a relatively
+dense uncountable Li-Yorke scrambled set for π rel. Rπ.
+Proof. Notice that Rπ is an M-flow with Rπ � ∆X. Let S = Trans(Rπ). Then S is a symmetric
+relation on X such that S ∩ ∆X = ∅ and it is residual in Rπ. Using Lemma 5.3 for the mappings
+Rπ
+(x,x′)�→πx
+−−−−−−−→ Y and π−1y × π−1y
+(x,x′)�→x
+−−−−−−→ π−1y, it follows that there exists a residual set YLY in Y
+such that for all y ∈ YLY, π−1y contains a relatively residual set (π−1y)LY with the property that
+for x ∈ (π−1y)LY the cell S [x] is relatively residual in π−1y.
+Given y ∈ YLY, put E = π−1y and R = S ∩ E × E. Then R is a symmetric relation on E with
+R ∩ ∆E = ∅. If E has an (relative) isolated point x0, then x0 ∈ (π−1y)LY and x0 ∈ S [x0] so that
+(x0, x0) is a transitive point for Rπ. This is impossible for Rπ � ∆X. This shows that E has no
+isolated points. Let A = (π−1y)LY. Then by Lemma 5.4, there exists a dense uncountable subset
+B of E such that (x, x′) ∈ R whenever x, x′ ∈ B with x � x′.
+Clearly, we have for all y ∈ YLY that B ⊆ π−1y is a Li-Yorke scrambled set for π rel. Rπ. The
+proof is complete.
+Therefore if π: X → Y is a nontrivial Bronstein extension of minimal metric flows and if π
+has no nontrivial equicontinuous factors, then π is Li-Yorke chaotic in the sense of Theorem 5.5
+(i.e., there is a residual set YLY in Y).
+By Theorem 3.7 and Theorem 5.5, it follows that if YLY is defined by Theorem 5.1, then there
+exists, for all y ∈ YLY, a uncountable Li-Yorke scrambled set in π−1y for π rel. M∞. When Y is a
+singleton set, the Bronstein condition on π in Theorem 5.5 may be removed as follows:
+22
+
+5.6 Proposition (cf. [14, Prop. 3.11]). Let X be a weak-mixing, nontrivial, metric dynamic.
+Then X has a dense uncountable Li-Yorke scrambled set (for π: X → {pt}) rel. X × X.
+Proof. Since X × X is a compact metric space, R := Trans(X × X ) is residual in X × X so that
+R ∩ ∆X = ∅. Moreover, as X × X → X defined by (x, x′) �→ x is an open onto map, it follows by
+Lemmas 5.3 and 5.4 that there exists a dense uncountable set B in X such that if x, x′ ∈ B and
+x � x′, then T(x, x′) = X × X. Since X � {pt}, ∆X � X × X and B is a Li-Yorke scrambled set for
+π: X → {pt} rel. X × X. The proof is complete.
+5.7 Proposition. Let π: X → Y be a nontrivial extension of minimal compact metric flows.
+Suppose π is Ellis weakly mixing and non-proximal. Then there exists a residual set YLY in Y such
+that for every y ∈ YLY, π−1y contains a uncountable Li-Yorke scrambled set for π rel. M∞.
+Proof. This follows easily from Theorem 5.5 and Lemma 3.11.
+Notice that if X is an Ellis weak-mixing minimal flow, then it is weakly mixing (cf. [25, 3]).
+However, M∞ ⊊ X × X in general even if X is weakly mixing.
+6. M-flows need not be point-transitive and examples
+If X is an M-dynamic, then it is syndetically transitive; that is, NT(U, V) is a syndetic set in
+T for all non-empty open sets U, V in X. It is well known that if X is not separable, then a T.T.
+flow need not be point-transitive (cf. [15, Exa. 4.17]). As a matter of fact, an M-flow need not be
+point-transitive too.
+6.1 Example. Let X = YT be the space of all functions f : T → Y, continuous or not, equipped
+with the pointwise convergence topology, where Y is a compact Hausdorff space and T is an
+infinite discrete group with identity e. Then X is a compact Hausdorff space. Given t ∈ T and
+f ∈ X, define f t : T → Y by τ �→ f(τt). We now define the right-translation flow X on X with
+the phase group T as follows: T × X → X, (t, f) �→ t f := f t. Then:
+(1) X is syndetically and thickly transitive.
+Proof. For all non-empty open subsets U, V of Y and τ1, . . ., τn, s1, . . ., sn in T, let
+U = [{τ1, . . ., τn}, U] = { f ∈ X | f(τi) ∈ U, i = 1, . . ., n}
+V = [{s1, . . ., sn}, V] .
+Let
+T0 =
+�
+s−1
+i τj | i = 1, . . ., n; j = 1, . . ., n} ∪ {s−1
+i sj | i = 1, . . ., n; j = 1, . . ., n
+�
+and
+T1 = T \ T0.
+Since T is an infinite discrete group and T0 is finite, it is easy to check that T1 is syndetic and
+thick in T. Indeed, if K = {e}∪t0T −1
+0 for some t0 ∈ T1, then Kt∩T1 � ∅ for all t ∈ T = T0 ∪T1 so
+T1 is syndetic in T. If F is any finite subset of T, then there exists some t ∈ T such that Ft ⊆ T1;
+otherwise, there exists a k ∈ F and t � s in T with kt, ks ∈ T0 and kt = ks so that t = s.
+Thus for all t ∈ T1, we have {s1t, . . ., snt} ∩ {τ1, . . ., τn, s1, . . ., sn} = ∅. Now choose f ∈ X
+such that f(τi) ∈ U and f(sit) ∈ V for 1 ≤ i ≤ n. Thus f ∈ U and f t ∈ V so that NT(U, V) ⊇ T1.
+This proves our assertion.
+23
+
+Since every syndetic subset of T intersects non-voidly every thick subset of T, thus we have in
+fact concluded the following fact:
+(2) X is weakly mixing.
+(3) Let Y be a non-separable space and T a countable discrete group. Then X is not point-
+transitive, namely, Trans(X ) = ∅; and, Equi (X ) = ∅.
+Proof. First X has no countable dense subset. Because X is not separable and T is countable, X
+is not point-transitive. Since X is T.T., Equi(X ) ⊆ Trans(X ) = ∅.
+(4) Let T = Z or R with e = 0, then X is an M-flow.
+Proof. Let f ∈ X and k > 1. Define a periodic function ˜f : T → Y with period 2k + 1 such that
+˜f (t) = f(t) for −k ≤ t ≤ k. So X has a dense periodic points set and X is an M-flow by (1).
+Moreover, as a matter of fact we have concluded the following fact:
+(5) Let T = Z or R with e = 0, then X × X is an M-flow.
+Here R is thought of as a discrete group. This completes the construction of our Example 6.1.
+We notice that even for X is a compact metric space, a point-transitive semiflow need not be
+T.T. in general. Let us consider a simple counterexample as follows.
+6.2 Example. Let T = Q+ be the additive semigroup of nonnegative rational numbers, and,
+let X = R+ ∪ {+∞} be equipped with the usual one-point compactification topology. Define a
+compact semiflow X as follows: T × X → X, (t, x) �→ t + x. Clearly, Trans(X ) = {0} so X
+is point-transitive. But X is not T.T., for Trans(X ) is not a dense Gδ-subset of X. In fact, for
+open subsets U = (10, 12) and V = (5, 6) of X, we have NT(U, V) = ∅.
+6.3 Example. Let
+k ≥ 2 be an integer and Z
+k = Z/ kZ endowed with the discrete topology. Let
+Γ be an infinite set, ϕ: Γ → Γ a function, and w = (wγ)γ∈Γ ∈ ZΓ
+k with wγ � 0. Then we shall
+consider the “weighted shift” induced by ϕ and w:
+σ: ZΓ
+k → ZΓ
+k,
+x = (xγ)γ∈Γ �→ σx = (wγxϕγ)γ∈Γ, i.e., σxγ = wγxϕγ ∀γ ∈ Γ.
+See [41]. Here the phase space ZΓ
+k is equipped with the product topology. The classical case is
+that Γ = Z+ or Z, wγ ≡ 1, and ϕ: γ �→ γ + 1 (see Example 6.1). Note that
+σnxγ = wγ · · · wϕn−1γxϕnγ
+∀x ∈ ZΓ
+k, γ ∈ Γ and n ≥ 1.
+An element k ∈ Z
+k is said to be “invertible” iff there exists an element in Z
+k, denoted k−1, such
+that kk−1 = 1 (mod
+k). For example, 2−1 = 3 and 3−1 = 2 in Z5. We say w ∈ ZΓ
+k is “invertible” if
+wγ is invertible for all γ ∈ Γ. A point γ ∈ Γ is called “periodic” under (ϕ, Γ), denoted γ ∈ Per(ϕ),
+iff ϕτγ = γ for some integer τ ≥ 1. By Per (σ) we denote the set of periodic points under (σ, ZΓ
+k).
+6.3a Lemma (cf. [41, Thm. 4.1] by a complicated proof). Let ϕ: Γ → Γ is 1-1 and let w be
+invertible. Then:
+(1) σ: ZΓ
+k → ZΓ
+k is continuous onto.
+(2) Per(σ) is dense in ZΓ
+k.
+24
+
+Proof.
+(1): The continuity is obvious. Now let y = (yγ)γ∈Γ ∈ ZΓ
+k. Put xγ = 0 if γ � ϕ(Γ), and, xγ = w−1
+α yα
+if α ∈ Γ and γ = ϕα. Then x = (xγ)γ∈Γ ∈ ZΓ
+k such that σx = y.
+(2): Let z = (zγ)γ∈Γ ∈ ZΓ
+k and let γ1, . . . , γn ∈ Γ be pairwise distinct. It suffices to show there is a
+σ-periodic point in the clopen cylinder neighborhood of z:
+U := [zγ1]γ1 ∩ · · · ∩ [zγn]γn = {x = (xγ)γ∈Γ | xγ1 = zγ1, . . ., xγn = zγn}.
+Since ϕ is 1-1, there is no loss of generality in considering only the special case that n = 4 and
+that γ1 is a periodic point of ϕ and γ2, γ3, γ4 are non-periodic points with some integer m ≥ 1
+such that {γ3, ϕγ3, . . . , ϕmγ3 = γ2, ϕm+1γ3 = ϕγ2, ϕm+2γ3 = ϕ2γ2, ϕm+3γ3 = ϕ3γ2, . . . } and
+{γ4, ϕγ4, ϕ2γ4, . . . } have no common elements, for the other cases may be proved analogously.
+Notice that r ∈ Z
+k is invertible iff there exists an integer ℓ ≥ 1 such that rℓ = 1 (mod
+k). 1
+Then we can find an integer τ such that
+τ ≥ m + 1,
+ϕτγ1 = γ1,
+wϕτ−1γ1 · · · wγ1 = 1.
+Let Γ0 = {γ1, . . . , ϕτ−1γ1} ∪ {ϕiτγ j | j = 2, 3, 4; i ≥ 0}. We now define a point x = (xγ)γ∈Γ ∈ ZΓ
+k as
+follows:
+(a) xγ = zγ for γ ∈ {γ1, ϕγ1, . . . , ϕτ−1γ1};
+(b) xγ2 = zγ2 and xϕiτγ2 = (wϕiτ−1γ2 · · ·wγ2)−1zγ2 for i = 1, 2, . . .;
+(c) xγ3 = zγ3 and xϕiτγ3 = (wϕiτ−1γ3 · · ·wγ3)−1zγ3 for i = 1, 2, . . .;
+(d) xγ4 = zγ4 and xϕiτγ4 = (wϕiτ−1γ4 · · ·wγ4)−1zγ4 for i = 1, 2, . . .;
+(e) if γ � Γ0 such that ϕτγ = γ j for some j = 2, 3, 4, then xγ = (wϕτ−1γ · · ·wγ)−1zγj;
+(f) xγ = 0 for the other γ in Γ.
+It is clear that στx = x and x ∈ U. The proof is complete.
+6.3b Theorem (cf. [41, Thm. 5.2] for proving (σ, ZΓ
+k) to be T.T.). Let ϕ: Γ → Γ be 1-1 without
+periodic points; let w be invertible. Then (σ, ZΓ
+k) is weakly mixing as a Z+-semiflow and (σ, ZΓ
+k ×
+ZΓ
+k) is an M-semiflow.
+Proof. Let
+U1 = [k1]γ1 ∩ · · · ∩ [kn]γn,
+U2 = [i1]α1 ∩ · · · ∩ [in]αn,
+and
+V1 = [j1]δ1 ∩ · · · ∩ [jm]δm,
+V2 = [ℓ1]λ1 ∩ · · · ∩ [ℓm]λm
+be cylinder subsets of ZΓ
+k. To show (σ, ZΓ
+k) is weakly mixing, it suffices to prove that there exists
+an integer τ > 0 such that στ(U1 × U2) ∩ (V1 × V2) � ∅. We notice that since ϕ is 1-1 and has no
+periodic points, there exists an integer τ ≥ 1 such that
+{γ1, . . ., γn, α1, . . . , αn} ∩
+�
+{ϕiδ j | i ≥ τ, 1 ≤ j ≤ m} ∪ {ϕiλ j | i ≥ τ, 1 ≤ j ≤ m}
+�
+= ∅.
+1Sufficiency is obvious. Now conversely, assume r is invertible. Clearly, rℓ, for ℓ ≥ 1, is invertible. Moreover, there
+exists a sequence ℓ1, ℓ2, . . . → ∞ such that rℓ1 = rℓ2 = · · · (mod
+k). Thus, rℓ = 1 (mod
+k) for some ℓ ≥ 1.
+25
+
+Let x = (xγ)γ∈Γ ∈ U1 and y = (yγ)γ∈Γ ∈ U2 be defined as follows:
+xγ =
+
+
+
+
+
+ki
+if γ = γi for 1 ≤ i ≤ n,
+�
+wϕτ−1δi · · · wδi
+�−1 ji
+if γ = ϕτδi for 1 ≤ i ≤ m,
+0
+for other γ ∈ Γ.
+and
+yγ =
+
+
+
+
+
+i j
+if γ = α j for 1 ≤ j ≤ n,
+�
+wϕτ−1λj · · · wλj
+�−1 ℓ j
+if γ = ϕτλ j for 1 ≤ j ≤ m,
+0
+for other γ ∈ Γ.
+Clearly, στx ∈ V1 and στy ∈ V2. Thus (x, y) ∈ U1 × U2 such that στ(x, y) ∈ V1 × V2.
+This proves that (σ, ZΓ
+k) is weakly mixing. By Lemma 6.3a, (σ, ZΓ
+k × ZΓ
+k) has a dense set of
+periodic points. Thus (σ, ZΓ
+k × ZΓ
+k) is an M-semiflow.
+In fact, it is easy to verify that if (σ, ZΓ
+k) is weakly mixing, then σ is onto so that ϕ is 1-1
+without periodic points and w is invertible.
+6.3c Proposition. Let Γ be countable. Let ϕ: Γ → Γ be 1-1 without periodic points; let w be
+invertible. Then there exists a dense uncountable Li-Yorke scrambled subset of ZΓ
+k rel. ZΓ
+k × ZΓ
+k.
+Proof. This follows easily from Theorem 6.3b and Proposition 5.6.
+Recall that (x, y) ∈ ZΓ
+k × ZΓ
+k is called an “asymptotic pair” if for all ε ∈ UZΓ
+k there exists an
+integer τ > 0 such that (σnx, σny) ∈ ε for all n ≥ τ.
+6.3d Proposition. Let ϕ: Γ → Γ be 1-1 having a non-periodic point and let w be invertible.
+Then there exists a uncountable subset Θ of ZΓ
+k such that (x, y) is a proximal non-asymptotic pair
+under (σ, ZΓ
+k) for all x, y ∈ Θ with x � y.
+Proof. Let θ ∈ Γ be a non-periodic point of ϕ. Since ϕ is 1-1, then Γθ := {γ, ϕγ, ϕ2γ, . . .} is a
+countable infinite ϕ-invariant pairwise distinct subset of Γ. By considering the naturally defined
+system (σ, ZΓθ
+k ) based on Γθ
+γ�→ϕγ
+−−−−→ Γθ and Γθ
+γ�→wγ
+−−−−→ Z
+k, and by Proposition 6.3c, it follows that
+there exists a uncountable Li-Yorke scrambled set Ξ in ZΓθ
+k rel. ZΓθ
+k ×ZΓθ
+k . For all x = (xγ)γ∈Γθ ∈ Ξ,
+define x′ ∈ ZΓ
+k by x′
+γ = xγ if γ ∈ Γθ and x′
+γ = 0 if γ ∈ Γ \ Γθ. Let Θ = {x′ | x ∈ Ξ}. Clearly Θ
+possesses the desired property. The proof is complete.
+6.3e Remark. The condition w is invertible is critical for Lemma 6.3a, Theorem 6.3b, Proposi-
+tions 6.3c and 6.3d. Otherwise we can construct a counterexample as follows: Let
+k = 4, wγ ≡ 2
+(� 0) for all γ ∈ Γ, and o = (oγ)γ∈Γ ∈ ZΓ
+4 with oγ = 0 for γ ∈ Γ. Now let x ∈ ZΓ
+4. Since
+2Z4 = {0, 2}, then σx ∈ {0, 2}Γ, σ2x = o, and σnx = 0 for all n ≥ 3, for all 1-1 function
+ϕ: Γ → Γ. Thus, (σ, ZΓ
+k) has no any chaotic dynamics.
+Acknowledgments. This work was supported by National Natural Science Foundation of China
+(Grant No. 11790274) and PAPD of Jiangsu Higher Education Institutions.
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf,len=2381
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='05441v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='DS]  13 Jan 2023  On M-dynamics and Li-Yorke chaos of extensions of minimal  dynamics  Xiongping Dai  Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China  Abstract  Let π: (T, X) → (T, Y) be an extension of minimal compact metric flows with discrete phase  group T such that Rπ � ∆X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' A subflow of Rπ is called an M-flow if it is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' and it contains a  dense set of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' In this paper we mainly prove the following:  (1) π is PI iff every M-flow containing ∆X in Rπ is just equal to ∆X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (2) If π is not PI, then there exists a canonical Li-Yorke chaotic M-flow in Rπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' In particular, an  Ellis weak-mixing non-proximal extension is non-PI and so Li-Yorke chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (3) If (T, X) is a unbounded M-flow or a locally compact noncompact M-flow, then it is sensitive  on initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  In addition, we show that every syndetically distal flow is pointwise Bohr a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Keywords: Minimal flow · PI-extension · M-flow · Li-Yorke chaos · Syndetic distability  2010 MSC: 37B05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Introduction  We begin by reviewing briefly the basic notions needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let T be a Hausdorff topological  group with identity e and X a Hausdorff space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Unless specified otherwise, by (T, X) we mean a  flow [28, 23, 44, 8, 2, 15] with phase group T and with phase space X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' that is to say, there is a  continuous phase mapping T × X → X, denoted (t, x) �→ tx, such that ex = x and (st)x = s(tx)  for all x ∈ X and s, t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If T is only a topological monoid, then (T, X) will be called a semiflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  If X is compact (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' metrizable, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' ), then (T, X) will be called a compact (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' metric, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' )  dynamic where T may be a group or monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  The dynamics (T, X), (T, Y), (T, Z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' will be sometimes written as X , Y , Z , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' , respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Write X × X for (T, X × X) defined by (t, (x1, x2)) �→ (tx1, tx2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Let X be a dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' A point x ∈ X is said to be almost periodic (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=') under X if T x is a  minimal subset of X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', Ty = T x ∀y ∈ T x (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1 for the precise definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X  is called topologically transitive (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=') if for all non-empty open sets U, V in X,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1) NT(V, U) := {t ∈ T | U ∩ tV � ∅} is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Email address: xpdai@nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='cn (Xiongping Dai)  Preprint submitted to JDE (28 Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 2022)  January 16, 2023  We say X is weakly mixing if X × X is a T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Following Glasner-Weiss [26]:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2) X is referred to as an M-dynamic if X is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' and X has a dense set of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  See Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2 for a sufficient condition of M-dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Moreover, we will prove  that every syndetically distal flow is pointwise Bohr a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1 and Theo-  rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Let X and Y be two compact dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' A continuous map π: X → Y is called an ex-  tension of dynamics, denoted by π: X → Y , if πX = Y and πtx = tπx for all t ∈ T and  x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Extensions are important elements in the structure theory of minimal topological dynam-  ics (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [23, 44, 8, 2, 15, 9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' For π: X → Y , we write  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3) Rπ = {(x, x′) | x, x′ ∈ X s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' πx = πx′},  which is an invariant closed equivalence relation on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' So Rπ = (T, Rπ) is a dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Usually  one is only interested to the dynamics of Rπ that is driven by Y , for example, in skew-product  flows and random dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Let π: X → Y be an extension of minimal compact dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' As usual we say that:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4) π is proximal if Tz ∩ ∆X � ∅ ∀z ∈ Rπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5) π is almost periodic (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=') if there is no (x, x′) ∈ Rπ \\ ∆X such that T(U × V ∩ Rπ) ∩ ∆X � ∅  for all U ∈ Nx and V ∈ Nx′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Here Nx stands for the neighborhood system of x in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='6) π is a PI-extension [18, 23, 44, 8, 2, 22], provided that there exists a minimal proximal  extension ρ: X ′ → X such that π′ = π ◦ ρ: X ′ → Y can be built by successive  proximal and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' extensions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' that is, there exists a “PI-tower”:  X ′  Y  π′  ∨  < ψ1  0  Y1 < ψ2  1  π′  1  >  Y2 <  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  π′  2  >  · · <  Yγ <  ψγ+1  γ  π′  γ  >  Yγ+1 <  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  π′  γ+1  >  · · <  Yϑ  π′  ϑ  >  such that:  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' π′  ϑ is an isomorphism,  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' ψγ+1  γ  is proximal or a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' for all ordinal γ < ϑ, and  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' if γ with γ ≤ ϑ is a limit ordinal then Yγ = lim  ←−−λ<γYλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  In the special case that ρ is an isomorphism, we say that π is strictly PI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  In particular, if X is a nontrivial minimal weakly mixing compact dynamics with T nilpotent  ([9, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='8]), then X → {pt} is not PI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Here {pt} stands for the one-point dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  A non-minimal compact M-flow is sensitive on initial conditions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [26, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3] and [14,  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' This implies that every non-minimal T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' compact flow with a dense set of periodic  points is Devaney chaotic (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', [6, 11, 38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' See [46, 45] on sensitivity and its variations of  extensions of minimal metric flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  In this paper, we shall study sensitivity of noncompact M-flows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' moreover, we will define and  study the Li-Yorke chaos of extensions of minimal compact metric dynamics from the viewpoint  of M-dynamics (see §§4 and 5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' and prove that every non-PI extension of minimal compact  dynamics can canonically induce M-dynamics (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We shall improve Bronstein’s  intrinsic characterization of PI-extensions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Moreover we shall consider three  examples in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  We now conclude our Introduction with a remark on an important open problem of Robert  Ellis (1970s), which is relative closely to the definition of M-dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X be a T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' pointwise a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' compact dynamic, where the phase space X is  non-metrizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Question (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Ellis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [15, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 263]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Is X a minimal dynamic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Using metric approaches there are some confirmative answers ((1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7a), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7b), and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7c)) to Ques-  tion above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7a) If T is σ-compact, then X is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' See [9, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1] for T a semigroup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [15, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='23] for T a countable  group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7b) If X is distal (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='0f), then X is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' See [4, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='31] for T a semigroup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [16, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='9] and [15, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='24] for T  a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7c) If θ: X → Z is a distal extension with Z a minimal compact dynamic (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='0f),  then X is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' See [9, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7d) If X is weakly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' (Gottschalk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', x �→ T x is continuous), then X is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7e) If X is locally a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1), then X is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' First we claim that P(X ) = RP(X ) (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='0a) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='0b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' In fact, if X is a flow,  this is [17, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 13(4)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Now assume X is a semiflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let Z ⊂ X and x ∈ X such that  x is distal from Z (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', there is an index α ∈ UX with (tx, tz) � α ∀t ∈ T, z ∈ Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Take  β ∈ UX with β3 ⊆ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Pick U ∈ Nx and a syndetic set A in T such that AU ⊆ β[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Clearly,  we have for all a ∈ A, y ∈ U and z ∈ Z that (ay, az) � β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Next, select a compact set  K in T such that Kt ∩ A � ∅ ∀t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Take γ ∈ UX so small that Kγ ⊆ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Now for all  y ∈ U, z ∈ Z and t ∈ T, we have that (ty, tz) � γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' for otherwise, we can pick some k ∈ K  with a = kt ∈ A so that (ay, az) ∈ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' This implies that if (x, x′) � P(X ), then there exist  U ∈ Nx and V ∈ Nx′ such that U is distal from V under X , and further, (x, x′) � RP(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Thus, P(X ) = RP(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If M1 � M2 are two minimal sets in X, then by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' it follows that  there is a pair (x1, x2) ∈ M1 × M2 such that (x1, x2) ∈ RP(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then (x1, x2) ∈ P(X ) and  T x1 = T x2, contrary to M1 � M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus, X is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7f) If X is a pointwise regularly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' flow (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1), then X is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By [28, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='18], (T, T x) is minimal regularly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' for all x ∈ X so that X is  distal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7b), X is a minimal regularly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Standing notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' In a non-metric space, the convergence “x j → x” is always under the sense of net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By Nx we will denote the neighborhood system of a point x in the ambit containing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is a Hausdorff uniform space, UX stands for a compatible symmetric uniformity structure  of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' AP-Transitive relations and stability  We shall present a sufficient necessary condition for a subdynamic of Rπ to be an M-dynamic  in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Moreover, we shall study in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2 the stability of noncompact flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' AP-Transitive relations  Let π: X → Y be an extension of compact dynamics, where Y is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We first need  to introduce a relation on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let L ⊆ Rπ, L � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We say that  L is an AP-transitive relation for π on X, provided that L is a reflexive symmetric relation on  X such that if (x1, x2), (x2, x3) ∈ L are both a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points, then (x1, x3) ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Clearly, an invariant closed AP-transitive relation for π on X need not be an equivalence  relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Of course, if X is a π-distal extension of Y , then every AP-transitive relation for π is  an equivalence relation on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Let I be a minimal left ideal of the Stone-ˇCech compactification βT and J = {u ∈ I | u2 = u}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If  L = {z ∈ Rπ | z is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='} such that whenever u, v ∈ J with ux = vx for some x ∈ X then ux′ = vx′  for all x′ ∈ L[x] (= {x′ ∈ X | (x, x′) ∈ L}), then L is an AP-transitive relation for π on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let (x1, x2) and (x2, x3) ∈ L be a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then there are u, v ∈ J with u(x1, x2) = (x1, x2) and  v(x2, x3) = (x2, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' So ux2 = vx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then ux1 = vx1 and (x1, x3) = v(x1, x3) ∈ Rπ is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' so that  (x1, x3) ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Let Pπ and RPπ be the π-relative proximal and regionally proximal relations on X, respec-  tively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' that is,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='0a)  Pπ = {˜x ∈ Rπ | ∃ t j ∈ T, ˜x′ ∈ ∆X s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' t j ˜x → ˜x′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='0b)  RPπ = {˜x ∈ Rπ | ∃ ˜x j ∈ Rπ, t j ∈ T, ˜x′ ∈ ∆X s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' ˜x j → ˜x, t j ˜x j → ˜x′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  In the case of Y = {pt}, we will write P(X ) = Pπ and RP(X ) = RPπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let L ⊆ Rπ, we then define  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='0c)  Pπ|L = Pπ ∩ L  and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='0d)  RPπ|L = {˜x ∈ L | ∃ ˜x j ∈ L, t j ∈ T, ˜x′ ∈ ∆X s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' ˜x j → ¯x, t j ˜x j → ˜x′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  In the case L = Rπ, Pπ|L = Pπ and RPπ|L = RPπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Then:  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='0e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' π is proximal iff Pπ = Rπ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='0f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' a point x ∈ X is π-distal iff Pπ[x] ∩ T x = {x};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' π is distal iff Pπ = ∆X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' in the case P = {pt},  X is distal iff P(X ) = ∆X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='0g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' π is distal-equicontinuous (d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [9]) iff RPπ = ∆X iff π is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Here the π-proximal cell of x ∈ X is defined by Pπ[x] = {x′ | x′ ∈ X, (x, x′) ∈ Pπ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Notice that d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  extension is also called a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' extension or isometric extension in flows (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', [23, 44, 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1 Theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [33] [44] for L = Rπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let π: X → Y be an extension of minimal compact  flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Suppose L ⊆ Rπ is a closed invariant AP-transitive relation on X such that  (a) L has a dense set of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points and  (b) RPπ|L = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Then (T, L) is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' that is, (T, L) is an M-flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Since L is not necessarily an equivalence relation on X, so T × X/L → X/L, defined by  (t, L[x]) �→ L[tx], need not be a well-defined phase mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' In view of this, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1 is not  a corollary of the classical McMahon-Veech theorem of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' In fact, using Ellis’ algebraic theory  we can obtain the semiflow version of the above theorem as follows:  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2 Theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [9] for L = Rπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let π: X → Y be an extension of minimal compact semi-  flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Suppose L ⊆ Rπ is a closed invariant AP-transitive relation on X such that  (a) L has a dense set of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points and  (b) RPπ|L = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Then (T, L) is an M-semiflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Before proving Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2 we need to introduce some terms for our convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' In the  sequel, T is thought of as a discrete monoid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' and let I be any fixed minimal left ideal in βT and  J = {u ∈ I | u2 = u}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By Aut (I ) we denote the set of automorphisms of the universal minimal  compact dynamic I = (T, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Put  Γ[α] = {(m, αm) | m ∈ I}  ∀α ∈ Aut (I ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  The so-called τ-topology on Aut (I ) is defined as follows: Let αi ∈ Aut (I ) be a net and let  α ∈ Aut (I ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We say αi →τ α iff for every m ∈ I there exists a net mi → m in I such that  αimi → αm in I (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [9, Appendix A]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Under the τ-topology Aut (I ) is a compact T1-space (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [23, 2] and [9, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let F  be a τ-closed subgroup of Aut (I );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' then its derived subgroup is defined as follows:  F′ = {α ∈ F | ∃ δi ∈ F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' δi →τ α & δi →τ idI}  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [9, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Here F′ measures clearly the degree to which the τ-topology on F fails to  be a Hausdorff space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' At first we can construct a CD of minimal compact semiflows and ho-  momorphisms as follows:  I  πX > X  Y  π  ∨  πY  >  and set  F = {α ∈ Aut (I ) | πY = πY ◦ α};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  that is, the Ellis group of Y rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' πY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let  L = {α ∈ F | (πX × πX)Γ[α] ⊆ L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  5  Notice that Γ[α] consists of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points of Rπ for every α ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since L is an AP-transitive relation  on X, (α−1m, m) = (α−1m, α(α−1m)) ∀m ∈ I, and F is a group, hence L is a subgroup of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Moreover, L is τ-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Indeed, if α j ∈ L and α j →τ α, then there exists a net m j → m in I  such that α jm j → αm so (πX × πX)(m, αm) ∈ L and α ∈ L, for L is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus, L is a τ-closed  subgroup of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Let (x, x′) ∈ L be an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' There exists u ∈ J with u(x, x′) = (x, x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since (x, x′) ∈ RPπ|L  and L has a dense set of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points, by a standard argument (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', [9, Proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3])  we can select (m, m′), (m j, m′  j), (n, n′) in RπY ⊆ I × I and t j ∈ T with  (πX × πX)(m, m′) = (x, x′), (πX × πX)(m j, m′  j) ∈ L, πXn = πXn′,  and  (m j, m′  j) → (m, m′), t j(m j, m′  j) → (n, n′)  such that (m, m′) and (m j, m′  j) are all a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' under (T, I × I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then by regularity of I , there exist  α j, γ ∈ L such that  m′ = γm, m′  j = α jm j, (m j, α jm j) → (m, γm), (t jm j, α jt jm j) → (n, n′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Now since πXn = πXn′, we can take n′′ ∈ I such that (n′, n′′) is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', n is proximal with n′′  and πXn′′ = πXn′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus, there exists some ξ ∈ Aut (I ) with n′′ = ξn′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then πXξn′ = πXn′ so  πX ◦ ξ = πX, ξ ∈ L, and (m j, ξα jm j) → (m, ξγm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Furthermore, there is a net sj ∈ T such that  (sjm j, ξα jsjm j) → (n, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus, ξα j →τ ξγ, ξα j →τ idI, ξα j ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' So ξγ ∈ L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  To sum up, we have concluded that if (x, x′) ∈ L is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', then there exists an α ∈ L with  (x, x′) ∈ (πX × πX)Γ[α] such that there is a net δ j ∈ L with δ j →τ α and δ j →τ idI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' that is, α ∈ L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Since L′ is a τ-closed subgroup (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [9, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2]), hence if α, γ ∈ L′, then there exists a  net δ j ∈ L such that δ j →τ α and δ j →τ γ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [9, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Now let ¯x, ¯w be two a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points in L, and, let U and V be two neighborhoods of ¯x and ¯w in  L, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then we can take α, γ ∈ L′ and m, n ∈ I such that (πX × πX)(m, αm) = ¯x and  (πX × πX)(n, γn) = ¯w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' So there exists a net δ j ∈ L such that δ j →τ α and δ →τ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Further, there  are nets m j → m and n j → n in I such that δ jm j → αm and δn j → γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since (πX × πX)(m j, δ jm j)  and (πX ×πX)(n j, δ jn j) lie in a same minimal subset of L, this implies that NT(U, V) � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' As L has  already a dense set of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points, (T, L) is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' and an M-semiflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  As a matter of fact, if (T, L) is an M-semiflow, then conditions (a) and (b) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2 are  clearly fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Lyapunov stability and Bohr/regular almost periodicity  Let X be an arbitrary flow, not necessarily minimal, with T a topological group not nec-  essarily discrete and with X a uniform Hausdorff space not necessarily compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We will now  consider conditions under which an (regularly) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point is Lyapunov stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Basic definitions  We first introduce some notions needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Let S and A be subsets of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then S is said to be thick if for all compact subset K of T there  exists an element t ∈ T such that Kt ⊆ S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' A is called a syndetic subset of T if there exists a  compact subset K of T such that T = K−1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  6  It is easy to check  A subset of T is syndetic iff it intersects non-voidly every thick subset of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Recall that:  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' A point x ∈ X is almost periodic (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=') under X iff NT(x, U) is a syndetic subset of T for  every U ∈ Nx in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' A point x ∈ X is locally a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' under X iff for every U ∈ Nx there exists a V ∈ Nx and a  syndetic subset A of T such that AV ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We say X is locally a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' if it is pointwise locally  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' under X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X is called a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' if for every α ∈ UX there exists a syndetic set A in T such that Ax ⊆ α[x]  for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' A point x ∈ X is Bohr a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' under X if (T, T x) is an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We say X is  pointwise Bohr a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' iff each point of X is a Bohr a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point under X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X is called equicontinuous if given ǫ ∈ UX and x ∈ X there exists an index δ ∈ UX such  that t(δ[x]) ⊆ ǫ[tx] for all t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X is said to be uniformly equicontinuous if for all ε ∈ UX  there exists an index δ ∈ UX with Tδ ⊆ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X is regularly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' at x ∈ X, or x is a regularly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point under X , iff NT(x, U) contains a  syndetic normal closed subgroup of T for all U ∈ Nx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We say X is regularly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' iff given  ε ∈ UX there exists a syndetic normal closed subgroup A of T such that Ax ⊆ ε[x] for all  x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X is called point-regularly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' iff there exists a regularly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point x such that  T x = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  If X is compact, then x ∈ X is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' under X iff T x is minimal under X ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' and, X is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' iff it  is uniformly equicontinuous iff it is equicontinuous iff RP(X ) = ∆X (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  We say that  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X is thickly stable iff for every index ε ∈ UX and all point x ∈ X there exists an index  δ ∈ UX and a thick subset S of T such that s(δ[x]) ⊆ ε[sx] for all s ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Here δ and S  depend on the choice of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X is thickly regularly stable iff for every index ε ∈ UX and all point x ∈ X there exists an  index δ ∈ UX and a thick subsemigroup S of T such that s(δ[x]) ⊆ ε[sx] for all s ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X is uniformly thickly regularly stable iff for every index ε ∈ UX there exists an index  δ ∈ UX and a thick subsemigroup S of T such that S δ ⊆ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X is syndetically stable provided that for every index ε ∈ UX and all point x ∈ X there  exists an index δ ∈ UX and a syndetic subset S of T such that s(δ[x]) ⊆ ε[sx] for all s ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  X is uniformly syndetically stable iff for every index ε ∈ UX there exists an index δ ∈ UX  and a syndetic subset S of T such that S δ ⊆ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X is syndetically distal if for every x ∈ X and ε ∈ UX there exists a syndetic subset S of  T and an index δ ∈ UX such that if y � ε[x] then ty � δ[tx] for all t ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X is uniformly  syndetically distal if for every ε ∈ UX there exists a syndetic subset S of T and an index  δ ∈ UX such that if y � ε[x] then ty � δ[tx] for all t ∈ S and x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Let S ⊆ T and X0 ⊆ X be non-empty sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Following [19, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 2] and [8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='11] we  say that:  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X is Lyapunov S -stable iff for every index ε ∈ UX and all x ∈ X there exists an index δ ∈ UX  such that t(δ[x]) ⊆ ε[tx] for all t ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X is uniformly Lyapunov S -stable iff for every index ε ∈ UX there exists an index δ ∈ UX  such that S δ ⊆ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  7  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X is Lyapunov S -stable at L ⊆ X w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X0 iff for every index ε ∈ UX and all x ∈ L there  exists an index δ ∈ UX such that t(δ[x] ∩ X0) ⊆ ε[tx] for all t ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Clearly the uniformly Lyapunov T-stable is exactly the uniformly equicontinuous as in IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If  X is compact, then a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Notice that comparing with “Lyapunov S -stable” and “uniformly Lyapunov S -stable” of  England [19, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 2] and Bronstein [8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='11], S may vary with ε in our thick and  syndetic stability cases i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' In addition, if T is not abelian, the inverse of  a syndetic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' thick) set need not be syndetic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' thick).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  The following important result is due to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Gottschalk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' However, we will give an alterna-  tive simple proof here for reader’s convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3 Lemma (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [27];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' see also [8, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='23] and [2, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' A compact flow X is equicon-  tinuous (or equivalently Lyapunov T-stable) iff it is an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Assume X is an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Clearly, it is uniformly syndetically stable and distal so that  RP(X ) = P(X ) = ∆X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus X is equicontinuous for RP(X ) = �  α∈UX Tα and X compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Conversely, suppose X is equicontinuous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' and then, X is distal for P(X ) = ∆X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let ε ∈ UX  and β ∈ UX with β3 ⊂ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then there exists an index δ ∈ UX with δ ⊂ β such that Tδ ⊂ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since X  is compact, there exists a finite set {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', xn} in X such that X = δ[x1]∪· · ·∪δ[xn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' As (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', xn)  is an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point in Xn, there is a syndetic subset A of T such that A(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', xn) ⊆ δ[x1]×· · ·×δ[xn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Now for y ∈ X we have for some 1 ≤ i ≤ n that (y, xi) ∈ δ, A(y, xi) ⊂ β and Axi ⊂ δ[xi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus  Ay ⊂ β3[y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Hence Ay ⊂ ε[y] for all y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thick stability  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X be a minimal flow and S a thick subset of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is Lyapunov S -stable at  some point x ∈ X w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X, then X is thickly stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5 Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X be a thickly stable flow with T an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If x ∈ X is an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point  under X , then (T, T x) is an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' subflow of X with discrete phase group T, and moreover,  (T, T x) is an equicontinuous subflow of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since X is a regular space so that the orbit closure of an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point is minimal (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [8,  Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3]), we can assume X = (T, T x) is minimal without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Moreover,  we may assume x is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' under X with T a discrete group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let ε, α ∈ UX with α2 ⊆ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since  X is thickly stable at x by hypothesis, there exists an index γ ∈ UX and a thick subset S of  T such that s(γ[x]) ⊆ α[sx] for all s ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Take an index δ ∈ UX with δ2 ⊆ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since x is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  under X , there exists a discretely syndetic subset A of T such that Ax ⊆ δ[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' To show that  X is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', it suffices to prove that Atx ⊆ ε[tx] for all t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' For this, let t ∈ T, a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' There  exists an index σ ∈ UX with σ ⊂ δ such that a(σ[x]) ⊆ δ[ax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then there is a syndetic subset  B of T such that Bx ⊆ σ[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since S is thick, so is S −1 in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since t−1B is syndetic in T, thus  t−1B ∩ S −1 � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then there are elements b ∈ B and s ∈ S such that t−1b = s−1 so b−1t = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By  bx ∈ Bx ⊆ σ[x], it follows that abx ∈ δ[ax] ⊆ γ[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By b−1t = s ∈ S and commutativity of T,  it follows that atx = (b−1t)(ab)x ∈ α[b−1tx].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Moreover, b ∈ B implies bx ∈ δ[x] ⊆ γ[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Hence  tx = (b−1t)bx ∈ α[b−1tx].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Therefore, (atx, tx) ∈ α2 ⊆ ε for all t ∈ T and all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Next we shall prove that X is equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let ε ∈ UX there exists an index δ1 ∈ UX and  a syndetic set A ⊆ T such that Aδ1 ⊆ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Select a compact set K ⊆ T such that KA = T = AK  8  (for T is abelian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Given x0 ∈ X, there exists an δ ∈ UX such that kδ[x0] ⊆ δ1[kx0] for all k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Thus, tδ[x0] ⊆ ε[tx0] for all t ∈ T and then X is equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5a Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is a locally a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' thickly stable compact flow with T abelian, then X is an  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since X is locally a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', X is pointwise a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' and P(X ) = RP(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then RP(X ) = ∆X by  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' This together with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3 proves Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5b Corollary (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 2] or [8, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X be a flow with T abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let S be  a thick subset of T and x ∈ X an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point under X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is Lyapunov S -stable at x w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' T x,  then x is a Bohr a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point under X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  When T is not necessarily an abelian topological group, we can conclude the following result  with “x regularly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.” instead of “a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.” under X :  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='6 Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X be a flow with X a Baire space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is thickly stable and it is regularly  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' at some point x ∈ X, then (T, T x) is a pointwise regularly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' and equicontinuous subflow  of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We can assume X = T x without loss of generality for T x is a minimal subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let  ε, α ∈ UX with α2 ⊆ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' There exists an index γ ∈ UX and a thick subset S of T such that  sγ[x] ⊆ α[sx] for all s ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Take an index δ ∈ UX with δ2 ⊆ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since x is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' under X , there is a  discretely syndetic set A in T with Ax ⊆ δ[x] and int Ax � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' To show X is Lyapunov T-stable  at x w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X, it suffices to prove that tax ∈ ε[tx] for all t ∈ T and every a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  For that, let t ∈ T and a ∈ A be arbitrarily given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Select an index σ ∈ UX with aσ[x] ⊆ δ[ax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Since x is regularly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' under X , there exists a syndetic normal closed subgroup B of T such  that Bx ⊆ σ[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus aBx ⊆ δ[ax] ⊆ γ[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since S is thick and tB is syndetic in T, tB ∩ S � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Thus tb = s for some b ∈ B and s ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then  tax = (tb)(b−1a)x ∈ (tb)(aB)x ⊆ α[tbx]  and  tx = (tb)b−1x ∈ (tb)Bx ⊆ (tb)γ[x] ⊆ α[tbx].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Whence (tax, tx) ∈ α2 ⊆ ε for all t ∈ T and all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Further by minimality of X, X is Lyapunov T-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' On the other hand, X is regularly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  at each point of T x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then by Lyapunov stability, each point of X = T x is regularly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' under X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='6a Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is a point-regularly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' thickly stable compact flow, then X is a (uni-  formly) regularly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Use Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='6 and the uniform equicontinuity of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='6b Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X be a flow with X a Baire space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let S be a thick subset of T and x ∈ X  a regularly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point under X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is Lyapunov S -stable at x w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' T x, then X is Lyapunov  T-stable at x w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' T x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Note that a point-regularly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' compact flow need not be a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' for example, [28, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  So condition “X is thickly stable” is critical for Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Lyapunov thick-subsemigroup stability  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7 (P-limit set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let x ∈ X, and P ⊆ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The P-limit set of x, denoted Px, is defined to be  �  t∈T Ptx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' that is, y ∈ Px iff for every t ∈ T there is a net p j ∈ P such that p jtx → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Each point of  Px is called a P-limit point of x under X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' See [28, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='33] or [8, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  If x ∈ X is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' under X and P a thick set in T, then x ∈ Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Note here that Px is different from the proximal cell P[x] under X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7a Lemma (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [28, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='07] for T a generative group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If P is a thick semigroup in T and  K a compact subset of T with e ∈ K, then �  k∈K kP is a thick semigroup in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Write Q = �  k∈K kP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since P ∩ kP is a semigroup for all k ∈ K, Q is a semigroup in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Let C be a compact subset of T and set D = C ∪ K−1C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since D is compact, we can select t ∈ T  such that Dt ⊆ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Now we have for k ∈ K that C ⊆ D ∩ kD and Ct ⊆ Dt ∩ kDt ⊆ P ∩ kP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus  Ct ⊆ Q and Q is thick in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7b Lemma (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [8, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10] for T an abelian group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let P be a normal thick subsemigroup  of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then based on X :  (1) Px is closed and T-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (2) If Px is compact, then Px � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (3) Px = �  t∈P Ptx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Here P ‘normal’ means ‘Pt = tP ∀t ∈ T’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (1): Clearly, Px is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let y ∈ Px and s, t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since y ∈ Ps−1tx, so sy ∈ sPs−1tx = Ptx and  sy ∈ Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus Px is T-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (2): Suppose Px is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', tn ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7a, P ∩ t1P ∩ · · · ∩ tnP is a thick  semigroup in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus �n  i=1Ptix = �n  i=1tiPx ⊇  ��n  i=1tiP  �  x � ∅ and Px � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (3): Since P is normal thick in T, P−1 is also thick in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then for all t ∈ T, there is some st ∈ P  with stt ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' So  Px ⊆  �  p∈PPpx ⊆  �  t∈T Psttx ⊆  �  t∈T Ptx ⊆ Px,  The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='8 Lemma (a special case of [4, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X be a compact flow and S a subsemigroup  of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is Lyapunov S -stable, then it is Lyapunov ⟨S ⟩-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Here ⟨S ⟩ is the subgroup of T  generated by S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='9 Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X be a compact flow and S a thick subsemigroup of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is Lyapunov  S -stable, then X is an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='8, (⟨S ⟩, X) is an equicontinuous flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since S is thick in T, so is ⟨S ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' How-  ever, every thick subgroup of T coincides with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus ⟨S ⟩ = T and X is equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since  X is compact, X is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='9a Corollary (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [32] [36, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='12] for T = R and S = R+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 3] [8, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='12]  for T abelian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X be a flow and let S be a thick normal subsemigroup of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let x ∈ X such  that S −1x is compact and that (T, T x) is a uniformly Lyapunov S -stable subflow of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then  (T, T x) is a compact minimal a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' subflow of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  10  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let P = S −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' then P ⊳ T and P is thick in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7b, Px is an T-invariant closed  non-empty compact subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then we can take a T-minimal set M with M ⊆ Px ⊆ T x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Since (T, T x) is uniformly Lyapunov S -stable, (T, T x) is also uniformly Lyapunov S -stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let  y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since M ⊆ S −1x, there is a net sn ∈ S such that s−1  n x → y in T x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By uniform S -stability,  sny → x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus T x = M is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='9, (T, T x) is equicontinuous and so  an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' minimal compact subflow of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='9b Corollary (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 4] [8, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='14] for T abelian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X be a flow and let P be  a thick normal subsemigroup of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then following statements are equivalent:  (1) x is Bohr a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' under X with T x compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (2) x ∈ Px, Px is compact, and X is uniformly Lyapunov P-stable at T x w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Assume (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Clearly, Px = T x is compact minimal by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7b, and (T, T x) is  equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus x ∈ Px and T x is uniformly Lyapunov P-stable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then (1) implies  (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Conversely assume (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7b, T x ⊆ Px and T x = Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Now applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='9  with T x and P instead of X and S respectively, (T, Px) is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since x ∈ Px, so x is Bohr a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' and  (2) implies (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='9c Corollary (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 5] for T abelian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X be a flow and let P be a thick normal  subsemigroup of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Suppose y ∈ X such that:  (1) The P-limit set of y, Py, is compact non-empty with Px = Py for all x ∈ Py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (2) X is Lyapunov P-stable at Ty w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Then Py is the closure of a Bohr a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Condition (1) implies that Py is a minimal set for X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since Py is compact, condition (2)  implies that (T, Py) is uniformly Lyapunov P-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='9, (T, Py) is (Bohr) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  so that Py is the closure of a Bohr a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' orbit of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Syndetic distality  It is easy to verify that every uniformly distal flow is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [4, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7(2)] and [9,  Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We now improve this result as follows:  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10 Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is a syndetically distal compact flow, then it is pointwise Bohr a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' First we claim that X is distal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let x ∈ X and y ∈ P[x] with x � y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then there is an  index ǫ ∈ UX with (x, y) � ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since X is syndetically distal, there exists an index γ ∈ UX  and a syndetic subset A of T such that t−1(γ[tx]) ⊆ ǫ[x] for all t ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' However, by y ∈ P[x],  B := {t ∈ T | (tx, ty) ∈ γ} is a thick subset of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By A ∩ B � ∅, take t0 ∈ A ∩ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then t0y ∈ γ[t0x]  and t−1  0 (γ[t0x]) ⊆ ǫ[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' So (x, y) ∈ ǫ, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus P(X ) = ∆X and X is a distal flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  So to show X pointwise Bohr a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', we can assume X is minimal distal, and we shall show  X is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' For this, define the strong regionally proximal relation on X by  U(X ) = {(x, y) | ∃ yn ∈ X → y and tn ∈ T s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' (tnx, tnyn) → ∆X}  and  U[x] = {y ∈ X | (x, y) ∈ U(X )}  ∀x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  11  Then  U[x] =  �  α∈UX  �  t∈Tt−1(α[tx])  ∀x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Since X is minimal distal, X admits an invariant Borel probability measure (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then it  follows from [34] that RP(X ) = U(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Therefore,  RP[x] = U[x] =  �  α∈UX  �  t∈Tt−1(α[tx])  for every x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Let x ∈ X and ε ∈ UX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' There exists an index δ ∈ UX and a syndetic subset S of T such that  s−1(δ[sx]) ⊆ ε[x] for all s ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since S is syndetic, there is a compact subset K of T such that  T = KS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Further, since X is compact and the phase mapping T × X → X is jointly continuous,  there exists an index α ∈ UX such that K−1α ⊆ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let t ∈ T be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then we can write t = ks  for some k ∈ K and some s ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus  t−1(α[tx]) = s−1(k−1(α[ksx])) ⊆ s−1(δ[sx]) ⊆ ε[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  This shows that RP[x] ⊆ ε[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since ε is arbitrary, so RP[x] = {x} and RP(X ) = ∆X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then X is  equicontinuous and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10a Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is a syndetically distal and syndetically stable compact flow, then it is an  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since X is syndetically stable, so P(X ) = RP(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10, RP(X ) = ∆X  so X is an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' flow by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10b Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is a syndetically distal T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' compact flow, then it is an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' minimal flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Recall that X is said to be uniformly syndetically stable/equicontinuous if for every index  ε ∈ UX there exists an index δ ∈ UX and a syndetic subset S of T such that S δ ⊆ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10c Corollary ([13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is a uniformly syndetically stable compact flow with T an abelian  group, then X is an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since T is abelian, X is syndetically distal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then X is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10d Question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is a uniformly syndetically stable compact flow with T non-abelian, is X  an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' flow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  We say that  X is uniformly syndetically regularly stable if for every index ε ∈ UX there exists an index  δ ∈ UX and a syndetic subsemigroup A of T such that Aδ ⊂ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  X is uniformly left-syndetically stable if for every index ε ∈ UX there exists an index δ ∈ UX  and a “left-syndetic” subset A of T such that Aδ ⊂ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  X is uniformly left-thickly stable if for every index ε ∈ UX there exists an index δ ∈ UX and  a “left-thick” subset A of T such that Aδ ⊂ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  12  Here A is left-syndetic iff there exists a compact subset K of T such that A ∩ tK � ∅ for all t ∈ T,  and, A is left-thick iff for every compact subset K of T there exists an element t ∈ T such that  tK ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10e Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is a uniformly syndetically regularly stable compact flow, then it is an  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let ε ∈ UX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We can take an index α ∈ UX such that ¯α ⊆ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Further there exists an index  δ ∈ UX and a syndetic subsemigroup A of T such that Aδ ⊆ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let S = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then S is a closed syn-  detic subsemigroup of T such that S δ ⊆ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Noting that S is a subgroup of T (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [8, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='17]),  X is syndetically distal and syndetically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus X is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof  is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10f Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is a uniformly left-syndetically stable compact flow, then it is an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let ε ∈ UX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then there exists an index δ ∈ UX and a left-syndetic subset A of T such that  Aδ ⊂ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since A is left-syndetic, there is a compact subset K of T such that T = AK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Further,  there exists an index α ∈ UX such that Kα ⊆ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus Tα = AKα ⊆ Aδ ⊂ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Hence X is an  equicontinuous and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10g Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is a uniformly syndetically distal compact flow, then it is an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X be uniformly syndetically distal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since the inverse of a syndetic set is left-syndetic,  X is clearly a uniformly left-syndetic stable flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus X is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof  is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10f is comparable with Clay’s theorem ([12, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 14] [8, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='21]): X is  uniformly left-thickly stable iff it is equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Moreover, following [4, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3], X is  equicontinuous iff for every α ∈ UX there exists a syndetic subset A of T such that A = A−1 (so  A is left-syndetic) and Ax ⊆ α[x] for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Canonical M-dynamics of non-PI extensions  Let π: X → Y be a nontrivial extension of minimal compact dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Recall that π is  weakly mixing if Rπ is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' and π is RIC if π is open such that  Rn  π = {(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' , xn) ∈ Xn | πx1 = · · · = πxn}  has a dense set of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points for all n ≥ 2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Here we shall define canonical M-dynamics  in Rπ, which will be useful for describing the π-relative unpredicted dynamics of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Moreover,  we will characterize intrinsically PI-extensions of minimal dynamics in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  In the sequel, T is thought of as a discrete space and let I be a minimal left ideal in βT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Let I = (T, I) as in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let πX : I → X and πY : I → Y be two extensions such that  πY = π ◦ πX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Associated to πX and πY we can define two groups in Aut (I ):  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' A = {α ∈ Aut (I ) | πX = πX ◦ α} and F = {α ∈ Aut (I ) | πY = πY ◦ α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Then A and F are τ-closed subgroups of Aut (T, I) with A ⊆ F, and, A = F iff π is proximal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  see [9, Appendix A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Put F0 = F, F1 = F′, F2 = F′  1, and for every ordinal o, Fo+1 = F′  o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If o is a  limit ordinal, put Fo = �  γ<o Fγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then by transfinite induction there there exists a least ordinal o  such that F0 ⊋ F1 ⊋ F2 ⊋ · · · ⊋ Fo = Fo+1 = · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Write  13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' F∞ = Fo, which is called the core of F and which is a τ-closed subgroup of F such that  AF∞ = F∞A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Then:  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' π is a PI-extension iff F∞ ⊆ A (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [23, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2] or [2, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' it may follow from  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='6 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Notice that the PI condition of π is independent of the choice of πX and πY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' So PI-extension is an  intrinsic concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Every PI-extension has “good” structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  There are many intrinsic characterizations of PI-extensions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' see [7, 35, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' In particular, “π  is PI iff every M-dynamic in Rπ is minimal” (due to Bronstein, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [7, 42, 22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' In fact, we will  show that π is PI iff every M-dynamic containing ∆X in Rπ is just equal to ∆X (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10  below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' So if π is not PI, then Rπ contains a non-minimal M-dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  However, we shall be only interested to the non-minimal M-dynamics that contains ∆X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' This  type of M-dynamic is useful for us to define chaos of non-PI extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' For this, we now put  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' M∞ = (πX × πX)Γ[F∞], where Γ[F∞] = �  α∈F∞Γ[α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Clearly, M∞ is a closed invariant symmetric subset of Rπ containing a dense set of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points of  RPπ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [9, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Although we have not shown that M∞ is an AP-transitive relation on X here, yet we can first  obtain the following:  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5 Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let π: X → Y be an extension of minimal dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If π is not PI, then (T, M∞)  is an M-dynamic in Rπ with ∆X ⊊ M∞ and RPπ|M∞ = M∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since π is not PI, F∞ ⊈ A by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3 and idI ∈ F∞ ∩ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus, ∆X ⊊ M∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let U and V be two  non-empty open sets in M∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' There exist two elements α, γ ∈ F∞ such that U ∩ (πX × πX)Γ[α] � ∅  and V ∩(πX ×πX)Γ[γ] � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' As F∞ = F′  ∞, it follows that (T, M∞) is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2)  so that (T, M∞) is an M-dynamic and then RPπ|M∞ = M∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Therefore, in view of Bronstein’s theorem and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5, if Rπ has a non-minimal M-  subdynamic, then (T, M∞) is an M-dynamic in Rπ with ∆X ⊊ M∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='6 (Ellis-Glasner-Shapiro tower).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Based on the classical Ellis-Glasner-Shapiro structure theo-  rem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [18, 23, 44, 2] and [22]), for π: X → Y , there exists the following canonical CD of  minimal compact dynamics and T-extensions:  I  X <  φ  πX  >  X∞  πX∞  >  Y  π  ∨  <  ψ  πY  >  Y∞  π∞  ∨  πY∞ − − − −  >  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ψ has a PI-tower,  φ is proximal,  π∞ is RIC weakly mixing,  g(X∞) = A,  g(Y∞) = AF∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Here A, F are as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1 and  g(X∞) = {α ∈ Aut (I ) | πX∞ = πX∞ ◦ α},  g(Y∞) = {α ∈ Aut (I ) | πY∞ = πY∞ ◦ α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Notice that (φ×φ)Rπ∞ ⊊ Rπ in general, for ψ need generally not be proximal and π need not satisfy  the Bronstein condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Moreover, π is PI iff π∞ is 1-1 onto iff π∞ is proximal iff A = AF∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  14  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7 Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let π: X → Y be a non-PI extension of minimal compact dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then  M∞ = (φ × φ)Rπ∞ and (πX × πX)Γ[F∞] is the set of all a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points of (T, M∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4 and F∞ ⊆ g(Y∞), it follows that M∞ ⊆ (φ×φ)Rπ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let (w, w′) ∈ Rπ∞ be an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since π∞ satisfies the Bronstein condition and (w, w′) ∈ RPπ∞, so by [9, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3]  we have that (w, w′) ∈ (πX∞ × πX∞)Γ[α] for some α ∈ (AF∞)′ ⊆ AF∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' This implies that (φ ×  φ)(w, w′) ∈ M∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus M∞ ⊇ (φ × φ)Rπ∞ and M∞ = (φ × φ)Rπ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Let (x1, x2) ∈ M∞ be an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By M∞ = (φ × φ)Rπ∞, we can take an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point (w1, w2)  in Rπ∞ such that φw1 = x1 and φw2 = x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Further, there exist elements α ∈ A and γ ∈ F∞ such  that (w1, w2) = (πX∞ × πX∞)(m, αγm) = (πX∞ × πX∞)(m, γm) for some m ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' So by πX = φ ◦ πX∞,  (x1, x2) = (πX × πX)(m, γm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='8 Lemma (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [5, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10] for T a group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let φ: X → Y be a proximal extension of  compact dynamics with X having a dense set of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then X is an M-dynamic iff so is  Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Necessity is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Conversely, suppose Y is an M-dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let K be an invariant  closed set in X with int K � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We need to prove K = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Suppose the contrary that K � X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then  X \\ K � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points are dense in X, so (X \\ K) ∩ t−1int K = ∅ for all t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' It is clear  that L := X \\ K = X \\ int K and t(X \\ K) ⊆ L for every t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus L is an invariant closed set  in X with int L � ∅ and K ∪ L = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' As φK ∪ φL = φX = Y, φK or φL must have a non-empty  interior in Y, and so, by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' of Y , φK = Y or φL = Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Assume φK = Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let x ∈ X \\ K be an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then we can find x′ ∈ K with φx = φx′ such  that (x, x′) is an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since φ is proximal, so x = x′ and K = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If φL = Y, then similarly it  follows that L = X, contrary to L ∩ int K = ∅ and int K � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Hence K = X and X is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' and  an M-dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='9 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let φ: Z → Y be an extension of minimal compact dynamics, let (T, N) be an  M-dynamic in Rφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then:  (1) If φ is proximal, then N = ∆Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (2) If φ is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', then N is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' (1) is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' For (2), we consider θ: N  → Y , which is induced canonically by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Then N is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' and θ-distal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7c), N is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='10 Theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [7, 42, 22] for (1) ⇔ (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let π: X → Y be an extension of minimal  compact dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then the following are pairwise equivalent:  (1) π is a PI-extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (2) Every M-dynamic in Rπ is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (3) Every M-dynamic containing ∆X in Rπ is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (1) ⇒ (2): Clearly by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3, X∞ � Y∞ in the CD of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Set ψ∞ = ψ ◦ π∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let (T, M) be an  M-dynamic in Rπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' There exists an invariant closed subset N of Rψ∞, which has a dense set of  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points such that M = (φ × φ)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='8, (T, N) is an M-dynamic in Rψ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since  ψ∞ is strictly PI, thus N is minimal by using the PI-tower of ψ∞ and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='9, and M is also  minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then (1) implies (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  15  (2) ⇒ (3): Obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (3) ⇒ (1): This follows easily from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='11 (Ellis weak-mixing extension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Recall that for π: X → Y :  X is called an π-Ellis-weak-mixing extension of Y or simply π is Ellis weak-mixing [25, 3] if  F = AF′, where A and F are defined as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  It is well known that if there is no the Bronstein condition and Y � {pt}, an Ellis weak-mixing  extension of minimal compact flows need not be a weak-mixing extension (see [25]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' and, an Ellis  weak-mixing minimal compact semiflow need not be weakly mixing (see [9, Exa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Moreover, by using the canonical Ellis-Glasner-Shapiro construction of RIC extensions [18,  23, 44, 2] we can easily obtain the following:  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='12 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If π is Ellis weak-mixing non-proximal, then it is not PI so that there exist non-  minimal canonical M-flows (T, M∞) in Rπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Notice that the definition of Ellis weak-mixing is independent of the choice of πX and πY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Moreover, a proximal minimal flow is Ellis weak-mixing and PI (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [23, 9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Li-Yorke chaos of extension, sensitivity and asymptotically a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' motions  We will introduce the definition of Li-Yorke chaotic extension in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1, provide a sufficient  condition for sensitivity on initial conditions in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2, and consider asymptotically a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' motions  in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1 (Li-Yorke chaos of extension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' In this part let π: X → Y be a nontrivial extension of  minimal compact metric dynamics, unless specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let ρ be a metric on the compact  metric space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then (x, x′) ∈ Pπ iff (x, x′) ∈ Rπ and there exists a sequence {tn}∞  n=1 in T such that  limn→∞ ρ(tnx, tnx′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Since π is nontrivial, so Rπ � ∆X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Recall that whenever T = Z+ and (x, x′) ∈ X × X with  x � x′, we say (x, x′) is a Li-Yorke pair iff  lim infn→∞ρ(nx, nx′) = 0  and  lim supn→∞ρ(nx, nx′) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  In view of this, the definition of Li-Yorke chaos has been generalized to group actions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=',  [30, 37, 14, 1] and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Moreover, Devaney’s chaos implies the Li-Yorke’s chaos (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', [29]  for Z-flow, [14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='21] and [1] for T an abelian group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We now introduce the Li-Yorke  chaos of extensions of minimal dynamics as follows:  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let L be a closed subset of X × X such that ∆X ⊊ L ⊆ Rπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (a) We say (x, x′) ∈ X × X is a Li-Yorke pair for π rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' L if (x, x′) ∈ Rπ and L ⊆ T(x, x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (b) A set S in X is called Li-Yorke scrambled for π rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' L if every pair (x, x′) ∈ S × S \\ ∆X is a  Li-Yorke pair for π rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1b Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let L = (T, L) be an M-dynamic in Rπ, and define Trans(L ) to be the set  Trans(L ) = {˜x ∈ L | T ˜x = L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  If ∆X ⊊ L, then each point of Trans(L ) is a Li-Yorke pair for π rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  16  Since (T, L) is an M-dynamic such that ∆X ⊊ L ⊆ Rπ, so if (x, x′) ∈ Trans(L ), then the set  of t ∈ T at which T x and T x′ are sufficiently close and the set of t ∈ T at which T x and T x′  are far away are both very big (it is at least syndetic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' In view of this, the Li-Yorke chaos here is  much stronger than the classical sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2 (Sensitive to initial conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X be a dynamic with X a uniform Hausdorff space not  necessarily compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X is said to be bounded if given α ∈ UX there exists a point x0 ∈ X and a compact set K  in T such that α[Kx0] = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is not bounded, then it called unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Clearly, if (X, ρ) is a metric space such that there is a point x0 ∈ X such that {ρ(x0, x) | x ∈ X}  is unbounded in R, then X is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Moreover, we can easily construct a compact metric  space that can support a unbounded pointwise a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By virtue of these, our “unbounded”  condition is then much general than the one introduced in [40, (D1)] for only metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  In fact,  (i) If X is bounded having a dense set of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points, then X is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Moreover, if X is  bounded, then it is totally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Suppose this is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then X would be bounded and thus it would be T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Indeed, let U, V be non-empty open sets in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then there exists a point x0 ∈ X and s, t ∈ T  such that sx0 ∈ U and tx0 ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Further, there exists an open W ∈ Nx0 such that sW ⊆ U  and tW ⊆ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points are dense in X, we can pick an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point, say x1, in sW  and some s1 ∈ T such that s1x1 ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' So ts1x1 ∈ V and NT(U, V) � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (ii) Let T be σ-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X is a pointwise a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' non-minimal compact dynamic, then X is  unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If this is false, then X is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' by (i) and thus, X would be minimal by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7a) in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We say that X is sensitive to initial conditions on X0, where X0 ⊆ X, if there exists an  index ε ∈ UX such that for every x ∈ X0 and all U ∈ Nx(X) there exists some y ∈ U and some  t ∈ T with (tx, ty) � ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2c Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If X0 is dense in X and X is sensitive to initial conditions on X0, then X is  sensitive to initial conditions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2d Theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [40, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1] for T = R+ or Z+ with X a unbounded metric space,  under a stronger definition of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X be a unbounded M-flow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' then X is sensitive  to initial conditions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X0 be the set consist of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then X0 is dense in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2c, it  is enough to show that X is sensitive to initial conditions on X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Suppose this is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let α,  ε ∈ UX with ε2 ⊆ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then there is an index δ ∈ UX and a point x0 ∈ X0 such that for all y ∈ δ[x0],  (ty, tx0) ∈ ε for all t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Moreover, since x0 is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' under X , there is a compact set K in T such  that KNT(x0, δ[x0]) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus  Tδ[x0] ⊆ ε[Kδ[x0]] ⊆ ε2[Kx0]  and  Tδ[x0] ⊆ α[Kx0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Since X is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', X = Tδ[x0] and X = α[Kx0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' This is contrary to that X is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof  is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  17  It should be noted that the unboundedness of X is critical for Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' for example, a  minimal equicontinuous compact flow is a nonsensitive M-flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2e Corollary (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [40, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2] for T = Z+ and R+ and RE(X ) dense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X be an M-flow  with X a complete noncompact space (like X = Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then X is sensitive to initial conditions on  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' First note that X is unbounded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' for otherwise, X would be totally bounded complete so  that X would be compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2d, X is sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' A point x ∈ X is called a Birkhoff recurrent point under X [36, 10] if given ε ∈ UX there  exists a compact set K in T such that T x ⊆ ε[Ktx] for all t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The set of Birkhoff recurrent  points of X is denoted RE(X ) in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  A Birkhoff recurrent point must be an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' In fact, in compact dynamics a point is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  iff it is Birkhoff recurrent (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [10, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' in locally compact metric flows, a point is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' iff  it is Birkhoff recurrent (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [10, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' However, in locally compact metric semiflows an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  point need not be Birkhoff recurrent (see [4, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='15(c)] for a counterexample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  So in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2d, our condition “a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points are dense in X” is generally weaker than  Seifert’s condition “RE(X ) is dense in X” in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Let X be a flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If x ∈ X such that given ε ∈ UX there exists a compact set L ⊆ T with  ε[tx] ∩ Latx � ∅ for all t, a ∈ T, then x ∈ RE(X ) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [40, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1721] for T = R+ and Z+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let s ∈ T and t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Select a ∈ T with s = at or equivalently t = a−1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then tx ∈ ε[Lsx].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  This shows that x ∈ RE(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Thus, x is Birkhoff recurrent under X iff (T, T x) is a weakly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' subflow of X (in the sense  of Gottschalk [4]: Given ε ∈ UX there exists a compact set L in T such that for all y ∈ T x,  Lt ∩ NT(y, ε[y]) � ∅ ∀t ∈ T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By Equi(X ) we denote the set of points at which X is equicontinuous/T-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' That is,  x ∈ Equi(X ) iff given ε ∈ UX there is a member U ∈ Nx such that (tx, ty) ∈ ε for all t ∈ T and  all y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' See Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2h Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X be a unbounded T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If Equi(X ) � ∅, then the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points are not  dense in X (and consequently RE(X ) is not dense in X and X is an M-flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Suppose this is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2d, X would be sensitive to initial conditions  on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' So there would be an index ε ∈ UX such that for x ∈ Equi(X ) and U ∈ Nx there exists  a point y ∈ U with (tx, ty) � ε for some t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' This contradicts equicontinuity of X at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The  proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2i Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If “a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.” points are replaced by “Birkhoff recurrent” points, then Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2d  and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2h still hold for T a semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By virtue of this, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2h is in conflict  with an open question in [40, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1725].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2j Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Using “X locally compact noncompact” instead of “X unbounded”, by the same  argument as that of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2d, we can conclude the following:  Let X be an M-flow with X a locally compact noncompact space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then X is sensitive to  initial conditions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  18  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3 (Asymptotically a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' motions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' In this part, let X be a semiflow with X a uniform Hausdorff  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We first define  (i) the ω-limit set ω[x] = {w ∈ X | ∀ U ∈ Nw and s ∈ T, ∃ t ∈ T s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' tsx ∈ U};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Clearly, for  T = R+ or Z+ and X a metric space, w ∈ ω[x] iff ∃ tn → ∞ such that tnx → w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Moreover,  the P-limit set of x, Px = ω[x], for P = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (ii) AAP(X ) = {x ∈ X | ∃ an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point y s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' ∀ ε ∈ UX, ∃ s ∈ T s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' (tsx, tsy) ∈ ε ∀t ∈ T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Clearly, for T = R+ or Z+ and (X, ρ) a metric space, x ∈ AAP(X ) iff ∃ an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' point y such  that ρ(tx, ty) → 0 as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  We call AAP(X ) the set of asymptotically a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By virtue of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2f, this is equivalent  to that defined by Seifert [39, (iv)] under compact ambit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3a Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let T be such that �  s∈F T s � ∅ for every finite set F ⊂ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If x ∈ X such that T x  is compact, then ω[x] � ∅ is compact invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By ω[x] = �  s∈T T sx and compactness, it follows easily that ω[x] � ∅ is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The  invariance is evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3b Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let T be such that �  s∈F T s � ∅ for every finite set F ⊂ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let x ∈ X such  that T x is compact and ω[x] ⊆ Equi(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then (T, ω[x]) is a minimal subsemiflow of X ,  ω[x] ⊆ RE(X ), and x ∈ AAP(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If, in addition, T is an amenable semigroup or T is a group,  then (T, ω[x]) is an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' subdynamic of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3a, ω[x] is an invariant compact nonempty subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let M be a minimal  subset of ω[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If M � ω[x], then we can take a point w ∈ ω[x]\\ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Further, there exists an index  ε ∈ UX such that ε[w] ∩ ε[M] = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Select U ∈ Ny with U ⊂ ε[M] such that if sx ∈ U  then tsx ∈ ε[M] for all t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' This is contrary to w ∈ ω[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus ω[x] is a minimal set in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' This  also shows that ω[x] ⊆ RE(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Now we will prove that x ∈ AAP(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By a theorem of Auslander-Ellis (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', [21,  Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7]), we can find a point y ∈ ω[x] such that x is proximal to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let ε ∈ UX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We note  here that since ω[x] ⊆ Equi(X ) is compact, there is an index δ ∈ UX such that (z, w) ∈ δ  and w ∈ ω[x] implies that (tz, tw) ∈ ε for all t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Now, we can take some s ∈ T such that  (sx, sy) ∈ δ, and further, (tsx, tsy) ∈ ε for all t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus x ∈ AAP(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Finally, If T is an amenable semigroup or T is a group, then by [4, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='15] it follows that  (T, ω[x]) is an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' subdynamic of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3c Corollary (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [39, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1] for d = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let T = Rd  + or Zd  + where d ≥ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' let T x be  compact such that ω[x] ⊆ Equi(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then x ∈ AAP(X ), ω[x] ⊂ RE(X ), and (T, ω[x]) is an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  subsemiflow of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Clearly, Rd  + and Zd  + both satisfy the condition of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3a (by Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3g below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Then Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3b follows Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3d Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let x ∈ X such that  (1) ω[x] � ∅ is compact with ω[y] = ω[x] for all y ∈ ω[x] and  (2) ω[x] ∩ Equi(X ) � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Then x ∈ AAP(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  19  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By condition (1), ω[x] is a compact minimal set of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then from condition (2), it follows  that ω[x] ⊆ Equi (X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Now using Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3b, we have that x ∈ AAP(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3e Remark (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [39, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1] for T = R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3b, if T is an amenable semigroup  or T is a group, then for x ∈ AAP(X ), the corresponding shadowing a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' motion as in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3(ii) is  unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Suppose this is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then there are two distinct points y1, y2 ∈ ω[x] such that for every  ε ∈ UX there are s1, s2 ∈ T such that (tsix, tsiyi) ∈ ε ∀t ∈ T, for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By T s1 ∩ T s2 � ∅  for all s1, s2 ∈ T, it follows that y1 is proximal to y2 under X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' However, since y1, y2 ∈ ω[x] and  (T, ω[x]) is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', thus (y1, y2) is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' under X × X so that y1 = y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  It should be mentioned that our unicity proof here is much simpler than Seifert’s proof for  the special case that T = R+ and X a complete metric space [39, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' There Seifert need to  use y1, y2 ∈ RE(X ) instead of our “proximality” argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3f Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Recall that a topological semigroup T is called a “C-semigroup” [31] if T \\ T s is  compact for all s ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' For example, (R+, +) and (Z+, +) are C-semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then:  If T is a noncompact C-semigroup, then �n  i=1 T si � ∅ for all n ≥ 2 and s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' , sn ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let n ≥ 2 and s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', sn ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If �n  i=1 T si = ∅, then (T \\ T s1) ∪ · · · ∪ (T \\ T sn) = T so that  T is compact, contrary to the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus �n  i=1 T si � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3g Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let T = Rd  + and Zd  +, where d ≥ 1 is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then �n  i=1(T + si) � ∅ for all  n ≥ 2 and s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' , sn ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Given n ≥ 2, let K = {o, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' , sn}, which is a compact subset of G, where G = Rd or Zd  and o = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Noting that T is a thick semigroup in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7a, �n  i=1(T + si)  is a thick set in G so that �n  i=1(T + si) � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Li-Yorke chaos of non-PI extensions  If π: X → Y is a non-PI extension of minimal compact flows, then we can define a canoni-  cal M-flow (T, M∞) such that ∆X ⊊ M∞ ⊆ Rπ as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4 in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Moreover, M∞ has been described  by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We can then state our main result of this section as follows:  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1 Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let π: X → Y be a non-PI extension of minimal compact metric flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then  there exists a residual set YLY in Y such that for every y ∈ YLY and for generic x ∈ π−1y there is a  residual set S [x] in M∞[x] with the property that (x, x′) is a Li-Yorke pair for π rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' M∞ for each  x′ ∈ S [x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Recall that a mapping is said to be semi-open if the image under this mapping of every open  non-empty set contains an open non-empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If φ: X → Y is an extension of compact flows  with Y minimal and X having a dense set of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points, then φ is semi-open ([8, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='15],  [2, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  To prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1, we shall need two auxiliary lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The first is a special case of the  mentioned semi-openness theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let π: X → Y be an extension of compact flows with Y minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If (T, L) is an  M-flow in Rπ, then p: L → Y defined by (x, x′) �→ πx is semi-open onto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  20  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3 Lemma (a topological ‘Fubini theorem’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [43, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1], [24, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2] for other ver-  sions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let φ: W → Z be a semi-open continuous onto map of Polish spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Suppose K is a  residual subset of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let  ZK = {z ∈ Z | K ∩ φ−1z is a residual subset of φ−1z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Then ZK is residual in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since K is residual in W, there exists a sequence F1, F2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' of closed nowhere dense sets  in W such that W \\ K ⊆ �∞  i=1 Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let Kz = K ∩ φ−1z for all z ∈ Z, and we notice that  φ−1z \\ Kz ⊆  �∞  i=1(Fi ∩ φ−1z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Now let  B = {z ∈ Z | ∃ i s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' intφ−1z(Fi ∩ φ−1z) � ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  So if z � B, then Kz is of course residual in φ−1z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then Z \\ ZK ⊆ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' It will therefore suffice to  prove B is of the first category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Let U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' be a basis for the topology on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If z ∈ B, then we have for some integers m  and i that ∅ � Um ∩ φ−1z ⊆ Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Put  Cmi = {z ∈ B | ∅ � Um ∩ φ−1z ⊆ Fi}  for 1 ≤ m, i < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Then B = �∞  m,i=1 Cmi and B is a first category set in Z if each set Cmi is nowhere dense in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Let Dmi = intZclsZCmi, and suppose Dmi � ∅ for some two positive integers m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since  Cmi ∩ Dmi � ∅, and, Um ∩ φ−1z � ∅ whenever z ∈ Cmi ∩ Dmi, hence the set  F := Um ∩ φ−1Dmi  is an open non-empty subset of Um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We shall prove F ⊆ Fi, contrary to that Fi is nowhere dense,  then Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3 will follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Let Wo = {w ∈ W | φ is open at w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since φ is semi-open and W is a Polish space, Wo is  a residual subset of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Indeed, let Gn = {x | x ∈ W s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' x ∈ B1/n(x) ∩ φ−1intWφB1/n(x)} for  n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' then Gn is open dense in W, and, Wo = �  n Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Now put Fo = F ∩ Wo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then Fo is dense in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If w ∈ Fo and z = φw ∈ Cmi, then by  definition w ∈ Um ∩ φ−1z ⊆ Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Now let w ∈ Fo but z = φw � Cmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then at least z ∈ clsZCmi, or,  z = limn→∞ zn, zn ∈ Cmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' So there exists a sequence wn → w in F such that φwn = zn for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Eventually, wn ∈ Um, and since zn ∈ Cmi, we have wn ∈ Fi for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Therefore w ∈ Fi = Fi and  Fo ⊆ Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since Fi is closed and Fo is dense in F, so F ⊆ Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Notice here that comparing with Veech [43, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1] and Glasner [24, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2], the point  of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3 is that W need not be a minimal flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Moreover, as a result of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3, we can  conclude the following useful result:  Corollary ([29, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let E be a Polish space and R a relation on E such that R contains  a residual subset of E × E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then there exists a residual subset A of E such that for all x ∈ A there  exists a residual subset Kx of E with {x} × Kx ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  21  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since M∞ := (T, M∞) is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5 and M∞ is a compact metric  space, Trans(M∞) is residual in M∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2, each point of Trans(M∞) is a Li-Yorke  pair for π rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' M∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let λ: M∞ → X be defined by (x, x′) �→ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Clearly λ is an extension of flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Since λ: M∞ → X is semi-open onto by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2, there exists by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3 a residual set  XLY in X such that for every x ∈ XLY, the set S [x] = {x′ ∈ X | (x, x′) ∈ Trans(M∞)} is relatively  residual in λ−1x = M∞[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Furthermore, since π: X → Y is semi-open onto, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3 there  exists a residual set YLY in Y such that for every y ∈ YLY, the set π−1y ∩ XLY is relatively residual  in π−1y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4 Lemma (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [29, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1] for E having no isolated point & R ⊂ E × E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let E be a Polish  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let R ⊂ E × E \\ ∆E be a symmetric relation on E with the property that there exists a  residual subset A of E such that for each a ∈ A, R[a] contains a residual subset of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then there  is a dense uncountable subset B of E such that B ⊆ A and B × B \\ ∆E ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let B be the family of dense subsets B of E such that B × B \\ ∆E ⊆ R and B ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Clearly,  B � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Indeed, if B ⊂ A is at most countable with B × B \\ ∆ ⊆ R and B � E, then there exists an  element x ∈ A ∩ (E \\ B) ∩  ��  b∈B R[b]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' So B1 = B ∪ {x} ⊂ A such that B1 × B1 \\ ∆E ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then  by induction we can find a subset of A that belongs to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Now by Zorn’s lemma there exists a maximal element B in (B, ⊆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then B is uncountable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Otherwise, we can choose an element x ∈ A∩  ��  b∈B R[b]  �  such that B∪{x} ∈ B, contrary to that  B is a maximal element in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' This proves Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1 is an extension of [14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' As a matter of fact, using Rπ instead of M∞  in the above arguments, we can obtain the following:  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5 Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let π: X → Y be a nontrivial extension of metric flows with Y minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Suppose  that Rπ has a dense set of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' points and that π is weakly mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then there exists a residual  set YLY in Y such that for every y ∈ YLY, π−1y has no isolated points and it contains a relatively  dense uncountable Li-Yorke scrambled set for π rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Rπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Notice that Rπ is an M-flow with Rπ � ∆X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let S = Trans(Rπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then S is a symmetric  relation on X such that S ∩ ∆X = ∅ and it is residual in Rπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3 for the mappings  Rπ  (x,x′)�→πx  −−−−−−−→ Y and π−1y × π−1y  (x,x′)�→x  −−−−−−→ π−1y, it follows that there exists a residual set YLY in Y  such that for all y ∈ YLY, π−1y contains a relatively residual set (π−1y)LY with the property that  for x ∈ (π−1y)LY the cell S [x] is relatively residual in π−1y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Given y ∈ YLY, put E = π−1y and R = S ∩ E × E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then R is a symmetric relation on E with  R ∩ ∆E = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If E has an (relative) isolated point x0, then x0 ∈ (π−1y)LY and x0 ∈ S [x0] so that  (x0, x0) is a transitive point for Rπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' This is impossible for Rπ � ∆X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' This shows that E has no  isolated points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let A = (π−1y)LY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4, there exists a dense uncountable subset  B of E such that (x, x′) ∈ R whenever x, x′ ∈ B with x � x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Clearly, we have for all y ∈ YLY that B ⊆ π−1y is a Li-Yorke scrambled set for π rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Rπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The  proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Therefore if π: X → Y is a nontrivial Bronstein extension of minimal metric flows and if π  has no nontrivial equicontinuous factors, then π is Li-Yorke chaotic in the sense of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', there is a residual set YLY in Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5, it follows that if YLY is defined by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1, then there  exists, for all y ∈ YLY, a uncountable Li-Yorke scrambled set in π−1y for π rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' M∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' When Y is a  singleton set, the Bronstein condition on π in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5 may be removed as follows:  22  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='6 Proposition (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X be a weak-mixing, nontrivial, metric dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Then X has a dense uncountable Li-Yorke scrambled set (for π: X → {pt}) rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since X × X is a compact metric space, R := Trans(X × X ) is residual in X × X so that  R ∩ ∆X = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Moreover, as X × X → X defined by (x, x′) �→ x is an open onto map, it follows by  Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='4 that there exists a dense uncountable set B in X such that if x, x′ ∈ B and  x � x′, then T(x, x′) = X × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since X � {pt}, ∆X � X × X and B is a Li-Yorke scrambled set for  π: X → {pt} rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' X × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='7 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let π: X → Y be a nontrivial extension of minimal compact metric flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Suppose π is Ellis weakly mixing and non-proximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then there exists a residual set YLY in Y such  that for every y ∈ YLY, π−1y contains a uncountable Li-Yorke scrambled set for π rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' M∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' This follows easily from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='5 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Notice that if X is an Ellis weak-mixing minimal flow, then it is weakly mixing (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [25, 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  However, M∞ ⊊ X × X in general even if X is weakly mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' M-flows need not be point-transitive and examples  If X is an M-dynamic, then it is syndetically transitive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' that is, NT(U, V) is a syndetic set in  T for all non-empty open sets U, V in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' It is well known that if X is not separable, then a T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  flow need not be point-transitive (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [15, Exa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' As a matter of fact, an M-flow need not be  point-transitive too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let X = YT be the space of all functions f : T → Y, continuous or not, equipped  with the pointwise convergence topology, where Y is a compact Hausdorff space and T is an  infinite discrete group with identity e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then X is a compact Hausdorff space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Given t ∈ T and  f ∈ X, define f t : T → Y by τ �→ f(τt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We now define the right-translation flow X on X with  the phase group T as follows: T × X → X, (t, f) �→ t f := f t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then:  (1) X is syndetically and thickly transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' For all non-empty open subsets U, V of Y and τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', τn, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', sn in T, let  U = [{τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', τn}, U] = { f ∈ X | f(τi) ∈ U, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', n}  V = [{s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', sn}, V] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Let  T0 =  �  s−1  i τj | i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', n} ∪ {s−1  i sj | i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', n  �  and  T1 = T \\ T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Since T is an infinite discrete group and T0 is finite, it is easy to check that T1 is syndetic and  thick in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Indeed, if K = {e}∪t0T −1  0 for some t0 ∈ T1, then Kt∩T1 � ∅ for all t ∈ T = T0 ∪T1 so  T1 is syndetic in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' If F is any finite subset of T, then there exists some t ∈ T such that Ft ⊆ T1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  otherwise, there exists a k ∈ F and t � s in T with kt, ks ∈ T0 and kt = ks so that t = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Thus for all t ∈ T1, we have {s1t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', snt} ∩ {τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', τn, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', sn} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Now choose f ∈ X  such that f(τi) ∈ U and f(sit) ∈ V for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus f ∈ U and f t ∈ V so that NT(U, V) ⊇ T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  This proves our assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  23  Since every syndetic subset of T intersects non-voidly every thick subset of T, thus we have in  fact concluded the following fact:  (2) X is weakly mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (3) Let Y be a non-separable space and T a countable discrete group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then X is not point-  transitive, namely, Trans(X ) = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' and, Equi (X ) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' First X has no countable dense subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Because X is not separable and T is countable, X  is not point-transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since X is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', Equi(X ) ⊆ Trans(X ) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (4) Let T = Z or R with e = 0, then X is an M-flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let f ∈ X and k > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Define a periodic function ˜f : T → Y with period 2k + 1 such that  ˜f (t) = f(t) for −k ≤ t ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' So X has a dense periodic points set and X is an M-flow by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Moreover, as a matter of fact we have concluded the following fact:  (5) Let T = Z or R with e = 0, then X × X is an M-flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Here R is thought of as a discrete group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' This completes the construction of our Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  We notice that even for X is a compact metric space, a point-transitive semiflow need not be  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let us consider a simple counterexample as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let T = Q+ be the additive semigroup of nonnegative rational numbers, and,  let X = R+ ∪ {+∞} be equipped with the usual one-point compactification topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Define a  compact semiflow X as follows: T × X → X, (t, x) �→ t + x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Clearly, Trans(X ) = {0} so X  is point-transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' But X is not T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', for Trans(X ) is not a dense Gδ-subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' In fact, for  open subsets U = (10, 12) and V = (5, 6) of X, we have NT(U, V) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let  k ≥ 2 be an integer and Z  k = Z/ kZ endowed with the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let  Γ be an infinite set, ϕ: Γ → Γ a function, and w = (wγ)γ∈Γ ∈ ZΓ  k with wγ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then we shall  consider the “weighted shift” induced by ϕ and w:  σ: ZΓ  k → ZΓ  k,  x = (xγ)γ∈Γ �→ σx = (wγxϕγ)γ∈Γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', σxγ = wγxϕγ ∀γ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  See [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Here the phase space ZΓ  k is equipped with the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The classical case is  that Γ = Z+ or Z, wγ ≡ 1, and ϕ: γ �→ γ + 1 (see Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Note that  σnxγ = wγ · · · wϕn−1γxϕnγ  ∀x ∈ ZΓ  k, γ ∈ Γ and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  An element k ∈ Z  k is said to be “invertible” iff there exists an element in Z  k, denoted k−1, such  that kk−1 = 1 (mod  k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' For example, 2−1 = 3 and 3−1 = 2 in Z5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We say w ∈ ZΓ  k is “invertible” if  wγ is invertible for all γ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' A point γ ∈ Γ is called “periodic” under (ϕ, Γ), denoted γ ∈ Per(ϕ),  iff ϕτγ = γ for some integer τ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By Per (σ) we denote the set of periodic points under (σ, ZΓ  k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3a Lemma (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [41, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='1] by a complicated proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let ϕ: Γ → Γ is 1-1 and let w be  invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then:  (1) σ: ZΓ  k → ZΓ  k is continuous onto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (2) Per(σ) is dense in ZΓ  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  24  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (1): The continuity is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Now let y = (yγ)γ∈Γ ∈ ZΓ  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Put xγ = 0 if γ � ϕ(Γ), and, xγ = w−1  α yα  if α ∈ Γ and γ = ϕα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then x = (xγ)γ∈Γ ∈ ZΓ  k such that σx = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (2): Let z = (zγ)γ∈Γ ∈ ZΓ  k and let γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' , γn ∈ Γ be pairwise distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' It suffices to show there is a  σ-periodic point in the clopen cylinder neighborhood of z:  U := [zγ1]γ1 ∩ · · · ∩ [zγn]γn = {x = (xγ)γ∈Γ | xγ1 = zγ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', xγn = zγn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Since ϕ is 1-1, there is no loss of generality in considering only the special case that n = 4 and  that γ1 is a periodic point of ϕ and γ2, γ3, γ4 are non-periodic points with some integer m ≥ 1  such that {γ3, ϕγ3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' , ϕmγ3 = γ2, ϕm+1γ3 = ϕγ2, ϕm+2γ3 = ϕ2γ2, ϕm+3γ3 = ϕ3γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' } and  {γ4, ϕγ4, ϕ2γ4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' } have no common elements, for the other cases may be proved analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Notice that r ∈ Z  k is invertible iff there exists an integer ℓ ≥ 1 such that rℓ = 1 (mod  k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 1  Then we can find an integer τ such that  τ ≥ m + 1,  ϕτγ1 = γ1,  wϕτ−1γ1 · · · wγ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Let Γ0 = {γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' , ϕτ−1γ1} ∪ {ϕiτγ j | j = 2, 3, 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' i ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We now define a point x = (xγ)γ∈Γ ∈ ZΓ  k as  follows:  (a) xγ = zγ for γ ∈ {γ1, ϕγ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' , ϕτ−1γ1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (b) xγ2 = zγ2 and xϕiτγ2 = (wϕiτ−1γ2 · · ·wγ2)−1zγ2 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (c) xγ3 = zγ3 and xϕiτγ3 = (wϕiτ−1γ3 · · ·wγ3)−1zγ3 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (d) xγ4 = zγ4 and xϕiτγ4 = (wϕiτ−1γ4 · · ·wγ4)−1zγ4 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (e) if γ � Γ0 such that ϕτγ = γ j for some j = 2, 3, 4, then xγ = (wϕτ−1γ · · ·wγ)−1zγj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  (f) xγ = 0 for the other γ in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  It is clear that στx = x and x ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3b Theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' [41, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='2] for proving (σ, ZΓ  k) to be T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let ϕ: Γ → Γ be 1-1 without  periodic points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' let w be invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then (σ, ZΓ  k) is weakly mixing as a Z+-semiflow and (σ, ZΓ  k ×  ZΓ  k) is an M-semiflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let  U1 = [k1]γ1 ∩ · · · ∩ [kn]γn,  U2 = [i1]α1 ∩ · · · ∩ [in]αn,  and  V1 = [j1]δ1 ∩ · · · ∩ [jm]δm,  V2 = [ℓ1]λ1 ∩ · · · ∩ [ℓm]λm  be cylinder subsets of ZΓ  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' To show (σ, ZΓ  k) is weakly mixing, it suffices to prove that there exists  an integer τ > 0 such that στ(U1 × U2) ∩ (V1 × V2) � ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' We notice that since ϕ is 1-1 and has no  periodic points, there exists an integer τ ≥ 1 such that  {γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=', γn, α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' , αn} ∩  �  {ϕiδ j | i ≥ τ, 1 ≤ j ≤ m} ∪ {ϕiλ j | i ≥ τ, 1 ≤ j ≤ m}  �  = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  1Sufficiency is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Now conversely, assume r is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Clearly, rℓ, for ℓ ≥ 1, is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Moreover, there  exists a sequence ℓ1, ℓ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' → ∞ such that rℓ1 = rℓ2 = · · · (mod  k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus, rℓ = 1 (mod  k) for some ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  25  Let x = (xγ)γ∈Γ ∈ U1 and y = (yγ)γ∈Γ ∈ U2 be defined as follows:  xγ =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  ki  if γ = γi for 1 ≤ i ≤ n,  �  wϕτ−1δi · · · wδi  �−1 ji  if γ = ϕτδi for 1 ≤ i ≤ m,  0  for other γ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  and  yγ =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  i j  if γ = α j for 1 ≤ j ≤ n,  �  wϕτ−1λj · · · wλj  �−1 ℓ j  if γ = ϕτλ j for 1 ≤ j ≤ m,  0  for other γ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Clearly, στx ∈ V1 and στy ∈ V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus (x, y) ∈ U1 × U2 such that στ(x, y) ∈ V1 × V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  This proves that (σ, ZΓ  k) is weakly mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3a, (σ, ZΓ  k × ZΓ  k) has a dense set of  periodic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus (σ, ZΓ  k × ZΓ  k) is an M-semiflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  In fact, it is easy to verify that if (σ, ZΓ  k) is weakly mixing, then σ is onto so that ϕ is 1-1  without periodic points and w is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3c Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let Γ be countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let ϕ: Γ → Γ be 1-1 without periodic points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' let w be  invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Then there exists a dense uncountable Li-Yorke scrambled subset of ZΓ  k rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' ZΓ  k × ZΓ  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' This follows easily from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3b and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Recall that (x, y) ∈ ZΓ  k × ZΓ  k is called an “asymptotic pair” if for all ε ∈ UZΓ  k there exists an  integer τ > 0 such that (σnx, σny) ∈ ε for all n ≥ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3d Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let ϕ: Γ → Γ be 1-1 having a non-periodic point and let w be invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Then there exists a uncountable subset Θ of ZΓ  k such that (x, y) is a proximal non-asymptotic pair  under (σ, ZΓ  k) for all x, y ∈ Θ with x � y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let θ ∈ Γ be a non-periodic point of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since ϕ is 1-1, then Γθ := {γ, ϕγ, ϕ2γ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='} is a  countable infinite ϕ-invariant pairwise distinct subset of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' By considering the naturally defined  system (σ, ZΓθ  k ) based on Γθ  γ�→ϕγ  −−−−→ Γθ and Γθ  γ�→wγ  −−−−→ Z  k, and by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3c, it follows that  there exists a uncountable Li-Yorke scrambled set Ξ in ZΓθ  k rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' ZΓθ  k ×ZΓθ  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' For all x = (xγ)γ∈Γθ ∈ Ξ,  define x′ ∈ ZΓ  k by x′  γ = xγ if γ ∈ Γθ and x′  γ = 0 if γ ∈ Γ \\ Γθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Let Θ = {x′ | x ∈ Ξ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Clearly Θ  possesses the desired property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3e Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' The condition w is invertible is critical for Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3a, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3b, Proposi-  tions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3c and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Otherwise we can construct a counterexample as follows: Let  k = 4, wγ ≡ 2  (� 0) for all γ ∈ Γ, and o = (oγ)γ∈Γ ∈ ZΓ  4 with oγ = 0 for γ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Now let x ∈ ZΓ  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Since  2Z4 = {0, 2}, then σx ∈ {0, 2}Γ, σ2x = o, and σnx = 0 for all n ≥ 3, for all 1-1 function  ϕ: Γ → Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' Thus, (σ, ZΓ  k) has no any chaotic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' This work was supported by National Natural Science Foundation of China  (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
+page_content=' 11790274) and PAPD of Jiangsu Higher Education Institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE5T4oBgHgl3EQfHg75/content/2301.05441v1.pdf'}
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+Threat Models over Space and Time:
+A Case Study of E2EE Messaging Applications
+Partha Das Chowdhury∗, Maria Sameen∗, Jenny Blessing†, Nicholas Boucher†, Joseph Gardiner∗, Tom Burrows†,
+Ross Anderson†‡, Awais Rashid∗
+∗University of Bristol, UK {partha.daschowdhury, maria.sameen, joe.gardiner, awais.rashid}@bristol.ac.uk
+†University of Cambridge, UK. {jenny.blessing, nicholas.boucher, ross.anderson}@cl.cam.ac.uk, tom@tpmb.uk
+‡University of Edinburgh, UK ross.j.anderson@ed.ac.uk
+Abstract—Threat modelling is foundational to secure systems
+engineering and should be done in consideration of the context
+within which systems operate. On the other hand, the continuous
+evolution of both the technical sophistication of threats and the
+system attack surface is an inescapable reality. In this work,
+we explore the extent to which real-world systems engineering
+reflects the changing threat context. To this end we examine the
+desktop clients of six widely used end-to-end-encrypted mobile
+messaging applications to understand the extent to which they
+adjusted their threat model over space (when enabling clients
+on new platforms, such as desktop clients) and time (as new
+threats emerged). We experimented with short-lived adversarial
+access against these desktop clients and analyzed the results with
+respect to two popular threat elicitation frameworks, STRIDE
+and LINDDUN. The results demonstrate that system designers
+need to both recognise the threats in the evolving context within
+which systems operate and, more importantly, to mitigate them
+by rescoping trust boundaries in a manner that those within
+the administrative boundary cannot violate security and privacy
+properties. Such a nuanced understanding of trust boundary
+scopes and their relationship with administrative boundaries
+allows for better administration of shared components, including
+securing them with safe defaults.
+I. INTRODUCTION
+Threat modeling has become an integral part of secure
+software development. Using a framework such as STRIDE
+(Spoofing, Tampering, Repudiation, Information disclosure,
+Denial of service, Elevation of privilege) is a key step within
+the Microsoft Security Development Lifecycle (SDL) [1] to
+analyze application information flows against key classes of
+threats. Threat modeling is also a recommended best practice
+by OWASP [2] and within Agile [3] and DevOps processes [4].
+Researchers have developed similar frameworks to systemati-
+cally analyze threats to user privacy when developing software
+applications [5].
+However, threat modeling should not be a one-off activity.
+As threats evolve and new attacks come to light, developers
+must continuously reassess applications against these emerging
+threats. Furthermore, adding new features to an application
+creates new information flows, which in turn can cause trust
+boundaries to shift (e.g., due to including additional hard-
+ware or software components or third party services). The
+recommended best practice is to do threat modeling “little
+and often” [6].
+In this paper, we analyze whether the threat models that
+underpin security and privacy aspects of applications evolve
+through space (as new features are added) and time (as under-
+standing of threats changes). We use the case study of end-to-
+end-encrypted (E2EE) mobile messaging applications (such as
+Signal 1 and WhatsApp) that have found widespread uptake
+and adoption amongst users aiming to protect the privacy of
+their communications and mitigate large-scale surveillance [7],
+[8]. E2EE mobile apps were conceived to protect the message
+contents from potential eavesdroppers on the communication
+channel (which includes the E2EE messaging provider itself).
+Most of these messaging platforms have since launched their
+desktop clients to make it easier for users to communicate
+across multiple devices.
+At the same time, security and privacy threats against which
+users need to be protected have also evolved. For instance,
+the mobile app messaging threat model was largely predicated
+on an eavesdropper on the communication channel. However,
+research into intimate partner violence has highlighted that
+abusers often utilize monitoring technology or shared devices
+as a way to monitor and exert control over the victim [9], [10].
+In this scenario, the threat actor is not remote and has direct
+physical access to devices. Furthermore, for an abuse victim,
+the persistence of the access, which may last long after they
+have left their abuser, is of serious concern. There are other
+contexts in which even short-lived access to a desktop client
+can pose potential threats:
+• Official searches: The border and customs officials of
+many countries sometimes search travelers’ devices and
+make copies of data stored on them. Potential exploitation
+of the access to E2EE messaging enabled by desktop
+clients would pose a serious threats to activists and
+journalists – a major concern in the context of the
+ongoing Russian invasion of Ukraine and the use of
+E2EE messaging applications to coordinate activities by
+defenders and for protecting free speech.
+• Shared devices: Many households share devices, espe-
+cially desktop computers and laptops. There is a high
+1We refer to the Signal messenger application here, rather than the Signal
+protocol, and will continue to distinguish between the two throughout the
+paper.
+arXiv:2301.05653v1  [cs.CR]  13 Jan 2023
+
+t1
+t2
+S2
+TM1
+TM1
+TM2
+Space
+Time
+Evolution of design features of 
+E2EE applications
+Emerging threats based on 
+STRIDE and LINDDUN
+S1 = Mobile
+S2 = Multi-device
+S1
+TMΔ
+Fig. 1: Evolution of Threat Models
+likelihood that individuals living in the same household
+would be able to gain at least temporary access to the
+other’s machine or device backups. It is essential that the
+security and privacy aimed by E2EE messaging holds to
+avoid leakage of sensitive information. For instance, leak-
+age of medical communications are a particular concern
+following the overturning of Roe v. Wade in the US.
+• Corporate Machines: Machines provided by an em-
+ployer are typically managed remotely with company sys-
+admins having full access. Any desktop clients of E2EE
+messaging applications must ensure that the communica-
+tions are protected from adversarial sysadmins.
+Figure 1 represents a conceptualization of the importance
+of evolving threat models over space and time. In order to
+explore if this happens in practice, we systematically analyze
+six major E2EE messaging applications: Signal, WhatsApp,
+Element, Viber, Wickr Me, and Telegram. We start from the
+original threat model (TM 1) of these applications, i.e., a
+mobile app client with a remote attacker. We then develop
+threat models (TM 2) encompassing the expanded feature
+space and understanding of threats with respect to security
+(using the STRIDE threat modeling approach [1]) and privacy
+(using the LINDDUN threat modeling approach [5]). Using an
+experimental test setup, we then simulate adversarial short-
+lived access to the desktop clients of each of the six appli-
+cations. We use any resultant compromises to derive the net
+evolution (TM ∆) for each application. Our analysis shows
+the applications evolve their threat model to varying degrees
+to mitigate the threats resulting from such adversarial short-
+lived access.
+We argue based on our investigation that threat models
+(and hence protection mechanisms informed by them) need to
+evolve in space and time as threats change. For some desktop
+clients, TM ∆ reveals their vulnerability to spoofing, repudi-
+ation, information disclosure and elevation of privilege in the
+face of short-lived adversarial access. These vulnerabilities, in
+turn, give rise to privacy leakages: linkability of information
+and identifiability of the communicating parties. Our analysis
+also highlights that recognition of the change in threat context
+is useful but not enough in itself unless backed by appropriate
+countermeasures.
+II. BACKGROUND - E2EE MESSAGING APPLICATIONS &
+THREAT FRAMEWORKS
+In this section we give a brief background of the root
+assumption underpinning security of the E2EE applications
+we investigate. We summarize the key security properties of
+these applications and their desktop clients in Table I. This is
+followed by a high-level description of the security and privacy
+threat modeling frameworks we use for this investigation —
+STRIDE [1] and LINDDUN [5].
+A. E2EE Messaging Applications
+Every installation is tied to a particular user identity. This
+identity is then used as the root of trust to securely com-
+municate with other participants through the service and to
+configure additional devices for the same account. The steps
+to generate a long-term identity and its corresponding short-
+lived asymmetric key pairs are as follows:
+1) Every communication node generates their own long
+term public private key pair – namely, identity key IK.
+Any entity/device in possession of the secret component
+of IK and can prove so, is the legitimate owner of the
+account connected to this long term identity key pair.
+2) There are ephemeral asymmetric key pairs known as
+pre-keys, which are used to encrypt messages between
+communicating entities. The goal is to prevent compro-
+mising forward and backward secrecy even if one or a
+subset of the pre-keys is compromised.
+3) The public component of the pre-keys are communicated
+to the server for other communicating entities – this
+communication is signed by the private component of
+the long term identity key pair. The assumption is that
+only the owner of the account has access to the private
+part of IK and thus signed it.
+4) The secret component of IK and the pre-keys are not
+known or stored by anyone except their owners, so no
+intermediary (including the application provider server)
+can decrypt a message. This makes the communication
+end-to-end encrypted.
+1) Mobile Applications: Signal, WhatsApp, Element and
+Viber broadly adhere to the Signal protocol’s Double Ratchet
+algorithm [11], as shown in Table I. In essence, every com-
+munication sequence is encrypted with the pre-key listed
+against the identity of the intended recipient pertaining to
+the sequence. The process of deriving secrets from previously
+held secrets is termed as Ratcheting and is initialized with a
+shared secret known as Rootkey – which is further used to
+derive subsequent Rootkeys. Sequence numbers also help to
+decrypt out of turn messages. Element (in its Double Ratchet
+implementation) alternates between the even and odd message
+encryption keys for each session. The root of all trust is the
+identity key configured by participants during account set up.
+2
+
+Wickr Me extends the foundation of identity as the root
+of trust their adoption of the Wickr secure messaging proto-
+col [12] and pins to identifiers generated from devices typically
+in the form of SHA-256 values. The application/device is
+the primary actor in this protocol and the basis of all trust
+relationships between communicating entities. The important
+primitives are the identity keys IK, ephemeral asymmetric
+key pairs (similar to pre-keys) and a platform specific message
+encryption key.
+Telegram is a MTProto 2.0 based messenger and the
+client and the server share a device specific 2048 bit perma-
+nent authorisation key created by exchanging Diffie Hellman
+(DH) [13] keys. In Telegram communication can happen
+through a cloud server or users can switch to secret chat, i.e.,
+E2EE communication. Any short-lived key that is created is
+linked to the device specific permanent authorisation key.
+2) Desktop Clients: The desktop clients for E2EE apps also
+shaare some common characteristics:
+• Every user needs to have an account with their primary
+device which is the mobile application and the legitimate
+user is in control and possession of the primary device.
+• After standard installation of the desktop client for any of
+the messaging application, when launched, they generate
+their own identity key pair. This identity key pair is
+distinct from the primary device identity key pair.
+• The primary device scans the identity information of
+the desktop client and they authenticate each other. The
+desktop client proves legitimate ownership of the identity
+key to the primary device. The primary device then
+communicates to the corresponding application server
+that the desktop client is trusted and can communicate
+as the primary device.
+• The primary device retains a list of the linked companion
+devices.
+A common security assumption across all the E2EE ap-
+plications we investigate is that account holders will be able
+to keep their identity key and its corresponding secret away
+from attackers. Their documentation though does not specify
+specific mechanisms to do so, nevertheless these are secrets
+that are generated by the applications upon installation. In case
+of some of the applications, the security primitives and their
+implementations are geared towards preventing any leakage
+of secrets from the communication network only, while others
+consider threats to the end points as well. We elaborate on this
+further in our analysis.
+B. Threat Modeling
+1) STRIDE: STRIDE is a widely known and utilized se-
+curity threat modeling approach. The system is modeled in
+the form of Data flow Diagrams (DFDs) that capture the
+key processes, data stores and data flows between them.
+Trust boundaries delineate which partitions of the system are
+assumed to be free from adversarial interference. Data flows
+across trust boundaries and individual components (processes
+and data stores) are then evaluated for their susceptibility to
+six key threats: Spoofing, Tampering, Repudiation, Informa-
+tion disclosure, Denial of service, and Elevation of privilege.
+This susceptibility is derived through a negation of key se-
+curity properties, respectively, authentication, integrity, non-
+repudiation, confidentiality, availability and authorization.
+2) LINDDUN: LINDDUN follows a similar approach to
+STRIDE but focuses on threats to privacy. The system is
+modeled using DFDs. Then individual DFD elements and
+data flows are evaluated for potential privacy threats: Linkabil-
+ity, Identifiability, Non-repudiation, Detectability, information
+Disclosure, content Unawareness and Noncompliance. The
+threats are elicited through negation of key privacy properties,
+respectively, unlinkability, anonymity/pseudonymity, plausi-
+ble deniability, undetectability/unobservability, confidentiality,
+content awareness and policy/consent compliance.
+III. METHODOLOGY
+A. Choice of E2EE Applications
+We selected applications that are generally widely used and
+are diverse in how they establish trust between primary and
+companion devices. The applications studied can be divided
+into two broad categories: those that rely in some capacity
+on the Signal protocol, and those that do not. For Signal-
+based apps, we study two implementations of the Signal
+protocol, Signal’s own messaging app and WhatsApp, where
+Signal’s app is open-source while WhatsApp’s implementation
+is closed-source. This allows us to gain a comparative un-
+derstanding of the trust establishment (of companion devices
+by the primary device) between Signal and WhatsApp (as
+both utilize the Signal E2EE protocol as their foundation). As
+we note later in our findings, there are noticeable differences
+between their respective implementations. While not relying
+directly on the Signal protocol, Viber and Element both make
+use of the Double Ratchet algorithm in their implementations.
+Viber differs from other implementations in the way the Root
+ID is shared between the primary and companion devices.
+We examine whether this affects the ability to protect against
+threats from short-lived adversarial access. Element is also
+noteworthy due to its decentralized nature (i.e., it does not
+rely on a central communications server). We investigate if
+this decentralization has any bearing on trust establishment of
+the companion devices by the primary devices.
+We further study two messaging services, Wickr Me and
+Telegram, which rely entirely on their own messaging pro-
+tocols. Wickr documentation indicates that device-specific
+information is used in device enrollment. We evaluate whether
+this design is sufficient to prevent silent desktop cloning.
+Telegram uses a custom protocol which distinguishes between
+“cloud” chats and “secret” chats. The documentation does not
+discuss any measures for forward secrecy post-compromise.
+B. Creation of DFDs
+We created DFDs (using Microsoft’s Threat Modeling
+Tool) for each app before and after addition of the Desktop
+client by studying the security properties in their documen-
+tation [7], [12], [14]–[17] and through our experiments (see
+3
+
+Applications
+Protocol
+Primary Device (Phone) Parameters
+Desktop Client
+Signal
+Signal
+Curve25519 Key pair – Long term
+Identity Key
+Curve25519 Key pair – Pre-Keys
+Desktop ID authenticated by
+primary device.
+Can be used independently.
+WhatsApp
+Signal
+Curve25519 Key pair –
+Long term Identity Key
+Curve25519 Key pair – Pre-Keys
+Desktop ID authenticated by
+primary device
+Can be used independently
+Element
+Olm-
+Double Ratchet
+Implementation
+Curve25519 Key pair –
+Long term Identity Key
+Curve25519 Key pair – Pre-Keys
+Desktop ID authenticated by
+primary device.
+Can be used independently.
+Wickr Me
+Wickr Secure
+Messaging Protocol
+Curve P521 Key pairs
+SHA-256 Device Identifier
+Desktop ID authenticated
+by primary device
+Can be used independently.
+Viber
+Double
+Ratchet Implementation
+Curve25519 Key pair –
+Long term Identity Key
+Desktop client authenticated
+by primary device
+Can be used independently.
+Telegram
+MTProto 2.0 –
+Diffie
+Hellman Implementation
+Cloud chat – 2048 bit permanent key
+Secret Chat –
+DH keys between communicating entities.
+Desktop ID authenticated
+by primary device
+Can be used independently.
+TABLE I: Properties of Popular Messaging Applications
+Section III-C). For example, WhatsApp documentataion ex-
+plicitly states (page 25) [14]
+“WhatsApp defines end-to-end encryption as com-
+munications that remain encrypted from a device
+controlled by the sender to one controlled by the
+recipient, where no third parties, not even WhatsApp
+or our parent company Facebook, can access the
+content in between.”
+Figure 2 depicts the DFD at time t1 and space S1 (cf.
+Figure 1). Figures 3 and 4 show the DFD at time t2 and space
+S2 (cf. Figure 1) where a desktop client has been added by
+the apps.
+Fig. 2: DFD for Signal, WhatsApp, Element, Wickr Me, Viber,
+and Telegram mobile applications.
+C. Experimental Setup
+The experiments were conducted between test accounts
+registered to phone numbers provided by pay-as-you-go SIM
+cards purchased specifically for these experiments. Account
+registration was performed using a Samsung Galaxy A21
+smartphone and iPhone SE which were used to receive SMS
+messages required for registration. The desktop clients were
+installed through the following steps:
+Fig. 3: DFD for Signal, WhatsApp, and Telegram desktop
+applications.
+Fig. 4: DFD for Element, Wickr Me and Viber desktop
+applications.
+1) Hardware. The desktop clients were installed on Mac-
+Book Pro laptops with 2 GHz Quad-Core Intel Core i5
+and 16GB 3733 MHz LPDDR4X memory.
+2) Firing up the legitimate desktop client. We started a
+legitimate desktop version of the application through the
+4
+
+Mobile
+Application
+Eavesdropper
+Eavesdropping Channel
+Trust Boundary
+Credentials/Messages
+Credentials/Messages
+Applications
+Cloud ServerCloned Device/
+Companion
+Attacker
+Device
+Primary Device
+Short Lived Physical Access
+Credentials
+Credentials/Messages
+Credential-/Messages/Acknowledge
+Credentials/Messages
+compaxion device
+Credentials/Messages
+Credentials/Messages
+Credential./Messages/Acknowledge
+compaion device
+Trust
+Boundary
+Applications'
+Cloud ServerCloned Device/
+Companion
+Attacker
+Device
+Primary Device
+Short Lived Physical Access
+ Credentials
+Credentials/Messages
+Trust Boundary
+Credentials/Messages/Acknowledge
+Credentials/Messages
+Credentials/Messages
+compaion device
+Credentials/Messages
+Credential:/Messages/Acknowledge
+compaylon device
+Applications'
+Cloud Serverrequired setup mechanism, for example, by scanning a
+QR code.
+3) Firing up the attacker’s desktop client. We performed
+a standard installation of the desktop client in an at-
+tacker’s machine, configured with a second account,
+then copied the state from the victim’s machine and
+placed it in the attacker’s machine. Our goal was to
+evaluate if these systems have protections against simple
+cloning attacks so we copied state information from
+∼/Library/Application Support/ of the vic-
+tim’s machine to the same directory of the attacker’s
+machine. The victim’s machine and the attacker’s ma-
+chine were of the same specifications.
+D. Testing for threats
+1) STRIDE:
+• Spoofing. We performed a standard installation of the
+desktop client in an attacker’s machine, then copied the
+state from the victim’s machine and placed it in the
+attacker’s machine.
+• Tampering. Our focus was on the end points than the
+network. So we did not experiment with altering message
+content while in transit.
+• Repudiation. While we set up the attacker’s machine,
+we engaged in communication between the legitimate
+participant, the attacker and other legitimate parties. This
+was repeated between the communicating entities to
+understand if the recipients observe any difference while
+communicating with the victim and the attacker.
+• Information Disclosure. We used the cloned desktop
+across space and time to understand the implications on
+forward and backward secrecy.
+• Denial of Service. We tested if the cloned machine
+throws the victim out of the network or allows the victim
+to continue sending and receiving messages even when
+the clone is in operation.
+• Elevation of privilege. This was tested by capturing
+the credentials using a tls interceptor from the
+rooted device. Then the victim’s desktop client was de-
+linked from the primary device. Subsequently, the cloned
+desktop client was also de-linked. Then we used the
+captured credentials to restart the desktop client in the
+attacker’s device.
+2) LINDDUN: We focused on identifiability and linkability
+as information disclosure and non-repudiation (vice versa
+repudiation) were already evaluated as part of our STRIDE
+analysis. Since our focus was on end points, we did not
+perform analysis for detectability. Non-compliance and un-
+awareness are out of scope for us as the registration processes
+for the messaging applications were based on primary device
+credentials without any room for opting out.
+• Linkability. – The various artefacts from a victim were
+checked if they link potential entities connected to the
+victim.
+• Identifiability. – We checked if those artefacts revealed
+identifying information about the victims and indirect
+entities connected to the victim.
+We then analyzed the type(s) of data accessible through
+potential threats. We then modeled these data items as trees
+to depict the identifiability and linkability of an entity.
+IV. FINDINGS
+A longstanding security goal in establishing a commu-
+nication channel is to ensure that the only participants in
+the channel are the legitimate participants. To satisfy this
+requirement, desktop clients of E2EE messaging apps should
+be capable of identifying distinct participants in a channel,
+and thereby withstand (short-lived adversarial access) attacks
+pertaining to TM2. We subject the desktop client to short-
+lived adversarial access (TM2) to elicit the threats based on
+the tests particular to elements of the threat as per STRIDE
+and LINDDUN. Table II shows the threats which were not
+scoped while expanding from S1 to S2 between time t1 and
+t2.
+A. Signal Messenger
+Signal messenger assumes that only an eavesdropper can
+be or can attempt to be the mal-actor and implements its
+authentication and key sharing mechanisms accordingly. For
+any other mal-actors the expectation is to replace the device
+and/or account. This was perhaps appropriate for implement-
+ing the security property of authentication because the secrets
+never left the user’s device into the communication network.
+However, our experiments show that such assumptions cannot
+sustain when the potential mal-actors reside within the trust
+boundary and adversarial short-lived access can go undetected.
+a) Masquerading as the victim: An attacker can simply
+replace the configuration files of a standard Signal desktop
+installation with the versions stolen from a victim’s machine.
+The specifics of the attacker machine do not influence the
+success of the attack. Signal desktop uses an encrypted
+SQLite database to store Signal authentication credentials:
+login and device password (the uuid_id and password
+fields), received messages, and pre-keys. The relevant files are
+named config.json and databases/Databases.db.
+The database decryption key is stored in plaintext in the parent
+directory in config.json. For a knowledgeable adversary,
+the database is effectively stored in plaintext2. The collapse of
+the assumption of an only eavesdropper threat model leads
+to the violation of other security properties with privacy
+consequences.
+For Signal mobile application, upon new installation (either
+due to compromised device or for other reasons), the identity
+keys of the user change along with the pre-keys. This will
+be reflected for all the contacts of a particular account. A
+comparison between the legitimate desktop version and the
+cloned desktop showed the same keys against the sequence
+2This design decision has been discussed on multiple occasions on the
+Signal Community Forum and GitHub issue tracker since 2017.
+5
+
+Applications
+Emerging Threats (TM∆)
+S
+T
+R
+I
+D
+E
+L
+I
+N
+D
+D
+U
+N
+Signal
+✓
+-
+✓
+✓
+×
+✓
+✓
+✓
+✓
+-
+✓
+-
+-
+Whatsapp
+✓
+-
+✓
+✓
+×
+×
+✓
+✓
+✓
+-
+✓
+-
+-
+Element
+×
+-
+×
+✓
+×
+×
+✓
+×
+×
+-
+✓
+-
+-
+Wickr Me
+×
+-
+×
+×
+×
+×
+×
+×
+×
+-
+×
+-
+-
+Viber
+×
+-
+×
+×
+×
+×
+×
+×
+×
+-
+×
+-
+-
+Telegram
+✓
+-
+✓
+✓
+×
+×
+✓
+✓
+✓
+-
+✓
+-
+-
+TABLE II: (TM∆) based on STRIDE and LINDDUN threat models. In this table, (-) indicates not being tested; (×) indicates
+attack not possible; and (✓) indicates attack is possible.
+number of the pre-keys. This shows that the DH ratchet is
+not effective at rendering the cloned version obsolete after the
+existing key material is exhausted.
+The mobile application assumes that the security primitives
+will prevent an eavesdropper to capture enough state informa-
+tion to pretend as the victim – there are message sequence
+numbers which will prevent an eavesdropper to replay past
+messages pretending to be the victim. Thus, an instance of
+improper communication will be detected. There is no such
+protection against an adversary with short-lived access to
+the desktop client. One caveat is that the attacker’s Signal
+desktop instance will work with delays or messages will be
+dropped when the victim’s mirrored desktop installation is
+actively online. Our observation was that it was dependent on
+the session established either with the victim or the attacker.
+Signal protocol documentation [11] advises setting time limit
+of retention of skipped message keys to protect against an
+eavesdropper.
+b) Compromising Forward Secrecy: The desktop client
+state information contains private pre-key material that will
+let the attacker break forward secrecy for the Signal account
+it represents. Copying it results in the attack computer’s
+Signal installation exactly mirroring the victim installation.
+The attacker instance contains all historical messages from the
+victim installation, and is able to receive and decrypt all future
+incoming messages as well as send encrypted messages that
+appear to be from the victim. The mobile device and legitimate
+desktop client do not have separate ratchets.
+We tested de-linking the legitimate Signal desktop client
+from the phone, which initially made the cloned desktop
+unable to send or receive messages. Since the database de-
+cryption key was easily available, we loaded the signal app
+into an adb emulated android device running on Ubuntu. Then
+loaded a tls interceptor app into the android device to
+capture the traffic. We were able capture the primary device ID
+and password using the tls-interceptor, and use them
+to update and reconnect the cloned (but de-linked) desktop
+client. We were then able to send and receive messages on the
+de-linked desktop client using the primary credentials. We note
+that no details of the attacker machine appear to impact the
+success of this attack. This experiment was performed only for
+Signal and the database decryption keys were not so readily
+identifiable for other desktop clients3.
+c) Un-scoped Threats, Security and Privacy Conse-
+quences: TM ∆ for Signal desktop client reveals that it was
+not scoped for protection against spoofing, repudiation, infor-
+mation disclosure, denial of service and elevation of privilege
+with respect to STRIDE and linkability and identifiability for
+LINDDUN. As shown by the example tree in Figure 5, the
+messages that an adversary has access to can be used to
+infer the inter-relationship between the contacts of the victim
+and with the victim. This can lead to secondary and tertiary
+linkability between the victim and entities through their direct
+contacts. Furthermore, as shown in Figure 6, a direct conse-
+quence of compromised DFD elements is that the attacker has
+further access to personally identifiable information and can
+identify the subjects and pair them with attributes.
+For Signal messenger desktop it is difficult to detect
+short-lived adversarial access and recover from a com-
+promise. Access to the database decryption keys can po-
+tentially render de-linking inconsequential. Adhering to
+the (mobile app) threat model of external eavesdroppers
+betrays the reality where other participants with access to
+the devices (for legitimate reasons or due to proximity)
+might misuse such access.
+Fig. 5: Linkability of an Entity due to cloning of a device
+B. WhatsApp
+WhatsApp desktop clients consider malicious entities other
+than an eavesdropper. However, our experiments show that
+3We did not undertake an exhaustive search, observing the requirement
+that the adversarial access is short-lived, hence an attacker won’t be able to
+undertake such exhaustive search or investigation.
+6
+
+Linkability of an Entity
+Contact
+Information
+Messages
+Display
+Region
+Status(s)
+Name(s)
+Profession
+Picture(s)
+Associated
+Behavioral
+Location
+Gender
+Age
+Psychology
+People
+PatternsFig. 6: Identifiability of an Entity due to cloning of a device
+their protection mechanisms were not commensurate against
+such an adversary. An attacker with short-lived access can
+steal the credentials and communicate as the victim. How-
+ever unlike Signal the key is not stored in plain text. This
+has a bearing on re-configuring a stolen desktop once it is
+de-linked from the primary device. Another implementation
+improvement over Signal is that all companion devices have
+an expiry date. WhatsApp explicitly alerts the primary device
+for new companion devices as well as existing companion
+devices in use. While spoofing is possible similar to Signal,
+there are implementation improvements which can limit the
+consequences.
+The messages are synchronized across devices, thus com-
+promising confidentiality. An attacker is able to send and
+receive messages as the victim. This breaks forward secrecy
+and the recipients are not able to distinguish if the messages
+are from an attacker or the victim.
+Though there is a limit in the form of expiry dates, yet there
+are privacy consequences namely linkability and identifiability.
+TM ∆ for WhatsApp desktop client reveals that it was not
+scoped for protection against spoofing, repudiation, informa-
+tion disclosure with respect to STRIDE and linkability and
+identifiability for LINDDUN. However, WhatsApp desktop
+client was scoped for protection against elevation of privilege
+and denial of service through short-lived adversarial access.
+The scoping for spoofing is a marginal improvement over
+Signal desktop with the provision of expiry dates and alert
+messages for companion devices.
+Short-lived adversarial access does not reveal the
+database keys which, coupled with the alerts and expiry
+dates, protects the more discerning WhatsApp desktop
+user. The desktop client threat model though considers
+that legitimate insiders can turn malicious but makes
+strong assumptions on users’ ability to note and act on
+warnings and protect themselves.
+C. Viber
+Our experiments show that Viber clones exited as soon as
+they were launched. This is because Viber explicitly pins the
+primary identity in the companion devices—configuring the
+companion device transfers the identity key pair to it. This is
+significant for both the mobile application and the desktop
+client. Any device willing to authenticate as a legitimate
+client needs to be explicitly authorized by the primary device.
+Though the state is stored in ∼/Library/Application
+Support/Viber yet using the state requires to explicit
+transfer of the primary identity key by the legitimate owner.
+When communicating entities are authenticated by each
+other without any interference of a man-in-the-middle, the
+trusted session is identified with a green lock. Any session
+otherwise is indicated with a red lock.
+For Viber, TM ∆ reveals that it was scoped for protection
+against spoofing, repudiation, information disclosure, denial
+of service and elevation of privilege with respect to STRIDE
+and linkability and identifiability for LINDDUN.
+Viber appears well-scoped for the dynamic nature of
+the threat landscape by explicitly mandating companion
+devices through a transfer of the mobile device primary
+ID. This eliminates the load on users to detect cloning at-
+tacks and/or communications by the attacker and engage
+in subsequent recovery.
+D. Wickr Me
+The cloning attacks were not possible for Wickr Me even
+when the victim’s state was set to remember password. This
+means that Wickr Me considered malicious participants be-
+yond just eavesdroppers. Their mobile messaging application
+verifies the association of an identity with their identity key
+pair and ephemeral key pairs. The association between identity
+key pair and the identity is managed by the Wickr app and
+is pinned with the device, making it difficult for an attacker
+to authenticate as a victim. To protect from an eavesdropper
+Wickr Me encrypts server requests using a rotated shared
+secret using AES 256 in CFB mode which is tunneled inside
+TLS. The security property of pinning the identity and key
+pair with the device also appears to be extended to the desktop
+client.
+TM ∆ for Wickr Me reveals that it was scoped for protection
+against spoofing, repudiation, information disclosure, denial of
+service and elevation of privilege with respect to STRIDE and
+linkability and identifiability for LINDDUN.
+Wickr Me is distinct in the way it ties the device to an
+instance of companion device identity establishment —
+any cloned instance of a Wickr Me account will not be
+able to masquerade as the victim. Such implementations
+acknowledge the changing nature of the threats from
+actors with access to the devices and also opens the
+possibility to integrate robust and verifiable bindings
+between cryptographic keys and real world entities.
+E. Element
+The desktop client considers malicious participants beyond
+eavesdroppers (to a larger extent than WhatsApp) but it still
+allows an attacker to find out who communicated with whom
+and when through our cloning attacks. While we moved the
+7
+
+Identifiability of an
+Entity
+PII
+Messages
+Contact
+Profile
+Typing
+Name(s)
+Workspace
+Title(s)
+Interest
+number
+picture
+Patterns
+Ethnicity
+Appearance
+Gender
+Agevictim state to the attacker machine, the attacker was able
+to fire up the desktop client. The attacker however, was not
+able to send and receive messages and could not connect to
+the server. The attacker could only see the user names of the
+entities with whom the victim communicated and when.
+The mobile version of Element generates a secret key
+pertaining to every user for a container particular to a device
+which we believe they extended for their desktop version.
+Matrix documentation states that they generate keys per device
+and not per user and the keys are never exported from the
+device [18]. The keys are not part of the state that can be
+stolen and replicated in another device through simple cloning
+attack. Though cloning is possible in Element, it does not
+compromise forward and backward secrecy. The ability to see
+who sent messages to whom and when could lead to linkability
+between the victim and their contacts. However, the message
+contents are not exposed due to cloning so an attacker cannot
+gather identifiable attributes of the communicating entities.
+In case of Element, TM ∆ reveals that the desktop client
+was scoped for protection against spoofing, repudiation, denial
+of service and elevation of privilege with respect to STRIDE
+and identifiability for LINDDUN. Element desktop client also
+appears to be scoped for information disclosure for forward
+and backward secrecy but not for linkability with respect to
+LINDDUN.
+A cloning attack against Element reveals only the
+identity of the communicating entities but lack of access
+to the keys prevents an adversary from reading the mes-
+sages. An attacker cannot compromise forward secrecy
+through short-lived adversarial access. Element considers
+the threat that insider access can be malicious but leaves
+users vulnerable to correlation (who communicated with
+whom) by an attacker.
+F. Telegram
+The mobile application has a cloud-based chat and an
+end-to-end secure chat, using MTProto 2.0. The protection
+primitives assume that a user is in control of the device.
+This assumption means that the user in control of the device
+can only authenticate with the long-term shared identity key.
+However, the desktop client’s state information can be cloned
+through short-lived access and thus spoofed by an adversary.
+It is difficult for a recipient to distinguish between a legitimate
+sender and an attacker using a cloned account.
+In the case of Telegram secret chats, message exchanges
+are not synchronized across legitimate and cloned devices. For
+secret chats, the client key pairs are replenished after every 100
+messages or after being in use for more than a week. This is
+to prevent any compromise of forward secrecy. Participants
+in a secret chat can initiate key generation if and when they
+detect any compromise of their keys. However attackers can
+also initiate secret chats with contacts of the victim without the
+contacts or the victim being able to detect it. The disclosure
+of contact information leads to inferences about the contacts
+of the victims and leads to identifying sensitive information.
+TM ∆ reveals that Telegram Desktop client was not scoped
+for spoofing, repudiation, information disclosure and denial of
+service with respect to STRIDE and linkability and identifia-
+bility for LINDDUN. The scoping for spoofing was a marginal
+improvement over Signal Desktop with the provision of multi-
+factor authentication and passwords (as well as the option to
+set an automatic logout after a period of time).
+Telegram desktop clients can be cloned and detection
+of a compromise is non-trivial for a victim. The ability
+to set expiry dates seems cognizant of malicious insiders
+but requires participation by users to protect themselves.
+The adoption of the eavesdropper-only threat model in a
+context where access to the account state information is
+easier, leaves users vulnerable to cloning attacks.
+V. THREAT MODELLING TO ALIGN TRUST BOUNDARIES
+WITH ADMINISTRATIVE BOUNDARIES
+Distinct individual or collective entities function within a
+logical space (personal space, departments or organizations).
+We refer to these asadministrative boundaries. System de-
+signers evaluate threats within and across these administrative
+boundaries, specify security policies to counter the threats
+and provision appropriate security controls to implement these
+policies. The artefacts protected by security controls are placed
+within a trust boundary in order to mitigate particular threats.
+The key question that arises is the extent to which trust
+and administrative boundaries should align. Delineating trust
+boundaries too restrictively will lead to legitimate users having
+to shoulder a heavy load from security tasks, e.g., frequent re-
+authentication, which would impact usability of the system.
+On the other hand, not scoping the trust boundary suitably
+can lead to scenarios where those falling within the admin-
+istrative boundary can compromise security and privacy. This
+is particularly pertinent because, as discussed above, threat
+models can change — previously un-scoped threats might
+emerge to an existing application [19], [20]. On the other
+hand, changes to an existing application by way of additional
+features or extensions can change the administrative boundary
+which in turn can lead to the emergence of new threats. A
+new threat model might render existing security controls weak
+consequently exposing previously protected elements to mal-
+actors. This phenomenon is observable in our case study.
+Mobile phones function within a administrative boundary
+where sharing of devices is not a widely prevalent reality.
+The threat model is focussed more on entities external to the
+administrative boundary like eavesdroppers and attackers in
+the middle. System designers accordingly defined the security
+policies for their E2EE services focusing on the mobile device
+and implemented controls to protect against such threat actors.
+This is reflected in the trust boundary of the DFD in Figure 2
+pertaining to TM 1, which contains the mobile phone.
+In case of the desktop clients of the messaging applica-
+tions, TM 2 represents scenarios where there is a shift in
+the administrative boundary — external participants in official
+environment or for statutory reasons or domestic settings have
+8
+
+Primary Device
+Companion Device
+Cloned Device
+(a) Mal-actors within the trust boundary
+Primary Device
+Companion Device
+Cloned Device
+(b) Primary device only in trust boundary
+Primary Device
+Companion Device
+Cloned Device
+(c) Mal-actors outside the trust boundary
+Fig. 7: Placement of trust boundary (red line) with respect to administrative boundary (blue dotted line).
+easier access to the devices with desktop clients. Threats can
+arise from otherwise legitimate insiders turning malicious. Our
+investigation reveals distinct positions of the trust boundary
+depending on how companion devices can be fired up pertain-
+ing to their corresponding user account. These are captured
+in Figure 7. The security controls of Signal messenger, What-
+sApp and Telegram do not prevent companion devices being
+set up without the primary device. There can be companion
+devices cloned from an initial companion device fired by
+the primary device — allowing mal-actors with short-lived
+adversarial access to do so. As we can observe from Figure 7a,
+this is because the trust boundary includes legitimate insiders
+who can turn malicious. For a comparative understanding we
+refer to a scenario where the trust boundary situates with only
+the mobile phone as in Figure 7b. Such a model would result in
+usability load requiring frequent authorization by the primary
+device to the secondary device. On the other hand, the security
+controls for Viber, Element and Wickr Me require that all
+companion devices are explicitly fired by the primary device.
+The trust boundary as depicted in Figure 7c includes the
+primary device and only those desktop clients explicitly fired
+by the primary device. This excludes any (potential malicious)
+insider who has short-lived access to a desktop client.
+The pertinent question is how security engineering can
+rescope the trust boundary with changes in the administrative
+boundary at threat models evolve over space and time. There
+have been several studies on the challenges of threat modeling
+in agile development environments [21] which acknowledge
+the emergence/dis-appearence of threats with the evolution of
+the system. A related study [22] reports operational challenges
+with respect to threat modelling in agile environments and
+recommends including security experts to help developers.
+The importance of security experts is further substantiated by
+Assal et al. who studied organized development teams and
+reported that absence of security knowledge is a significant
+blocker in adoption of secure coding practices [23].
+Our investigation shows that re-evaluation of trust bound-
+aries in the light of TM ∆ needn’t be a highly resource
+intensive task. For instance, such re-evaluation can build on
+lightweight interventions proposed in previous practice studies
+such as Weir et al. [24]. They designed low cost practical sup-
+port for development teams — Developer Security Essentials.
+Among the proposed interventions, threat assessment is one
+of the key yet low cost and easy to implement intervention
+with the aid of a facilitator. A specific output of the threat
+assessment (at various stages) can be the evaluation of TM ∆
+with respect to specific threat taxonomies. This in turn will
+determine the security policies and controls to rescope the
+trust boundaries of systems against any possible change to
+their administrative boundaries.
+VI. DISCUSSION
+a) Reconciliation of security requirements across com-
+ponents with shared state: The E2EE messaging applications
+were initially designed for mobile phones and the desktop
+clients followed later on — mobile phone application contin-
+ued to be the root of trust for the desktop clients. A pertinent
+issue is if ‘trust’ in the security of the mobile application is
+enough to trust the security of the desktop clients. Can the
+compromise of the desktop client, on the other hand, lower
+the security of the mobile application account as well?
+Setting up a desktop client involves sharing of credentials
+between the primary device and the secondary device —
+the credential is the shared state between the primary and
+companion device. As our analysis shows, for some of the
+desktop clients, the shared system state is open to compromise
+due to short-lived adversarial access.
+Research in secure systems development has considered the
+question of when the security of the components is sufficient to
+trust the larger system (i.e., the composability problem) [25].
+Threat modeling of individual system components can help
+identify the shared state between them and analyze the conse-
+quences of such sharing. If sharing is inescapable then security
+(of the larger system) is perhaps incumbent on the adminis-
+tration of the shared components. In the context of security
+of open distributed processing, Bull et al. suggest independent
+administration of the components [26]. Gong et al. proposed
+a model for autonomous administration of shared state for
+secure inter-operation of systems [27]. Minimal sharing of
+state information has been a longstanding security principle,
+e.g., least common mechanism in the Saltzer & Schroeder
+principles [28].
+b) Safe Defaults: Systems are designed with clear de-
+lineation of the participants in the system. We learn from
+our investigation of the desktop clients of the messaging
+applications that some of them assume that these participants
+have fixed behavior which does not change across space
+and time. The manner in which applications respond to the
+change in behavior of the participants determine the security
+of the applications. Fail safe defaults has been a long standing
+principle too [28] — participants with access to devices should
+not be able to use the access maliciously.
+Developers, administrators and end users find it challenging
+to make fail safe design and usage decisions [29]. How-
+9
+
+ever, there are proposal to aid in making such decisions.
+Prompting and nudging developers when committing code,
+for instance, may help developers make their code follow the
+principles [30]. There have been suggestions of contextualizing
+the principles to the extant development requirements of the
+application developers, for example, by further exploring the
+applicability of the principles with respect to composability,
+trustworthiness, and assurance [31]. However, there remains a
+need for further work to find systematic ways to make fail-
+safety decisions [29] as well as exploring new mechanisms for
+the suitability of deny access as a fail safe default [32].
+VII. RELATED WORK
+There is a sizable body of work investigating and demon-
+strating vulnerabilities in E2EE messaging protocols and their
+client implementations. Prior work has focused particularly on
+Telegram due to its use of a custom protocol, and researchers
+have demonstrated numerous attacks over the years on both
+the MTProto protocol and its implementation [33], [34]. Most
+recently, the Matrix protocol (which Element uses) suffered
+several severe vulnerabilities in which a malicious actor was
+able to break confidentiality of communications using a com-
+promised Matrix server [35].
+Research has also demonstrated the technical feasibility of
+device cloning as a means of accessing user accounts on E2EE
+messaging platforms. Cremers et al. [36] investigated practical
+post-compromise security measures in the mobile clients and
+some desktop clients of the main E2EE messaging platforms,
+finding that almost all clients studied are vulnerable to some
+degree of cloning attack. The authors identify challenges in
+consistently maintaining state across devices as the primary
+practical reason why these platforms generally do not attempt
+to prevent cloning: a post-compromise security mechanism
+can run the risk of being overly strict (cf. Figure 7b and
+lock legitimate users out of their devices if synchronization
+fails, prompting usability concerns. Cisco’s threat intelligence
+unit further experimented with cloning a handful of E2EE
+desktop clients (specifically, Signal, WhatsApp, and Telegram)
+in the wake of the 2018 ‘Telegrab’ malware strain that took
+advantage of Telegram’s vulnerability to desktop cloning [37].
+Similarly to Cremers et al., they found that all three desktop
+clients studied allowed a cloned client to masquerade as
+legitimate, sending and receiving messages with little to no
+indication to the original user.
+In this paper, we contribute to this area by analyzing the
+problem from a threat modeling perspective. We demonstrate
+that the problem arises from a lack of consideration of threats
+across space and time. Evolution of applications is a reality
+as is addition of new features. However, rescoping of the
+trust boundary as both the application and the understanding
+of threats evolves is critical to mitigating against emerging
+threats. Our analysis of the delineation of trust and administra-
+tive boundaries provides a basis for such rescoping and better
+administration of shared components, including securing them
+with safe defaults.
+VIII. CONCLUSION
+Software is an intersection between its functional require-
+ments and their security implications, where functionalities
+take a precedence. In our investigation, we observe two
+contrasting ways in which desktop clients deal with the threats
+they face. One set of applications anticipate the possibility of
+client cloning and implement various systems-level mitigation
+strategies, while a second set puts the onus squarely on the user
+to prevent device compromise in the first place. An additional
+observation from our experiments is that the addition of a
+less secure desktop client can substantially lower the security
+of the mobile client. While most of the desktop clients are
+based on the same or similar cryptographic primitives, their
+manifest diversity is a reflection of their varying perceptions
+of an attacker. The larger lesson here is that securing evolving
+threats with assumptions made in a different context will result
+in a mismatch leaving users exposed to attacks. We make a
+case for nuanced and in-depth modelling of an attacker in
+appropriate contexts as integral to the software development
+lifecycle. Application features evolve – existing features are
+deprecated or updated and new ones added – and so do threat
+models. Our key take away for application designers is to
+not only do threat modeling little and often but also to pay
+close attention to TM ∆ when doing so. Understanding how a
+threat model evolves over space and time is key to rescoping
+trust boundaries to mitigate against emergent threats that may
+come into scope. Another key take away is that during threat
+modeling it is critical to evaluate if alignment of administrative
+and trust boundaries leads to incorrect or weak assumptions
+about the threats posed by those who may have (legitimate)
+access within an administrative boundary but ought not to be
+deemed trustworthy entities by default. Security and privacy
+is not a zero sum game. Delineation of trust boundaries and
+security controls to protect assets and data flows accordingly
+can be overly strict leading to usability challenges and hence
+causing effort overload for legitimate users. Such trade-offs
+between non-functional requirements are a well-known chal-
+lenge in software engineering. However, our analysis shows
+– as evident in some of the E2EE applications we studied –
+that, at least in our case study of E2EE applications, design
+choices are possible and implementable that account for the
+potential threats without significantly impacting usability for
+a legitimate user.
+Future work should consider how application developers
+(and teams) can maintain an ongoing view of TM ∆, for
+instance, as part of key steps in development processes. New
+requirements arising from such an analysis can be added to,
+e.g., future sprints in agile development processes. In cases
+where they cannot be addressed, the threat can be clearly
+communicated to end users and guidance provided on how to
+mitigate against them. Other future work can include addition
+of features within threat modeling tools to derive TM ∆ and
+means to model administrative and trust boundaries explicitly
+to make it easier for developers to explore the impact of their
+design choices wrt particular threats over space and time.
+10
+
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf,len=674
+page_content='Threat Models over Space and Time:  A Case Study of E2EE Messaging Applications  Partha Das Chowdhury∗, Maria Sameen∗, Jenny Blessing†, Nicholas Boucher†, Joseph Gardiner∗, Tom Burrows†,  Ross Anderson†‡, Awais Rashid∗  ∗University of Bristol, UK {partha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='daschowdhury, maria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='sameen, joe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='gardiner, awais.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='rashid}@bristol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='uk  †University of Cambridge, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' {jenny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='blessing, nicholas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='boucher, ross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='anderson}@cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='uk, tom@tpmb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='uk  ‡University of Edinburgh, UK ross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='anderson@ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='uk  Abstract—Threat modelling is foundational to secure systems  engineering and should be done in consideration of the context  within which systems operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' On the other hand, the continuous  evolution of both the technical sophistication of threats and the  system attack surface is an inescapable reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' In this work,  we explore the extent to which real-world systems engineering  reflects the changing threat context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' To this end we examine the  desktop clients of six widely used end-to-end-encrypted mobile  messaging applications to understand the extent to which they  adjusted their threat model over space (when enabling clients  on new platforms, such as desktop clients) and time (as new  threats emerged).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We experimented with short-lived adversarial  access against these desktop clients and analyzed the results with  respect to two popular threat elicitation frameworks, STRIDE  and LINDDUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The results demonstrate that system designers  need to both recognise the threats in the evolving context within  which systems operate and, more importantly, to mitigate them  by rescoping trust boundaries in a manner that those within  the administrative boundary cannot violate security and privacy  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Such a nuanced understanding of trust boundary  scopes and their relationship with administrative boundaries  allows for better administration of shared components, including  securing them with safe defaults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' INTRODUCTION  Threat modeling has become an integral part of secure  software development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Using a framework such as STRIDE  (Spoofing, Tampering, Repudiation, Information disclosure,  Denial of service, Elevation of privilege) is a key step within  the Microsoft Security Development Lifecycle (SDL) [1] to  analyze application information flows against key classes of  threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Threat modeling is also a recommended best practice  by OWASP [2] and within Agile [3] and DevOps processes [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Researchers have developed similar frameworks to systemati-  cally analyze threats to user privacy when developing software  applications [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  However, threat modeling should not be a one-off activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  As threats evolve and new attacks come to light, developers  must continuously reassess applications against these emerging  threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Furthermore, adding new features to an application  creates new information flows, which in turn can cause trust  boundaries to shift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=', due to including additional hard-  ware or software components or third party services).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The  recommended best practice is to do threat modeling “little  and often” [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  In this paper, we analyze whether the threat models that  underpin security and privacy aspects of applications evolve  through space (as new features are added) and time (as under-  standing of threats changes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We use the case study of end-to-  end-encrypted (E2EE) mobile messaging applications (such as  Signal 1 and WhatsApp) that have found widespread uptake  and adoption amongst users aiming to protect the privacy of  their communications and mitigate large-scale surveillance [7],  [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' E2EE mobile apps were conceived to protect the message  contents from potential eavesdroppers on the communication  channel (which includes the E2EE messaging provider itself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Most of these messaging platforms have since launched their  desktop clients to make it easier for users to communicate  across multiple devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  At the same time, security and privacy threats against which  users need to be protected have also evolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' For instance,  the mobile app messaging threat model was largely predicated  on an eavesdropper on the communication channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' However,  research into intimate partner violence has highlighted that  abusers often utilize monitoring technology or shared devices  as a way to monitor and exert control over the victim [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  In this scenario, the threat actor is not remote and has direct  physical access to devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Furthermore, for an abuse victim,  the persistence of the access, which may last long after they  have left their abuser, is of serious concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' There are other  contexts in which even short-lived access to a desktop client  can pose potential threats:  Official searches: The border and customs officials of  many countries sometimes search travelers’ devices and  make copies of data stored on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Potential exploitation  of the access to E2EE messaging enabled by desktop  clients would pose a serious threats to activists and  journalists – a major concern in the context of the  ongoing Russian invasion of Ukraine and the use of  E2EE messaging applications to coordinate activities by  defenders and for protecting free speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Shared devices: Many households share devices, espe-  cially desktop computers and laptops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' There is a high  1We refer to the Signal messenger application here, rather than the Signal  protocol, and will continue to distinguish between the two throughout the  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='05653v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='CR]  13 Jan 2023  t1  t2  S2  TM1  TM1  TM2  Space  Time  Evolution of design features of  E2EE applications  Emerging threats based on  STRIDE and LINDDUN  S1 = Mobile  S2 = Multi-device  S1  TMΔ  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' 1: Evolution of Threat Models  likelihood that individuals living in the same household  would be able to gain at least temporary access to the  other’s machine or device backups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' It is essential that the  security and privacy aimed by E2EE messaging holds to  avoid leakage of sensitive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' For instance, leak-  age of medical communications are a particular concern  following the overturning of Roe v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Wade in the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Corporate Machines: Machines provided by an em-  ployer are typically managed remotely with company sys-  admins having full access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Any desktop clients of E2EE  messaging applications must ensure that the communica-  tions are protected from adversarial sysadmins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Figure 1 represents a conceptualization of the importance  of evolving threat models over space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' In order to  explore if this happens in practice, we systematically analyze  six major E2EE messaging applications: Signal, WhatsApp,  Element, Viber, Wickr Me, and Telegram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We start from the  original threat model (TM 1) of these applications, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=', a  mobile app client with a remote attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We then develop  threat models (TM 2) encompassing the expanded feature  space and understanding of threats with respect to security  (using the STRIDE threat modeling approach [1]) and privacy  (using the LINDDUN threat modeling approach [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Using an  experimental test setup, we then simulate adversarial short-  lived access to the desktop clients of each of the six appli-  cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We use any resultant compromises to derive the net  evolution (TM ∆) for each application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Our analysis shows  the applications evolve their threat model to varying degrees  to mitigate the threats resulting from such adversarial short-  lived access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  We argue based on our investigation that threat models  (and hence protection mechanisms informed by them) need to  evolve in space and time as threats change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' For some desktop  clients, TM ∆ reveals their vulnerability to spoofing, repudi-  ation, information disclosure and elevation of privilege in the  face of short-lived adversarial access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' These vulnerabilities, in  turn, give rise to privacy leakages: linkability of information  and identifiability of the communicating parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Our analysis  also highlights that recognition of the change in threat context  is useful but not enough in itself unless backed by appropriate  countermeasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' BACKGROUND - E2EE MESSAGING APPLICATIONS &  THREAT FRAMEWORKS  In this section we give a brief background of the root  assumption underpinning security of the E2EE applications  we investigate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We summarize the key security properties of  these applications and their desktop clients in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This is  followed by a high-level description of the security and privacy  threat modeling frameworks we use for this investigation —  STRIDE [1] and LINDDUN [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' E2EE Messaging Applications  Every installation is tied to a particular user identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This  identity is then used as the root of trust to securely com-  municate with other participants through the service and to  configure additional devices for the same account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The steps  to generate a long-term identity and its corresponding short-  lived asymmetric key pairs are as follows:  1) Every communication node generates their own long  term public private key pair – namely, identity key IK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Any entity/device in possession of the secret component  of IK and can prove so, is the legitimate owner of the  account connected to this long term identity key pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  2) There are ephemeral asymmetric key pairs known as  pre-keys, which are used to encrypt messages between  communicating entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The goal is to prevent compro-  mising forward and backward secrecy even if one or a  subset of the pre-keys is compromised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  3) The public component of the pre-keys are communicated  to the server for other communicating entities – this  communication is signed by the private component of  the long term identity key pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The assumption is that  only the owner of the account has access to the private  part of IK and thus signed it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  4) The secret component of IK and the pre-keys are not  known or stored by anyone except their owners, so no  intermediary (including the application provider server)  can decrypt a message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This makes the communication  end-to-end encrypted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  1) Mobile Applications: Signal, WhatsApp, Element and  Viber broadly adhere to the Signal protocol’s Double Ratchet  algorithm [11], as shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' In essence, every com-  munication sequence is encrypted with the pre-key listed  against the identity of the intended recipient pertaining to  the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The process of deriving secrets from previously  held secrets is termed as Ratcheting and is initialized with a  shared secret known as Rootkey – which is further used to  derive subsequent Rootkeys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Sequence numbers also help to  decrypt out of turn messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Element (in its Double Ratchet  implementation) alternates between the even and odd message  encryption keys for each session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The root of all trust is the  identity key configured by participants during account set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  2  Wickr Me extends the foundation of identity as the root  of trust their adoption of the Wickr secure messaging proto-  col [12] and pins to identifiers generated from devices typically  in the form of SHA-256 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The application/device is  the primary actor in this protocol and the basis of all trust  relationships between communicating entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The important  primitives are the identity keys IK, ephemeral asymmetric  key pairs (similar to pre-keys) and a platform specific message  encryption key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Telegram is a MTProto 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='0 based messenger and the  client and the server share a device specific 2048 bit perma-  nent authorisation key created by exchanging Diffie Hellman  (DH) [13] keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' In Telegram communication can happen  through a cloud server or users can switch to secret chat, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=',  E2EE communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Any short-lived key that is created is  linked to the device specific permanent authorisation key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  2) Desktop Clients: The desktop clients for E2EE apps also  shaare some common characteristics:  Every user needs to have an account with their primary  device which is the mobile application and the legitimate  user is in control and possession of the primary device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  After standard installation of the desktop client for any of  the messaging application, when launched, they generate  their own identity key pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This identity key pair is  distinct from the primary device identity key pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  The primary device scans the identity information of  the desktop client and they authenticate each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The  desktop client proves legitimate ownership of the identity  key to the primary device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The primary device then  communicates to the corresponding application server  that the desktop client is trusted and can communicate  as the primary device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  The primary device retains a list of the linked companion  devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  A common security assumption across all the E2EE ap-  plications we investigate is that account holders will be able  to keep their identity key and its corresponding secret away  from attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Their documentation though does not specify  specific mechanisms to do so, nevertheless these are secrets  that are generated by the applications upon installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' In case  of some of the applications, the security primitives and their  implementations are geared towards preventing any leakage  of secrets from the communication network only, while others  consider threats to the end points as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We elaborate on this  further in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Threat Modeling  1) STRIDE: STRIDE is a widely known and utilized se-  curity threat modeling approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The system is modeled in  the form of Data flow Diagrams (DFDs) that capture the  key processes, data stores and data flows between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Trust boundaries delineate which partitions of the system are  assumed to be free from adversarial interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Data flows  across trust boundaries and individual components (processes  and data stores) are then evaluated for their susceptibility to  six key threats: Spoofing, Tampering, Repudiation, Informa-  tion disclosure, Denial of service, and Elevation of privilege.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  This susceptibility is derived through a negation of key se-  curity properties, respectively, authentication, integrity, non-  repudiation, confidentiality, availability and authorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  2) LINDDUN: LINDDUN follows a similar approach to  STRIDE but focuses on threats to privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The system is  modeled using DFDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Then individual DFD elements and  data flows are evaluated for potential privacy threats: Linkabil-  ity, Identifiability, Non-repudiation, Detectability, information  Disclosure, content Unawareness and Noncompliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The  threats are elicited through negation of key privacy properties,  respectively, unlinkability, anonymity/pseudonymity, plausi-  ble deniability, undetectability/unobservability, confidentiality,  content awareness and policy/consent compliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' METHODOLOGY  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Choice of E2EE Applications  We selected applications that are generally widely used and  are diverse in how they establish trust between primary and  companion devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The applications studied can be divided  into two broad categories: those that rely in some capacity  on the Signal protocol, and those that do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' For Signal-  based apps, we study two implementations of the Signal  protocol, Signal’s own messaging app and WhatsApp, where  Signal’s app is open-source while WhatsApp’s implementation  is closed-source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This allows us to gain a comparative un-  derstanding of the trust establishment (of companion devices  by the primary device) between Signal and WhatsApp (as  both utilize the Signal E2EE protocol as their foundation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' As  we note later in our findings, there are noticeable differences  between their respective implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' While not relying  directly on the Signal protocol, Viber and Element both make  use of the Double Ratchet algorithm in their implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Viber differs from other implementations in the way the Root  ID is shared between the primary and companion devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  We examine whether this affects the ability to protect against  threats from short-lived adversarial access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Element is also  noteworthy due to its decentralized nature (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=', it does not  rely on a central communications server).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We investigate if  this decentralization has any bearing on trust establishment of  the companion devices by the primary devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  We further study two messaging services, Wickr Me and  Telegram, which rely entirely on their own messaging pro-  tocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Wickr documentation indicates that device-specific  information is used in device enrollment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We evaluate whether  this design is sufficient to prevent silent desktop cloning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Telegram uses a custom protocol which distinguishes between  “cloud” chats and “secret” chats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The documentation does not  discuss any measures for forward secrecy post-compromise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Creation of DFDs  We created DFDs (using Microsoft’s Threat Modeling  Tool) for each app before and after addition of the Desktop  client by studying the security properties in their documen-  tation [7], [12], [14]–[17] and through our experiments (see  3  Applications  Protocol  Primary Device (Phone) Parameters  Desktop Client  Signal  Signal  Curve25519 Key pair – Long term  Identity Key  Curve25519 Key pair – Pre-Keys  Desktop ID authenticated by  primary device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Can be used independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  WhatsApp  Signal  Curve25519 Key pair –  Long term Identity Key  Curve25519 Key pair – Pre-Keys  Desktop ID authenticated by  primary device  Can be used independently  Element  Olm-  Double Ratchet  Implementation  Curve25519 Key pair –  Long term Identity Key  Curve25519 Key pair – Pre-Keys  Desktop ID authenticated by  primary device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Can be used independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Wickr Me  Wickr Secure  Messaging Protocol  Curve P521 Key pairs  SHA-256 Device Identifier  Desktop ID authenticated  by primary device  Can be used independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Viber  Double  Ratchet Implementation  Curve25519 Key pair –  Long term Identity Key  Desktop client authenticated  by primary device  Can be used independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Telegram  MTProto 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='0 –  Diffie  Hellman Implementation  Cloud chat – 2048 bit permanent key  Secret Chat –  DH keys between communicating entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Desktop ID authenticated  by primary device  Can be used independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  TABLE I: Properties of Popular Messaging Applications  Section III-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' For example, WhatsApp documentataion ex-  plicitly states (page 25) [14]  “WhatsApp defines end-to-end encryption as com-  munications that remain encrypted from a device  controlled by the sender to one controlled by the  recipient, where no third parties, not even WhatsApp  or our parent company Facebook, can access the  content in between.”  Figure 2 depicts the DFD at time t1 and space S1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Figures 3 and 4 show the DFD at time t2 and space  S2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Figure 1) where a desktop client has been added by  the apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' 2: DFD for Signal, WhatsApp, Element, Wickr Me, Viber,  and Telegram mobile applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Experimental Setup  The experiments were conducted between test accounts  registered to phone numbers provided by pay-as-you-go SIM  cards purchased specifically for these experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Account  registration was performed using a Samsung Galaxy A21  smartphone and iPhone SE which were used to receive SMS  messages required for registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The desktop clients were  installed through the following steps:  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' 3: DFD for Signal, WhatsApp, and Telegram desktop  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' 4: DFD for Element, Wickr Me and Viber desktop  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  1) Hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The desktop clients were installed on Mac-  Book Pro laptops with 2 GHz Quad-Core Intel Core i5  and 16GB 3733 MHz LPDDR4X memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  2) Firing up the legitimate desktop client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We started a  legitimate desktop version of the application through the  4  Mobile  Application  Eavesdropper  Eavesdropping Channel  Trust Boundary  Credentials/Messages  Credentials/Messages  Applications  Cloud ServerCloned Device/  Companion  Attacker  Device  Primary Device  Short Lived Physical Access  Credentials  Credentials/Messages  Credential-/Messages/Acknowledge  Credentials/Messages  compaxion device  Credentials/Messages  Credentials/Messages  Credential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content="/Messages/Acknowledge  compaion device  Trust  Boundary  Applications'  Cloud ServerCloned Device/  Companion  Attacker  Device  Primary Device  Short Lived Physical Access  Credentials  Credentials/Messages  Trust Boundary  Credentials/Messages/Acknowledge  Credentials/Messages  Credentials/Messages  compaion device  Credentials/Messages  Credential:/Messages/Acknowledge  compaylon device  Applications'  Cloud Serverrequired setup mechanism, for example, by scanning a  QR code." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  3) Firing up the attacker’s desktop client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We performed  a standard installation of the desktop client in an at-  tacker’s machine, configured with a second account,  then copied the state from the victim’s machine and  placed it in the attacker’s machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Our goal was to  evaluate if these systems have protections against simple  cloning attacks so we copied state information from  ∼/Library/Application Support/ of the vic-  tim’s machine to the same directory of the attacker’s  machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The victim’s machine and the attacker’s ma-  chine were of the same specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Testing for threats  1) STRIDE:  Spoofing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We performed a standard installation of the  desktop client in an attacker’s machine, then copied the  state from the victim’s machine and placed it in the  attacker’s machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Tampering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Our focus was on the end points than the  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' So we did not experiment with altering message  content while in transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Repudiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' While we set up the attacker’s machine,  we engaged in communication between the legitimate  participant, the attacker and other legitimate parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This  was repeated between the communicating entities to  understand if the recipients observe any difference while  communicating with the victim and the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Information Disclosure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We used the cloned desktop  across space and time to understand the implications on  forward and backward secrecy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Denial of Service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We tested if the cloned machine  throws the victim out of the network or allows the victim  to continue sending and receiving messages even when  the clone is in operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Elevation of privilege.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This was tested by capturing  the credentials using a tls interceptor from the  rooted device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Then the victim’s desktop client was de-  linked from the primary device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Subsequently, the cloned  desktop client was also de-linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Then we used the  captured credentials to restart the desktop client in the  attacker’s device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  2) LINDDUN: We focused on identifiability and linkability  as information disclosure and non-repudiation (vice versa  repudiation) were already evaluated as part of our STRIDE  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Since our focus was on end points, we did not  perform analysis for detectability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Non-compliance and un-  awareness are out of scope for us as the registration processes  for the messaging applications were based on primary device  credentials without any room for opting out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Linkability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' – The various artefacts from a victim were  checked if they link potential entities connected to the  victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Identifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' – We checked if those artefacts revealed  identifying information about the victims and indirect  entities connected to the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  We then analyzed the type(s) of data accessible through  potential threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We then modeled these data items as trees  to depict the identifiability and linkability of an entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' FINDINGS  A longstanding security goal in establishing a commu-  nication channel is to ensure that the only participants in  the channel are the legitimate participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' To satisfy this  requirement, desktop clients of E2EE messaging apps should  be capable of identifying distinct participants in a channel,  and thereby withstand (short-lived adversarial access) attacks  pertaining to TM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We subject the desktop client to short-  lived adversarial access (TM2) to elicit the threats based on  the tests particular to elements of the threat as per STRIDE  and LINDDUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Table II shows the threats which were not  scoped while expanding from S1 to S2 between time t1 and  t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Signal Messenger  Signal messenger assumes that only an eavesdropper can  be or can attempt to be the mal-actor and implements its  authentication and key sharing mechanisms accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' For  any other mal-actors the expectation is to replace the device  and/or account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This was perhaps appropriate for implement-  ing the security property of authentication because the secrets  never left the user’s device into the communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  However, our experiments show that such assumptions cannot  sustain when the potential mal-actors reside within the trust  boundary and adversarial short-lived access can go undetected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  a) Masquerading as the victim: An attacker can simply  replace the configuration files of a standard Signal desktop  installation with the versions stolen from a victim’s machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  The specifics of the attacker machine do not influence the  success of the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Signal desktop uses an encrypted  SQLite database to store Signal authentication credentials:  login and device password (the uuid_id and password  fields), received messages, and pre-keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The relevant files are  named config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='json and databases/Databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='db.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  The database decryption key is stored in plaintext in the parent  directory in config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='json.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' For a knowledgeable adversary,  the database is effectively stored in plaintext2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The collapse of  the assumption of an only eavesdropper threat model leads  to the violation of other security properties with privacy  consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  For Signal mobile application, upon new installation (either  due to compromised device or for other reasons), the identity  keys of the user change along with the pre-keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This will  be reflected for all the contacts of a particular account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' A  comparison between the legitimate desktop version and the  cloned desktop showed the same keys against the sequence  2This design decision has been discussed on multiple occasions on the  Signal Community Forum and GitHub issue tracker since 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  5  Applications  Emerging Threats (TM∆)  S  T  R  I  D  E  L  I  N  D  D  U  N  Signal  ✓ ✓  ✓  ×  ✓  ✓  ✓  ✓ ✓ Whatsapp  ✓ ✓  ✓  ×  ×  ✓  ✓  ✓ ✓ Element  × ×  ✓  ×  ×  ✓  ×  × ✓ Wickr Me  × ×  ×  ×  ×  ×  ×  × × Viber  × ×  ×  ×  ×  ×  ×  × × Telegram  ✓ ✓  ✓  ×  ×  ✓  ✓  ✓ ✓ TABLE II: (TM∆) based on STRIDE and LINDDUN threat models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' In this table, (-) indicates not being tested;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' (×) indicates  attack not possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' and (✓) indicates attack is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  number of the pre-keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This shows that the DH ratchet is  not effective at rendering the cloned version obsolete after the  existing key material is exhausted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  The mobile application assumes that the security primitives  will prevent an eavesdropper to capture enough state informa-  tion to pretend as the victim – there are message sequence  numbers which will prevent an eavesdropper to replay past  messages pretending to be the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Thus, an instance of  improper communication will be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' There is no such  protection against an adversary with short-lived access to  the desktop client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' One caveat is that the attacker’s Signal  desktop instance will work with delays or messages will be  dropped when the victim’s mirrored desktop installation is  actively online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Our observation was that it was dependent on  the session established either with the victim or the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Signal protocol documentation [11] advises setting time limit  of retention of skipped message keys to protect against an  eavesdropper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  b) Compromising Forward Secrecy: The desktop client  state information contains private pre-key material that will  let the attacker break forward secrecy for the Signal account  it represents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Copying it results in the attack computer’s  Signal installation exactly mirroring the victim installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  The attacker instance contains all historical messages from the  victim installation, and is able to receive and decrypt all future  incoming messages as well as send encrypted messages that  appear to be from the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The mobile device and legitimate  desktop client do not have separate ratchets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  We tested de-linking the legitimate Signal desktop client  from the phone, which initially made the cloned desktop  unable to send or receive messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Since the database de-  cryption key was easily available, we loaded the signal app  into an adb emulated android device running on Ubuntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Then  loaded a tls interceptor app into the android device to  capture the traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We were able capture the primary device ID  and password using the tls-interceptor, and use them  to update and reconnect the cloned (but de-linked) desktop  client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We were then able to send and receive messages on the  de-linked desktop client using the primary credentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We note  that no details of the attacker machine appear to impact the  success of this attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This experiment was performed only for  Signal and the database decryption keys were not so readily  identifiable for other desktop clients3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  c) Un-scoped Threats, Security and Privacy Conse-  quences: TM ∆ for Signal desktop client reveals that it was  not scoped for protection against spoofing, repudiation, infor-  mation disclosure, denial of service and elevation of privilege  with respect to STRIDE and linkability and identifiability for  LINDDUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' As shown by the example tree in Figure 5, the  messages that an adversary has access to can be used to  infer the inter-relationship between the contacts of the victim  and with the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This can lead to secondary and tertiary  linkability between the victim and entities through their direct  contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Furthermore, as shown in Figure 6, a direct conse-  quence of compromised DFD elements is that the attacker has  further access to personally identifiable information and can  identify the subjects and pair them with attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  For Signal messenger desktop it is difficult to detect  short-lived adversarial access and recover from a com-  promise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Access to the database decryption keys can po-  tentially render de-linking inconsequential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Adhering to  the (mobile app) threat model of external eavesdroppers  betrays the reality where other participants with access to  the devices (for legitimate reasons or due to proximity)  might misuse such access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' 5: Linkability of an Entity due to cloning of a device  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' WhatsApp  WhatsApp desktop clients consider malicious entities other  than an eavesdropper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' However, our experiments show that  3We did not undertake an exhaustive search, observing the requirement  that the adversarial access is short-lived, hence an attacker won’t be able to  undertake such exhaustive search or investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  6  Linkability of an Entity  Contact  Information  Messages  Display  Region  Status(s)  Name(s)  Profession  Picture(s)  Associated  Behavioral  Location  Gender  Age  Psychology  People  PatternsFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' 6: Identifiability of an Entity due to cloning of a device  their protection mechanisms were not commensurate against  such an adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' An attacker with short-lived access can  steal the credentials and communicate as the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' How-  ever unlike Signal the key is not stored in plain text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This  has a bearing on re-configuring a stolen desktop once it is  de-linked from the primary device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Another implementation  improvement over Signal is that all companion devices have  an expiry date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' WhatsApp explicitly alerts the primary device  for new companion devices as well as existing companion  devices in use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' While spoofing is possible similar to Signal,  there are implementation improvements which can limit the  consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  The messages are synchronized across devices, thus com-  promising confidentiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' An attacker is able to send and  receive messages as the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This breaks forward secrecy  and the recipients are not able to distinguish if the messages  are from an attacker or the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Though there is a limit in the form of expiry dates, yet there  are privacy consequences namely linkability and identifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  TM ∆ for WhatsApp desktop client reveals that it was not  scoped for protection against spoofing, repudiation, informa-  tion disclosure with respect to STRIDE and linkability and  identifiability for LINDDUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' However, WhatsApp desktop  client was scoped for protection against elevation of privilege  and denial of service through short-lived adversarial access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  The scoping for spoofing is a marginal improvement over  Signal desktop with the provision of expiry dates and alert  messages for companion devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Short-lived adversarial access does not reveal the  database keys which, coupled with the alerts and expiry  dates, protects the more discerning WhatsApp desktop  user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The desktop client threat model though considers  that legitimate insiders can turn malicious but makes  strong assumptions on users’ ability to note and act on  warnings and protect themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Viber  Our experiments show that Viber clones exited as soon as  they were launched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This is because Viber explicitly pins the  primary identity in the companion devices—configuring the  companion device transfers the identity key pair to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This is  significant for both the mobile application and the desktop  client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Any device willing to authenticate as a legitimate  client needs to be explicitly authorized by the primary device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Though the state is stored in ∼/Library/Application  Support/Viber yet using the state requires to explicit  transfer of the primary identity key by the legitimate owner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  When communicating entities are authenticated by each  other without any interference of a man-in-the-middle, the  trusted session is identified with a green lock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Any session  otherwise is indicated with a red lock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  For Viber, TM ∆ reveals that it was scoped for protection  against spoofing, repudiation, information disclosure, denial  of service and elevation of privilege with respect to STRIDE  and linkability and identifiability for LINDDUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Viber appears well-scoped for the dynamic nature of  the threat landscape by explicitly mandating companion  devices through a transfer of the mobile device primary  ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This eliminates the load on users to detect cloning at-  tacks and/or communications by the attacker and engage  in subsequent recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Wickr Me  The cloning attacks were not possible for Wickr Me even  when the victim’s state was set to remember password.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This  means that Wickr Me considered malicious participants be-  yond just eavesdroppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Their mobile messaging application  verifies the association of an identity with their identity key  pair and ephemeral key pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The association between identity  key pair and the identity is managed by the Wickr app and  is pinned with the device, making it difficult for an attacker  to authenticate as a victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' To protect from an eavesdropper  Wickr Me encrypts server requests using a rotated shared  secret using AES 256 in CFB mode which is tunneled inside  TLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The security property of pinning the identity and key  pair with the device also appears to be extended to the desktop  client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  TM ∆ for Wickr Me reveals that it was scoped for protection  against spoofing, repudiation, information disclosure, denial of  service and elevation of privilege with respect to STRIDE and  linkability and identifiability for LINDDUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Wickr Me is distinct in the way it ties the device to an  instance of companion device identity establishment —  any cloned instance of a Wickr Me account will not be  able to masquerade as the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Such implementations  acknowledge the changing nature of the threats from  actors with access to the devices and also opens the  possibility to integrate robust and verifiable bindings  between cryptographic keys and real world entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Element  The desktop client considers malicious participants beyond  eavesdroppers (to a larger extent than WhatsApp) but it still  allows an attacker to find out who communicated with whom  and when through our cloning attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' While we moved the  7  Identifiability of an  Entity  PII  Messages  Contact  Profile  Typing  Name(s)  Workspace  Title(s)  Interest  number  picture  Patterns  Ethnicity  Appearance  Gender  Agevictim state to the attacker machine, the attacker was able  to fire up the desktop client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The attacker however, was not  able to send and receive messages and could not connect to  the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The attacker could only see the user names of the  entities with whom the victim communicated and when.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  The mobile version of Element generates a secret key  pertaining to every user for a container particular to a device  which we believe they extended for their desktop version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Matrix documentation states that they generate keys per device  and not per user and the keys are never exported from the  device [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The keys are not part of the state that can be  stolen and replicated in another device through simple cloning  attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Though cloning is possible in Element, it does not  compromise forward and backward secrecy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The ability to see  who sent messages to whom and when could lead to linkability  between the victim and their contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' However, the message  contents are not exposed due to cloning so an attacker cannot  gather identifiable attributes of the communicating entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  In case of Element, TM ∆ reveals that the desktop client  was scoped for protection against spoofing, repudiation, denial  of service and elevation of privilege with respect to STRIDE  and identifiability for LINDDUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Element desktop client also  appears to be scoped for information disclosure for forward  and backward secrecy but not for linkability with respect to  LINDDUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  A cloning attack against Element reveals only the  identity of the communicating entities but lack of access  to the keys prevents an adversary from reading the mes-  sages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' An attacker cannot compromise forward secrecy  through short-lived adversarial access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Element considers  the threat that insider access can be malicious but leaves  users vulnerable to correlation (who communicated with  whom) by an attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Telegram  The mobile application has a cloud-based chat and an  end-to-end secure chat, using MTProto 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The protection  primitives assume that a user is in control of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  This assumption means that the user in control of the device  can only authenticate with the long-term shared identity key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  However, the desktop client’s state information can be cloned  through short-lived access and thus spoofed by an adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  It is difficult for a recipient to distinguish between a legitimate  sender and an attacker using a cloned account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  In the case of Telegram secret chats, message exchanges  are not synchronized across legitimate and cloned devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' For  secret chats, the client key pairs are replenished after every 100  messages or after being in use for more than a week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This is  to prevent any compromise of forward secrecy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Participants  in a secret chat can initiate key generation if and when they  detect any compromise of their keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' However attackers can  also initiate secret chats with contacts of the victim without the  contacts or the victim being able to detect it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The disclosure  of contact information leads to inferences about the contacts  of the victims and leads to identifying sensitive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  TM ∆ reveals that Telegram Desktop client was not scoped  for spoofing, repudiation, information disclosure and denial of  service with respect to STRIDE and linkability and identifia-  bility for LINDDUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The scoping for spoofing was a marginal  improvement over Signal Desktop with the provision of multi-  factor authentication and passwords (as well as the option to  set an automatic logout after a period of time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Telegram desktop clients can be cloned and detection  of a compromise is non-trivial for a victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The ability  to set expiry dates seems cognizant of malicious insiders  but requires participation by users to protect themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  The adoption of the eavesdropper-only threat model in a  context where access to the account state information is  easier, leaves users vulnerable to cloning attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' THREAT MODELLING TO ALIGN TRUST BOUNDARIES  WITH ADMINISTRATIVE BOUNDARIES  Distinct individual or collective entities function within a  logical space (personal space, departments or organizations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  We refer to these asadministrative boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' System de-  signers evaluate threats within and across these administrative  boundaries, specify security policies to counter the threats  and provision appropriate security controls to implement these  policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The artefacts protected by security controls are placed  within a trust boundary in order to mitigate particular threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  The key question that arises is the extent to which trust  and administrative boundaries should align.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Delineating trust  boundaries too restrictively will lead to legitimate users having  to shoulder a heavy load from security tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=', frequent re-  authentication, which would impact usability of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  On the other hand, not scoping the trust boundary suitably  can lead to scenarios where those falling within the admin-  istrative boundary can compromise security and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This  is particularly pertinent because, as discussed above, threat  models can change — previously un-scoped threats might  emerge to an existing application [19], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' On the other  hand, changes to an existing application by way of additional  features or extensions can change the administrative boundary  which in turn can lead to the emergence of new threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' A  new threat model might render existing security controls weak  consequently exposing previously protected elements to mal-  actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This phenomenon is observable in our case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Mobile phones function within a administrative boundary  where sharing of devices is not a widely prevalent reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  The threat model is focussed more on entities external to the  administrative boundary like eavesdroppers and attackers in  the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' System designers accordingly defined the security  policies for their E2EE services focusing on the mobile device  and implemented controls to protect against such threat actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  This is reflected in the trust boundary of the DFD in Figure 2  pertaining to TM 1, which contains the mobile phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  In case of the desktop clients of the messaging applica-  tions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' TM 2 represents scenarios where there is a shift in  the administrative boundary — external participants in official  environment or for statutory reasons or domestic settings have  8  Primary Device  Companion Device  Cloned Device  (a) Mal-actors within the trust boundary  Primary Device  Companion Device  Cloned Device  (b) Primary device only in trust boundary  Primary Device  Companion Device  Cloned Device  (c) Mal-actors outside the trust boundary  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' 7: Placement of trust boundary (red line) with respect to administrative boundary (blue dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  easier access to the devices with desktop clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Threats can  arise from otherwise legitimate insiders turning malicious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Our  investigation reveals distinct positions of the trust boundary  depending on how companion devices can be fired up pertain-  ing to their corresponding user account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' These are captured  in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The security controls of Signal messenger, What-  sApp and Telegram do not prevent companion devices being  set up without the primary device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' There can be companion  devices cloned from an initial companion device fired by  the primary device — allowing mal-actors with short-lived  adversarial access to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' As we can observe from Figure 7a,  this is because the trust boundary includes legitimate insiders  who can turn malicious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' For a comparative understanding we  refer to a scenario where the trust boundary situates with only  the mobile phone as in Figure 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Such a model would result in  usability load requiring frequent authorization by the primary  device to the secondary device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' On the other hand, the security  controls for Viber, Element and Wickr Me require that all  companion devices are explicitly fired by the primary device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  The trust boundary as depicted in Figure 7c includes the  primary device and only those desktop clients explicitly fired  by the primary device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This excludes any (potential malicious)  insider who has short-lived access to a desktop client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  The pertinent question is how security engineering can  rescope the trust boundary with changes in the administrative  boundary at threat models evolve over space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' There  have been several studies on the challenges of threat modeling  in agile development environments [21] which acknowledge  the emergence/dis-appearence of threats with the evolution of  the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' A related study [22] reports operational challenges  with respect to threat modelling in agile environments and  recommends including security experts to help developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  The importance of security experts is further substantiated by  Assal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' who studied organized development teams and  reported that absence of security knowledge is a significant  blocker in adoption of secure coding practices [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Our investigation shows that re-evaluation of trust bound-  aries in the light of TM ∆ needn’t be a highly resource  intensive task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' For instance, such re-evaluation can build on  lightweight interventions proposed in previous practice studies  such as Weir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' They designed low cost practical sup-  port for development teams — Developer Security Essentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Among the proposed interventions, threat assessment is one  of the key yet low cost and easy to implement intervention  with the aid of a facilitator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' A specific output of the threat  assessment (at various stages) can be the evaluation of TM ∆  with respect to specific threat taxonomies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' This in turn will  determine the security policies and controls to rescope the  trust boundaries of systems against any possible change to  their administrative boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' DISCUSSION  a) Reconciliation of security requirements across com-  ponents with shared state: The E2EE messaging applications  were initially designed for mobile phones and the desktop  clients followed later on — mobile phone application contin-  ued to be the root of trust for the desktop clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' A pertinent  issue is if ‘trust’ in the security of the mobile application is  enough to trust the security of the desktop clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Can the  compromise of the desktop client, on the other hand, lower  the security of the mobile application account as well?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Setting up a desktop client involves sharing of credentials  between the primary device and the secondary device —  the credential is the shared state between the primary and  companion device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' As our analysis shows, for some of the  desktop clients, the shared system state is open to compromise  due to short-lived adversarial access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Research in secure systems development has considered the  question of when the security of the components is sufficient to  trust the larger system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=', the composability problem) [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Threat modeling of individual system components can help  identify the shared state between them and analyze the conse-  quences of such sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' If sharing is inescapable then security  (of the larger system) is perhaps incumbent on the adminis-  tration of the shared components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' In the context of security  of open distributed processing, Bull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' suggest independent  administration of the components [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' proposed  a model for autonomous administration of shared state for  secure inter-operation of systems [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Minimal sharing of  state information has been a longstanding security principle,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=', least common mechanism in the Saltzer & Schroeder  principles [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  b) Safe Defaults: Systems are designed with clear de-  lineation of the participants in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We learn from  our investigation of the desktop clients of the messaging  applications that some of them assume that these participants  have fixed behavior which does not change across space  and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The manner in which applications respond to the  change in behavior of the participants determine the security  of the applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Fail safe defaults has been a long standing  principle too [28] — participants with access to devices should  not be able to use the access maliciously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Developers, administrators and end users find it challenging  to make fail safe design and usage decisions [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' How-  9  ever, there are proposal to aid in making such decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Prompting and nudging developers when committing code,  for instance, may help developers make their code follow the  principles [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' There have been suggestions of contextualizing  the principles to the extant development requirements of the  application developers, for example, by further exploring the  applicability of the principles with respect to composability,  trustworthiness, and assurance [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' However, there remains a  need for further work to find systematic ways to make fail-  safety decisions [29] as well as exploring new mechanisms for  the suitability of deny access as a fail safe default [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' RELATED WORK  There is a sizable body of work investigating and demon-  strating vulnerabilities in E2EE messaging protocols and their  client implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Prior work has focused particularly on  Telegram due to its use of a custom protocol, and researchers  have demonstrated numerous attacks over the years on both  the MTProto protocol and its implementation [33], [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Most  recently, the Matrix protocol (which Element uses) suffered  several severe vulnerabilities in which a malicious actor was  able to break confidentiality of communications using a com-  promised Matrix server [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Research has also demonstrated the technical feasibility of  device cloning as a means of accessing user accounts on E2EE  messaging platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Cremers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' [36] investigated practical  post-compromise security measures in the mobile clients and  some desktop clients of the main E2EE messaging platforms,  finding that almost all clients studied are vulnerable to some  degree of cloning attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The authors identify challenges in  consistently maintaining state across devices as the primary  practical reason why these platforms generally do not attempt  to prevent cloning: a post-compromise security mechanism  can run the risk of being overly strict (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Figure 7b and  lock legitimate users out of their devices if synchronization  fails, prompting usability concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Cisco’s threat intelligence  unit further experimented with cloning a handful of E2EE  desktop clients (specifically, Signal, WhatsApp, and Telegram)  in the wake of the 2018 ‘Telegrab’ malware strain that took  advantage of Telegram’s vulnerability to desktop cloning [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Similarly to Cremers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=', they found that all three desktop  clients studied allowed a cloned client to masquerade as  legitimate, sending and receiving messages with little to no  indication to the original user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  In this paper, we contribute to this area by analyzing the  problem from a threat modeling perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We demonstrate  that the problem arises from a lack of consideration of threats  across space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Evolution of applications is a reality  as is addition of new features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' However, rescoping of the  trust boundary as both the application and the understanding  of threats evolves is critical to mitigating against emerging  threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Our analysis of the delineation of trust and administra-  tive boundaries provides a basis for such rescoping and better  administration of shared components, including securing them  with safe defaults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' CONCLUSION  Software is an intersection between its functional require-  ments and their security implications, where functionalities  take a precedence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' In our investigation, we observe two  contrasting ways in which desktop clients deal with the threats  they face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' One set of applications anticipate the possibility of  client cloning and implement various systems-level mitigation  strategies, while a second set puts the onus squarely on the user  to prevent device compromise in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' An additional  observation from our experiments is that the addition of a  less secure desktop client can substantially lower the security  of the mobile client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' While most of the desktop clients are  based on the same or similar cryptographic primitives, their  manifest diversity is a reflection of their varying perceptions  of an attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' The larger lesson here is that securing evolving  threats with assumptions made in a different context will result  in a mismatch leaving users exposed to attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' We make a  case for nuanced and in-depth modelling of an attacker in  appropriate contexts as integral to the software development  lifecycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Application features evolve – existing features are  deprecated or updated and new ones added – and so do threat  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Our key take away for application designers is to  not only do threat modeling little and often but also to pay  close attention to TM ∆ when doing so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Understanding how a  threat model evolves over space and time is key to rescoping  trust boundaries to mitigate against emergent threats that may  come into scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Another key take away is that during threat  modeling it is critical to evaluate if alignment of administrative  and trust boundaries leads to incorrect or weak assumptions  about the threats posed by those who may have (legitimate)  access within an administrative boundary but ought not to be  deemed trustworthy entities by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Security and privacy  is not a zero sum game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Delineation of trust boundaries and  security controls to protect assets and data flows accordingly  can be overly strict leading to usability challenges and hence  causing effort overload for legitimate users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Such trade-offs  between non-functional requirements are a well-known chal-  lenge in software engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' However, our analysis shows  – as evident in some of the E2EE applications we studied –  that, at least in our case study of E2EE applications, design  choices are possible and implementable that account for the  potential threats without significantly impacting usability for  a legitimate user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='  Future work should consider how application developers  (and teams) can maintain an ongoing view of TM ∆, for  instance, as part of key steps in development processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' New  requirements arising from such an analysis can be added to,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=', future sprints in agile development processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' In cases  where they cannot be addressed, the threat can be clearly  communicated to end users and guidance provided on how to  mitigate against them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
+page_content=' Other future work can include addition  of features within threat modeling tools to derive TM ∆ and  means to model administrative and trust boundaries explicitly  to make it easier for developers to explore the impact of their  design choices wrt particular threats over space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
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+page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfjA_C/content/2301.05653v1.pdf'}
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diff --git a/odE3T4oBgHgl3EQf7Qv3/content/tmp_files/2301.04798v1.pdf.txt b/odE3T4oBgHgl3EQf7Qv3/content/tmp_files/2301.04798v1.pdf.txt
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+arXiv:2301.04798v1  [math.PR]  12 Jan 2023
+Coloured and Dependent Planar Matchings of Random
+Bipartite Graphs
+Ghurumuruhan Ganesan1
+1IISER, Bhopal
+E-mail: gganesan82@gmail.com
+Abstract: In this paper, we study two problems related to planar matchings in ran-
+dom bipartite graphs. First, we colour each edge of the complete bipartite graph Kn,n
+uniformly randomly from amongst r colours and show that if r grows linearly with n,
+then the maximum rainbow matching is a non-trivial fraction of r, with high probability,
+i.e. with probability converging to one as n → ∞. Next we consider planar matchings
+in a dependent setting where each vertex is forced to choose exactly one neighbour from
+amongst all possible choices. We obtain estimates for the largest size of a planar match-
+ing and also discuss the implication of our results to longest increasing subsequences in
+enlarged random permutations.
+Keywords: Rainbow Planar Matchings, Dependent Planar Matchings, Random Bipar-
+tite Graphs.
+2010 Mathematics Subject Classification: Primary: 60J10, 60K35; Secondary:
+60C05, 62E10, 90B15, 91D30.
+1
+Introduction
+Planar matchings in random graphs have applications in determining longest length of
+increasing subsequences in permutations and related topics. In [8, 6, 7] the authors stud-
+ied various properties of the largest size of a planar matching in random bipartite graphs
+with dependencies. In the first part of the paper, we study rainbow planar matchings
+of randomly coloured bipartite graphs. Specifically, we colour each edge of the complete
+bipartite graph with a random colour and use martingale difference methods and segmen-
+tation to obtain deviation bounds on the maximum size of a rainbow planar matching.
+In the second part of our paper, we study dependent planar matchings of random
+bipartite graphs with applications to enlarged random permutations. Random permuta-
+tions are of great interest from both theoretical and application perspectives. In particu-
+lar, the longest increasing subsequence Mn of a uniformly randomly chosen permutation
+1
+
+Planar Matchings
+2
+of {1, 2, . . . , n} has been well-studied and various properties of Mn are known. The pa-
+per [10] initiated the study of the longest increasing subsequence of a randomly chosen
+permutation and [4] used subadditive methods to show that the expected value of Mn
+√n
+converges to a constant c. That c = 2 was independently determined in [11] and [5].
+Later [1] comprehensively determined the asymptotics of Mn including a central limit
+theorem and for a detailed survey, we refer to [9].
+In this paper, we consider a variant of uniform permutations which we call as en-
+larged permutations. Roughly speaking, we increase the “alphabet size” of the n−tuples
+to k ≥ n and study the deviation and expectation properties of the longest increasing
+subsequence Tn = Tn(k). The change in the alphabet size affects the underlying distribu-
+tion due to an increase in size of the sample space and we use a segmentation approach
+to obtain the desired bounds for Tn. Throughout, for completeness, we state and prove
+our results in the form of planar matchings.
+The paper is organized as follows: In Section 2, we state and prove our main result
+regarding the maximum size of rainbow planar matchings and in Section 3, we state and
+prove our main result regarding the maximum size of dependent planar matchings and
+describe how our result applies to enlarged random permutations.
+2
+Rainbow Planar Matchings
+For i ≥ 1, let ui = (i, 0)2 and vi = (i, 1)2 be points in R2. We use the subscript 2 to
+differentiate from the two-tuple notations for edges introduced later. For each i ̸= j
+join the vertices ui and vj by an edge to obtain an infinite bipartite graph Ktot. For
+integer n ≥ 1 let Kn,n be the complete bipartite graph containing n bottom vertices X =
+{u1, . . . , un} and n top vertices Y = {v1, . . . , vn}. This is illustrated in Figure 1 where S
+denotes the line x = 1 that contains the vertices of Y and T denotes the line x = 0 that
+contains the vertices of X.
+An edge e ∈ Kn,n if and only if e has one end-vertex ui ∈ X and the other end-
+vertex vj ∈ Y. We denote the edge e as e = (ui, vj). A matching of size t in Kn,n is a set
+of vertex disjoint edges W = {e1, . . . , et}. Suppose ei has end-vertices al = (uil, 1)2 ∈ X
+and bl = (vjl, 0)2 ∈ Y. We say that W is a planar matching if i1 < i2 < . . . < it
+and j1 < j2 < . . . < jt. In other words, no two edges in W intersect each other. The size
+of W is defined to the number of edges t.
+In Figure 1, we illustrate the above definition with an example of a planar match-
+ing W0 containing six edges (u1, v2), (u4, v3), (u5, v5), (u7, v6) and (u9, v7).
+Each edge f of Kn,n is now coloured with a colour Xf chosen uniformly randomly
+from the set {1, 2, . . . , r}, independent of other edges. Let G be the resulting randomly
+coloured graph and let W = {h1, h2, . . . , ht} be a planar matching of G containing t
+edges.
+2
+
+Planar Matchings
+3
+Figure 1: Illustration of a planar matching.
+Definition 2.1. We say that W is a rainbow planar matching if Xhi ̸= Xhj for any hi ̸=
+hj.
+In other words, all the colours in W must be distinct.
+Letting Rn denote the maximum size of a rainbow planar matching of G, we have the
+following result.
+Theorem 2.2. Let r = αn for some α > 0. There exists 0 < α0 < 1 such that for
+every α > α0, there are constants 0 < β1 ≤ β2 < 1 strictly and β3 > 0 such that
+P (β1r ≤ Rn ≤ β2r) ≥ 1 − e−β3n and β1r ≤ ERn ≤ β2r.
+(2.1)
+Moreover, var(Rn) ≤ 2ERn.
+In words, the above result says that the largest size of the rainbow planar matching
+is a non-trivial fraction of the total number of colours r, if the parameter α is close to 1.
+On the other hand, the proof of our result can be also adapted to show that if r = o(n),
+then Rn
+r −→ 1 in probability as n → ∞. This hints at the possibility of a transition point
+for the size of the rainbow planar matching, with respect to the parameter α.
+Proof Theorem 2.2: The proof of the variance bound uses a pivotal edge argument and
+is exactly as in the strong matching case (see proof of Theorem 2.2, [3]). Next we show
+that the expectation bounds follow from the deviation bounds. Indeed from the deviation
+lower bound in (2.1), we get ERn ≥ β1r(1 − e−β3n) and using the fact that Rn ≤ r and
+the upper bound in (2.1), we get that ERn ≤ β2r + re−β3n. This obtains the expectation
+bounds in (2.1).
+We now begin with the proof of the upper deviation bound in (2.1).
+Let S =
+{(ui, vi)}1≤i≤t be any planar matching containing t edges.
+The probability that S is
+3
+
+Planar Matchings
+4
+a rainbow planar matching is �t−1
+i=1
+�r−i
+r
+�
+which is bounded above by
+exp
+�
+−1
+r
+t−1
+�
+i=1
+i
+�
+=
+exp
+�
+−t(t − 1)
+2r
+�
+=
+exp
+� t
+2r
+�
+exp
+�
+− t2
+2r
+�
+≤
+√e exp
+�
+− t2
+2r
+�
+,
+since t ≤ r. The number of choices for {ui}1≤i≤t is
+�n
+t
+�
+and an analogous bound holds
+for {vi}1≤i≤t. Therefore if Fn = Fn(ǫ), ǫ < 1
+2 denotes the event that every rainbow planar
+matching has size at most (1 − ǫ)r, then by the union bound we have that
+P(F c
+n) ≤ √e
+�
+t≥(1−ǫ)r+1
+�n
+t
+�2
+· exp
+�
+− t2
+2r
+�
+≤ √e
+�
+t≥(1−ǫ)r+1
+�n
+t
+�2
+e−(1−ǫ)2r/2.
+(2.2)
+Assuming α > 1
+2, we choose ǫ > 0 small enough to apply the monotonicity of the Bi-
+nomial coefficient and the Stirling approximation successively to get that
+�n
+t
+�
+≤
+�
+n
+(1−ǫ)r
+�
+≤
+nenH((1−ǫ)α) where H(x) = −x log x − (1 − x) log(1 − x) is the binary entropy function
+and the logarithms are natural throughout. From (2.2) we therefore get that
+P(F c
+n) ≤ √e
+�
+t≥(1−ǫ)r+1
+n2e2nH((1−ǫ)α)e−(1−ǫ)2αn/2.
+(2.3)
+We know that H(x) −→ 0 as x → 0. Therefore if α0 is the solution to the equa-
+tion 2H(x) =
+x
+2, then for every α > α0, we can choose ǫ > 0 small in (2.3) to get
+that P(F c
+n) ≤ e−δn for some constant δ > 0. Fixing such an ǫ, gives the upper deviation
+bound in (2.1).
+For the lower deviation bound in (2.1), we argue as follows: For 0 < ǫ < 1
+2, let En
+be the event that the maximum rainbow planar matching contains t ≤ ǫr edges. We
+bound P(En) by estimating the size of the colour set of the n vertical edges fi =
+(ui, vi), 1 ≤ i ≤ n. The probability that the colour Xfi of the edge fi belongs to the
+set {1, 2, . . . , t} equals t
+r ≤ ǫ. Therefore the probability that each fi, 1 ≤ i ≤ n is coloured
+with a colour from {1, 2, . . . , t} is at most ǫn. Since there are
+�r
+t
+�
+≤
+� r
+ǫr
+�
+≤ 2r = 2αn ways
+of choosing t colours from the set of all available colours, we get by the union bound that
+P(En) ≤
+�
+1≤t≤ǫr
+2αnǫn ≤ n(2αǫ)n ≤ e−γn
+for some constant γ > 0 provided ǫ > 0 is small enough. Fixing such an ǫ, we get the
+lower deviation bound in (2.1).
+4
+
+Planar Matchings
+5
+3
+Dependent Planar Matchings
+Let Ktot be the infinite bipartite graph described in Section 2 and for integers k ≥ n ≥ 1,
+let Kk,n ⊆ Ktot be the complete bipartite graph containing the k top vertices v1, . . . , vk
+and the n bottom vertices u1, . . . , un. Let Gk,n be the set of all 1−regular subgraphs
+of Kk,n; i.e., the set of all subgraphs of Kk,n satisfying the property that each bottom
+vertex ui, 1 ≤ i ≤ n is adjacent to exactly one top vertex vj, 1 ≤ j ≤ k and each top
+vertex is adjacent to at most one bottom vertex. Let G be uniformly randomly chosen
+from Gk,n and let Tn = Tn(k) be the largest size of a planar matching of G.
+For k = n, the quantity Tn could be interpreted as the length of the longest increasing
+subsequence in a uniformly randomly chosen permutation of n. In this case, it is well-
+known that Tn is of the order of √n with high probability [9], i.e.
+with probability
+converging to one as n → ∞. We could therefore think of the above dependent planar
+matching setup as an “enlarged permutation” where we choose n−tuples with distinct
+entries from the set {1, 2, . . . , k} with k ≥ n.
+For general k ≥ n, we have the following result regarding the mean and variance
+of Tn. As before, constants do not depend on n.
+Theorem 3.1. We have:
+(a) For every ǫ > 0, there are positive constants Ci, 0 ≤ i ≤ 3 and a constant 0 < D0 < 1
+such that for all n ≥ C0,
+µlow := (1 − e−1)√n − 17
+√n ≤ ETn ≤ (e + ǫ)√n + 1 =: µup,
+(3.1)
+P (Tn ≤ µup) ≥ 1 − e−C1
+√n and P (Tn ≥ (1 − ǫ)µlow) ≥ 1 − zn,
+(3.2)
+where zn := min
+�
+D0, C2
+√n + C2n
+k
+�
+.
+(b) For every 2 < b = b(n) ≤
+n
+(32)2 log n we have that
+P
+�
+Tn ≥
+�
+n
+b log n
+�
+≥ 1 − exp
+�
+−
+�b
+2 − 1
+�
+log n
+�
+.
+(3.3)
+From (3.2) we see that if k grows faster than n, then the maximum planar matching
+size Tn is of the order of √n with high probability, i.e. with probability converging to
+one as n → ∞.
+The proof of the upper bounds in Theorem 3.1 is analogous to the case of random per-
+mutations (see Lemma 1.5 in Romik (2014)). For completeness, we give small proofs in
+the next section. In our proof of the lower deviation bounds in Theorem 3.1, we use seg-
+mentation to split the given graph Kk,n into smaller subgraphs and use planar matchings
+within the smaller subgraphs to obtain an estimate on the overall quantity Tn. Therefore,
+in the next subsection, we collect preliminary results regarding the minimum size of a
+segmented planar matching of Kk,n and independent planar matchings, respectively. In
+the final subsection, we prove Theorem 3.1.
+5
+
+Planar Matchings
+6
+Segmented Planar Matchings
+For integer 1 ≤ t ≤ n and for 1 ≤ i ≤ I :=
+n
+t let Ti be the set of all integers l
+satisfying (i−1)t+1 ≤ l ≤ it. For simplicity we have assumed that n
+t is an integer; if not
+then we let I be the largest integer such that I · t ≤ n and allow the last “segment” TI
+to be the set of all integers j satisfying
+(I − 1)t + 1 ≤ i ≤ n. In this case, TI contains between n
+t and 2n
+t
+integers. Similarly
+for an integer 1 ≤ s ≤ k and for 1 ≤ j ≤ J := k
+s let Si be the set of all integers l
+satisfying (j − 1)s + 1 ≤ l ≤ js. As before, we assume that k
+s is an integer and we always
+choose s ≤ tk
+n so that the number of “bottom” segments I ≤ J, the number of “top”
+segments.
+Suppose s = tk
+n so that I = J and let W be a planar matching of G. We say that W is
+a t−segmented planar matching if for each edge e ∈ W, there exists an integer 1 ≤ i ≤ I
+such that one end-vertex of e belongs to Ti and the other end-vertex of e belongs to Si.
+For 1 ≤ i ≤ n
+t , let Ai be the event that there exists an integer z ∈ Ti such that π(z) ∈ Si.
+Defining the minimum size of a t−segmented planar matching of the random graph G
+to be
+Xt = Xt(s) :=
+n/t
+�
+i=1
+11(Ai),
+(3.4)
+we have the following estimates for the mean and variance of Xt.
+Lemma 3.1. Suppose s = kt
+n so that I = J. Letting µt := n
+t
+�
+1 − exp
+�
+− t2
+n
+��
+, we have
+that
+µt − 17
+t e− t2
+n ≤ EXt ≤ µt + 32n
+k2 e− t2
+n .
+(3.5)
+If θ1
+√n ≤ t ≤ θ2
+√n for some constants θ1, θ2 > 0, then there exists a positive con-
+stant D = D(θ1, θ2) such that
+var(Xt) ≤ D
+�√n + n2
+k
+�
+(3.6)
+In the proof of Theorem 3.1 in the next Section, we use the fact that Xt is a lower
+bound on Tn to obtain deviation and expectation estimates for Tn.
+Proof of (3.5) in Lemma 3.1: We use Lemma 4.2 in Appendix with s = kt
+n to get
+from the upper bound in (A.2) that
+P(Ac
+1) ≤ exp
+�
+−t2
+n + 4(s + t)2t
+k2
+�
+.
+The term
+(s + t)2t
+k2
+= (k + n)2t
+k2n2
+≤ 4t
+n2 ≤ 4
+n
+(3.7)
+6
+
+Planar Matchings
+7
+where the first inequality in (3.7) is true since k ≥ n and the second inequality in (3.7)
+follows from the fact that t ≤ n. Thus
+P(Ac
+1) ≤ exp
+�
+−t2
+n + 16
+n
+�
+≤ exp
+�
+−t2
+n
+� �
+1 + 17
+n
+�
+(3.8)
+for all n large and so
+EXt = n
+t (1 − P(Ac
+1)) ≥ n
+t
+�
+1 − exp
+�
+−t2
+n
+��
+− 17
+t e− t2
+n .
+This obtains the lower bound in (3.5).
+Similarly using the lower bound in (A.2) we get that
+P(Ac
+1) ≥ exp
+�
+−t2
+n − 8(s + t)2t
+k2
+�
+,
+where (s+t)2t
+k2
+= (k+n)2t
+k2n2
+≥ 4t
+k2 since k ≥ n. Thus
+P(Ac
+1) ≥ exp
+�
+−t2
+n
+�
+exp
+�
+−32t
+k2
+�
+≥ exp
+�
+−t2
+n
+� �
+1 − 32t
+k2
+�
+and consequently,
+EXt = n
+t (1 − P(Ac
+1)) ≤ n
+t
+�
+1 − exp
+�
+−t2
+n
+��
++ 32n
+k2 e− t2
+n .
+This obtains the upper bound in (3.5).
+Proof of (3.6) in Lemma 3.1: To obtain the deviation estimate for Xt, we let Y =
+n
+t − Xt and first estimate the variance of Y. Setting
+cov(11(Ac
+i), 11(Ac
+j)) := P(Ac
+i ∩ Ac
+j) − P(Ac
+i)P(Ac
+j)
+to be the covariance between the indicator functions of the events Ai and Aj, we have
+that
+var(Y )
+=
+n/t
+�
+i=1
+P(Ac
+i) − P2(Ac
+i) + 2
+�
+i<j
+cov(11(Ac
+i), 11(Ac
+j))
+≤
+n
+t
+�
+P(Ac
+1) − P2(Ac
+1)
+�
++ 2n2
+t2 cov(11(Ac
+1), 11(Ac
+2))
+≤
+n
+t P(Ac
+1) + 2n2
+t2 cov(11(Ac
+1), 11(Ac
+2))
+≤
+2n
+t e− t2
+n + 2n2
+t2 cov(11(Ac
+1), 11(Ac
+2))
+(3.9)
+7
+
+Planar Matchings
+8
+for all n large, using (3.8).
+For the second term in (3.9), we use the correlation estimate (A.3) in Lemma 4.2 to
+get that
+cov(11(Ac
+1), 11(Ac
+2)) ≤ P(Ac
+1)P(Ac
+2)
+�
+e
+5t2
+k − 1
+�
+.
+Since t ≤ θ2
+√n, we have that 5t2
+k ≤ 5θ2
+2n
+k
+≤ 5θ2
+2. For all x ≤ 5θ2
+2, we have that ex ≤ 1+D1x
+for some large constant D1 > 0 and so again using θ1
+√n ≤ t ≤ θ2
+√n, we get that
+cov(11(Ac
+1), 11(Ac
+2)) ≤ 5D1
+t2
+k P(Ac
+1)P(Ac
+2) ≤ D2n
+k e− 2t2
+n ≤ D3n
+k
+(3.10)
+for some constants D2, D3 > 0. Plugging (3.10) into (3.9) and using θ1
+√n ≤ t ≤ θ2
+√n,
+we get that
+var(Y ) ≤ 2n
+t e− t2
+n + n2
+t2 · D3n
+k
+≤ D4
+�√n + n2
+k
+�
+for some constant D4 > 0. Since var(Xt) = var(Y ), this completes the proof of (3.6).
+Proof of Theorem 3.1
+Proof of the upper bound in (3.1): For a given integer l ≥ 1 we first estimate the prob-
+ability of the event El that the random graph G contains a planar matching of size l.
+Indeed, let π(i) be the x−coordinate of the top vertex adjacent to the bottom ver-
+tex ui in the graph G and suppose that π(i1) < . . . < π(il) for some deterministic
+integers 1 ≤ i1 < i2 < . . . < il ≤ n. The number of choices for {i1, . . . , il} is
+�n
+l
+�
+and the
+number of choices for {π(i1), . . . , π(il)} is
+�k
+l
+�
+. From (A.4) and the union bound, we then
+get that
+P(El) ≤
+�n
+l
+�
+·
+�k
+l
+�
+·
+1
+k(k − 1) . . . (k − l + 1) =
+�n
+l
+�
+· 1
+l!.
+Using
+�n
+l
+�
+≤
+�ne
+l
+�l and l! ≥ lle−l, we get that P(El) ≤
+�
+ne2
+l2
+�l
+≤
+� e
+a
+�2l for l ≥ a√n.
+If a > e strictly then
+� e
+a
+�2l = e−2δl for some constant δ > 0 and so
+P(El) ≤ e−2δl. If Eup denotes the event that G contains a planar matching of length at
+least a√n, then from the union bound, we get that
+P(Eup) ≤ n · e−2aδ√n ≤ e−aδ√n
+(3.11)
+for all n large. To upper bound the expectation, we use the fact that if Eup occurs,
+then Tn ≤ a√n and if the complement event Ec
+up occurs, then Tn ≤ n. Therefore
+from (3.11), we get that ETn ≤ a√n + n · e−aδ√n ≤ a√n + 1 for all n large.
+This
+obtains the upper bound in (3.1).
+8
+
+Planar Matchings
+9
+Proof of the lower bound in (3.1): By definition the term Xt defined in (3.4) is a
+lower bound on the maximum size Tn of a planar matching of G. To see this is true
+suppose X = w and suppose Aj1, . . . , Ajw occur with j1 < . . . < jw. For each 1 ≤ u ≤ w,
+there exists an index pu ∈ Tju and an integer qu ∈ Sju such that π(pu) = qu. By
+definition π(p1) < π(p2) < . . . < π(pw) and so G contains a planar matching of size at
+least w.
+We therefore use Lemma 3.1 with t = √n to get that µt = √n(1 − e−1) and so from
+the lower bound in (3.5), we get that EXt ≥ (1−e−1)√n− 17
+√n. This completes the proof
+of the lower bound in (3.1).
+Proof of (3.2): The upper bound follows from (3.11). To prove the lower bound, we
+use Lemma 3.1 with t = √n. First from the upper bound in (3.5) and the fact that k ≥ n,
+we get that
+EXt ≤ (1 − e−1)√n + 32n
+k2 ≤ (1 − e−1)√n + 32
+n ≤ √n.
+From the variance estimate (3.6), we already know that
+var(Xt) ≤ D
+�√n + n2
+k
+�
+.
+(3.12)
+We now use the Paley-Zygmund inequality to show that Xt ≥ (1−ǫ)µlow with positive
+probability and then use the fact that Xt is a lower bound for Tn to get that Tn ≥
+(1 − ǫ)µlow with positive probability. From Paley-Zygmund inequality we have for ǫ > 0
+that
+P(Xt ≥ (1 − ǫ)EXt) ≥ ǫ2(EXt)2
+EX2
+t
+and so using EXt ≥ µlow ≥
+√n
+2 , we have P(Xt ≥ (1 − ǫ)µlow) ≥
+ǫ2n
+4EX2
+t . From (3.12) and
+the fact that k ≥ n, we then obtain EX2
+t ≤ D1n for some constant D1 > 0. Conse-
+quently Xt ≥ (1 − ǫ)µlow with probability at least
+ǫ2
+4D1 =: D0.
+Next, using the Chebychev’s inequality, we have for ǫ > 0 that
+P (|Xt − EXt| ≤ ǫEXt) ≥ 1 − var(Xt)
+ǫ2(EXt)2 .
+Since EXt ≥ µlow we get from (3.12) that P (Xt ≥ (1 − ǫ)µlow) ≥ 1 − D2
+√n − D3n
+k
+for some
+positive constants Di = Di(ǫ), i = 2, 3. This obtains the second lower bound in (3.2) and
+therefore completes the proof of the deviation lower bound in (3.2).
+Proof of (3.3): We set t := √bn log n and s := kt
+n so that the number of segments I =
+J and use Lemma 4.2 to estimate the probability of the event P(Ai) = P(A1). By the
+9
+
+Planar Matchings
+10
+definition of t and the fact that k ≥ n, we respectively get
+st
+k = t2
+n = b log n and s + t
+k
+= t
+k + t
+n ≤ 2t
+n = o(1)
+(3.13)
+and so the conditions in Lemma 4.2 are satisfied. Moreover, using k ≥ n again, we also
+get that
+(s + t)2t
+k2
+= t3(k + n)2
+k2n2
+≤ 4t3
+n2 = 4(b log n)3/2
+√n
+(3.14)
+Plugging (3.13) and (3.14) into the upper bound of (A.2), we get that
+P(Ac
+1)
+≤
+e−b log n · exp
+�
+16(b log n)3/2
+√n
+�
+=
+exp
+�
+−b log n
+�
+1 − 16
+�
+b log n
+n
+��
+≤
+exp
+�
+−b log n
+2
+�
+for all n large since b ≤
+n
+(32)2 log n (see statement of Theorem 3.1) and so by the union
+bound we get that �n/t
+i=1 Ai occurs with probability at least
+1 − n
+t · exp
+�
+−b
+2 log n
+�
+≥ 1 − n · exp
+�
+−b
+2 log n
+�
+.
+This in turn implies that Tn ≥ Xt = n
+t =
+�
+n
+b log n with probability least
+1 − n · exp
+�
+− b
+2 log n
+�
+, completing the proof of (3.3).
+4
+Conclusion
+In this paper, we have studied coloured and dependent planar matchings in random
+bipartite graphs. In the colouring part, we have shown that the largest rainbow matching
+is a non-trivial fraction of the total number of colours with high probability. This hints
+at a possible transition point with respect to the colouring parameter. In the dependent
+setting, we have obtained estimates on the largest size of a planar matching and also
+explained our results in terms of longest increasing subsequences in random enlarged
+permutations.
+10
+
+Planar Matchings
+11
+Appendix
+Throughout we use the following standard deviation estimate.
+Lemma 4.1. Let {Xj}1≤j≤r be independent Bernoulli random variables with
+P(Xj = 1) = 1 − P(Xj = 0) > 0.
+If Tr := �r
+j=1 Xj, θr := ETr and 0 < γ ≤ 1
+2, then
+P (|Tr − θr| ≥ θrγ) ≤ 2 exp
+�
+−γ2
+4 θr
+�
+(A.1)
+for all r ≥ 1.
+For a proof of (A.1), we refer to Corollary A.1.14, pp. 312 of Alon and Spencer (2008).
+To prove Lemma 3.1, we use the following preliminary result that obtains estimates
+on the probability of the events Ac
+i and Ac
+i ∩ Ac
+j.
+Lemma 4.2. Let 1 ≤ t ≤ n and 1 ≤ s ≤ k be any two integers satisfying
+s+t
+k
+≤ 1
+8. We have that
+exp
+�
+−st
+k − 8(s + t)2t
+k2
+�
+≤ P(Ac
+1) ≤ exp
+�
+−st
+k + 4(s + t)2t
+k2
+�
+(A.2)
+and
+P(Ac
+1 ∩ Ac
+2) ≤ P(Ac
+1)P(Ac
+2) exp
+�5t2
+k
+�
+.
+(A.3)
+Proof of (A.2) in Lemma 4.2: For any integer 1 ≤ l ≤ n and l distinct integers 1 ≤
+a1, . . . , al ≤ n, we have that
+P(u1 = a1, . . . , ul = al) =
+1
+k(k − 1) . . . (k − l + 1).
+(A.4)
+Let π(i) be the x−coordinate of the top vertex adjacent to ui in the graph G. If the
+event Ac
+1 occurs, then the number of choices for π(1) is k − s. Similarly, given π(1),
+the number of choices for π(2) is k − s − 1 and so on. Thus the number of choices for
+the t−tuple (π(1), . . . , π(t)) is
+(k − s) · (k − s − 1) · · · (k − s − t + 1)
+and so we get from (A.4) that
+P(Ac
+1)
+=
+(k − s) · (k − s + 1) . . . (k − s − t + 1)
+k(k − 1) · · · (k − t + 1)
+=
+�
+1 − s
+k
+�
+· · ·
+�
+1 − s+t−1
+k
+�
+�
+1 − 1
+k
+�
+· · ·
+�
+1 − t−1
+k
+� .
+(A.5)
+11
+
+Planar Matchings
+12
+Using s+t
+k
+≤ 1
+4 we get for all x ≤ 2(s+t)
+k
+that
+e−x
+1 − x
+=
+1 +
+1
+1 − x
+�x2
+2! − x3
+3! + . . .
+�
+≤
+1 + 2x2
+1 − x
+≤
+1 + 4x2.
+(A.6)
+Thus
+e−x ·
+�
+1 + 4(s + t)2
+k2
+�−1
+≤ 1 − x ≤ e−x.
+(A.7)
+Plugging the upper bound of (A.7) into the numerator of (A.5) and the lower bound
+of (A.7) into the denominator, we get that P(Ac
+1) ≤ I1 · I2 where
+I1 :=
+exp
+�
+− 1
+k
+�s+t−1
+l=s
+k
+�
+exp
+�
+− 1
+k
+�t−1
+l=1 k
+�
+= exp
+�
+−st
+k
+�
+(A.8)
+and
+I2 :=
+�
+1 + 4(s + t)2
+k2
+�t
+≤ exp
+�4(s + t)2t
+k2
+�
+.
+This obtains the upper bound in (A.2).
+Similarly, substituting the lower bound of (A.7) into the numerator of (A.5) and the
+upper bound of (A.7) into the denominator, we get that
+P(Ac
+1) ≥ e− st
+k ·
+�
+1 + 4(s + t)2
+k2
+�−t
+≥ e− st
+k ·
+�
+1 − 4(s + t)2
+k2
+�t
+.
+(A.9)
+Again using s+t
+k
+≤ 1
+2 and the fact that 1−x ≥ e−2x for all x < 1
+2, we get the lower bound
+in (A.2) from the final expression in (A.9).
+Proof of (A.3) in Lemma 4.2: We use a split set argument and as before, π(i), 1 ≤
+i ≤ n is the x−coordinate of the top vertex adjacent to ui, in the graph Kk,n. For a
+deterministic set Q1 ⊆ {1, 2, . . . , t}, let F1(Q1) be the event that π(j) ∈ {s + 1, . . . , 2s}
+for every integer j ∈ Q1 and π(l) /∈ {1, 2, . . . , 2s} for every l ∈ {1, 2, . . . , t}\Q1. Similarly
+for Q2 ⊆ {t+1, . . . , 2t}, let F2(Q2) be the event that π(j) ∈ {1, . . . , s} for every integer j ∈
+Q2 and π(j) /∈ {1, 2, . . . , 2s} for every j ∈ {t+1, . . . , 2t}\Q2. By definition we then have
+that
+Ac
+1 ∩ Ac
+2 =
+�
+Q1,Q2
+F1(Q1) ∩ F2(Q2),
+(A.10)
+12
+
+Planar Matchings
+13
+where the union is over all sets Q1 ⊆ {1, 2, . . . , t} and Q2 ⊆ {t + 1, . . . , 2t}. Thus
+P(Ac
+1 ∩ Ac
+2) =
+�
+j1,j2
+�
+#Q1=j1
+�
+#Q2=j2
+P (F1(Q1) ∩ F2(Q2)) .
+(A.11)
+Letting (a)b :=
+� a(a − 1) · · · (a − b + 1)
+if b ≥ 1
+1
+if b = 0,
+we have that
+P(F1(Q1)) = (s)j1(k − 2s)t−j1
+(k)t
+.
+A similar expression holds for F2(Q2) and the intersection F1(Q1)∩F2(Q2). Letting jlow :=
+min(j1, j2) and jup := max(j1, j2) and combining the above expressions, we get that
+P(F1(Q1) ∩ F2(Q2))
+P(F1(Q1))P(F2(Q2)) = f(k − 2s, t − jlow, t − jup)
+f(k, t, t)
+,
+(A.12)
+where f(a, b, c) := (a−b)c
+(a)c .
+Suppose c ≥ 1 and b+c
+a
+< 1
+2. We rewrite
+f(a, b, c) =
+c−1
+�
+j=0
+�
+1 − b + j
+a
+�
+·
+c−1
+�
+j=1
+�
+1 − j
+a
+�−1
+and use the estimate e−2x ≤ 1 − x ≤ e−x for x < 1
+2 to get that
+f(a, b, c) ≤
+c−1
+�
+j=0
+exp
+�
+−b + j
+a
++ 2j
+a
+�
+≤ e− bc
+a + c2
+a ≤ e
+c2
+a .
+(A.13)
+Similarly
+f(a, b, c) ≥
+c�
+j=1
+exp
+�
+−2b + 2j
+a
++ j
+a
+�
+= e− 2bc
+a exp
+
+−
+c
+�
+j=1
+j
+a
+
+ ≥ e− 2bc
+a − c2
+a .
+(A.14)
+We now use (A.13) with a = k − 2s, b = t − jlow and c = t − jup to evaluate f(k −
+2s, t − jlow, t − jup). We have that
+b + c
+a
+= 2t − j1 − j2
+k − 2s
+≤
+2t
+k − 2s ≤ 4t
+k < 1
+2
+(A.15)
+where the second inequality in (A.15) follows from s
+k < s+t
+k
+≤ 1
+4 and the third inequality
+in (A.15) again follows from t
+k < s+t
+k
+≤ 1
+8 (see statement of Lemma 4.2). From (A.13)
+we therefore have
+f(k − 2s, t − jlow, t − jup) ≤ exp
+�(t − jup)2
+k − 2s
+�
+≤ exp
+�
+t2
+k − 2s
+�
+≤ exp
+�2t2
+k
+�
+(A.16)
+13
+
+Planar Matchings
+14
+since s
+k ≤ s+t
+k
+≤ 1
+4. Similarly, using (A.14) with a = k, b = c = t we get that
+f(k, t, t) ≥ exp
+�
+−3t2
+k
+�
+(A.17)
+Plugging (A.16) and (A.17) into (A.12), we get that
+P(F1(Q1) ∩ F2(Q2))
+P(F1(Q1))P(F2(Q2)) ≤ exp
+�5t2
+k
+�
+.
+(A.18)
+Substituting (A.18) into (A.11) and summing over Q1 and Q2, we get (A.3).
+Acknowledgement
+I thank Professors Rahul Roy, C. R. Subramanian and the referee for crucial comments
+that led to an improvement of the paper. I also thank IMSc and IISER Bhopal for my
+fellowships.
+References
+[1] J. Baik, P. Deift and K. Johansson. (1999). On the Distribution of the Length of the
+Longest Increasing Subsequence of Random Permutations. Journal of the American
+Mathematical Society, 12, pp. 1119–1178.
+[2] B. Bollob´as. (2001). Random Graphs. Cambridge University Press.
+[3] G. Ganesan. (2021).
+Strong and Weighted Matchings in Inhomogenous Random
+Graphs. Electronic Communications in Probability, pp. 1–12.
+[4] J. M. Hammersley. (1972). A Few Seedlings of Research. Proceedings of the Sixth
+Berkeley Symposium on Mathematical Statistics and Probability, pp. 345–294.
+[5] B. F. Logan and L. A. Shepp. (1977). A Variational Problem for Random Young
+Tableaux. Advances in Mathematics, 26, pp. 206–222.
+[6] M. Kiwi and M. Loebl. (2002).
+Largest Planar Matching in Random Bipartite
+Graphs. Random Structures and Algorithms, 21, 162–181.
+[7] M. Kiwi and M. Loebl. (2008). Towards the Distribution of the Size of a Largest
+Planar Matching and Largest Planar Subgraphs in Random Bipartite Graphs. Elec-
+tronic Journal of Combinatorics, 15:R135, 1–20.
+[8] K. Johansson. (2000). Shape fluctuations and random matrices. Communications
+in Mathematical Physics, 209, 437–476.
+14
+
+Planar Matchings
+15
+[9] D. Romik. (2014). The Surprising Mathematics of Longest Increasing Subsequences.
+Cambridge University Press.
+[10] S. Ulam. (1961). Monte Carlo Calculations in Problems of Mathematical Physics.
+Modern Mathematics For the Engineer (McGraw-Hill), pp. 261–281.
+[11] A. M. Vershik and S. V. Kerov. (1977). Asymptotics of the Plancherel Measure of the
+Symmetric Group and the Limiting Shape of Young Tableaux. Soviet Mathematics
+Doklady, 18, pp. 527–531.
+15
+
diff --git a/odE3T4oBgHgl3EQf7Qv3/content/tmp_files/load_file.txt b/odE3T4oBgHgl3EQf7Qv3/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..e419fc8f1e0d45fe9573f01e4c0dfe81152e046b
--- /dev/null
+++ b/odE3T4oBgHgl3EQf7Qv3/content/tmp_files/load_file.txt
@@ -0,0 +1,523 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf,len=522
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='04798v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='PR]  12 Jan 2023  Coloured and Dependent Planar Matchings of Random  Bipartite Graphs  Ghurumuruhan Ganesan1  1IISER, Bhopal  E-mail: gganesan82@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='com  Abstract: In this paper, we study two problems related to planar matchings in ran-  dom bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' First, we colour each edge of the complete bipartite graph Kn,n  uniformly randomly from amongst r colours and show that if r grows linearly with n,  then the maximum rainbow matching is a non-trivial fraction of r, with high probability,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' with probability converging to one as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Next we consider planar matchings  in a dependent setting where each vertex is forced to choose exactly one neighbour from  amongst all possible choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' We obtain estimates for the largest size of a planar match-  ing and also discuss the implication of our results to longest increasing subsequences in  enlarged random permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Keywords: Rainbow Planar Matchings, Dependent Planar Matchings, Random Bipar-  tite Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  2010 Mathematics Subject Classification: Primary: 60J10, 60K35;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Secondary:  60C05, 62E10, 90B15, 91D30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  1  Introduction  Planar matchings in random graphs have applications in determining longest length of  increasing subsequences in permutations and related topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' In [8, 6, 7] the authors stud-  ied various properties of the largest size of a planar matching in random bipartite graphs  with dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' In the first part of the paper, we study rainbow planar matchings  of randomly coloured bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Specifically, we colour each edge of the complete  bipartite graph with a random colour and use martingale difference methods and segmen-  tation to obtain deviation bounds on the maximum size of a rainbow planar matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  In the second part of our paper, we study dependent planar matchings of random  bipartite graphs with applications to enlarged random permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Random permuta-  tions are of great interest from both theoretical and application perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' In particu-  lar, the longest increasing subsequence Mn of a uniformly randomly chosen permutation  1  Planar Matchings  2  of {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , n} has been well-studied and various properties of Mn are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' The pa-  per [10] initiated the study of the longest increasing subsequence of a randomly chosen  permutation and [4] used subadditive methods to show that the expected value of Mn  √n  converges to a constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' That c = 2 was independently determined in [11] and [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Later [1] comprehensively determined the asymptotics of Mn including a central limit  theorem and for a detailed survey, we refer to [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  In this paper, we consider a variant of uniform permutations which we call as en-  larged permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Roughly speaking, we increase the “alphabet size” of the n−tuples  to k ≥ n and study the deviation and expectation properties of the longest increasing  subsequence Tn = Tn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' The change in the alphabet size affects the underlying distribu-  tion due to an increase in size of the sample space and we use a segmentation approach  to obtain the desired bounds for Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Throughout, for completeness, we state and prove  our results in the form of planar matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  The paper is organized as follows: In Section 2, we state and prove our main result  regarding the maximum size of rainbow planar matchings and in Section 3, we state and  prove our main result regarding the maximum size of dependent planar matchings and  describe how our result applies to enlarged random permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  2  Rainbow Planar Matchings  For i ≥ 1, let ui = (i, 0)2 and vi = (i, 1)2 be points in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' We use the subscript 2 to  differentiate from the two-tuple notations for edges introduced later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' For each i ̸= j  join the vertices ui and vj by an edge to obtain an infinite bipartite graph Ktot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' For  integer n ≥ 1 let Kn,n be the complete bipartite graph containing n bottom vertices X =  {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , un} and n top vertices Y = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , vn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' This is illustrated in Figure 1 where S  denotes the line x = 1 that contains the vertices of Y and T denotes the line x = 0 that  contains the vertices of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  An edge e ∈ Kn,n if and only if e has one end-vertex ui ∈ X and the other end-  vertex vj ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' We denote the edge e as e = (ui, vj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' A matching of size t in Kn,n is a set  of vertex disjoint edges W = {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , et}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Suppose ei has end-vertices al = (uil, 1)2 ∈ X  and bl = (vjl, 0)2 ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' We say that W is a planar matching if i1 < i2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' < it  and j1 < j2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' < jt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' In other words, no two edges in W intersect each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' The size  of W is defined to the number of edges t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  In Figure 1, we illustrate the above definition with an example of a planar match-  ing W0 containing six edges (u1, v2), (u4, v3), (u5, v5), (u7, v6) and (u9, v7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Each edge f of Kn,n is now coloured with a colour Xf chosen uniformly randomly  from the set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , r}, independent of other edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Let G be the resulting randomly  coloured graph and let W = {h1, h2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , ht} be a planar matching of G containing t  edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  2  Planar Matchings  3  Figure 1: Illustration of a planar matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' We say that W is a rainbow planar matching if Xhi ̸= Xhj for any hi ̸=  hj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  In other words, all the colours in W must be distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Letting Rn denote the maximum size of a rainbow planar matching of G, we have the  following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Let r = αn for some α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' There exists 0 < α0 < 1 such that for  every α > α0, there are constants 0 < β1 ≤ β2 < 1 strictly and β3 > 0 such that  P (β1r ≤ Rn ≤ β2r) ≥ 1 − e−β3n and β1r ≤ ERn ≤ β2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1)  Moreover, var(Rn) ≤ 2ERn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  In words, the above result says that the largest size of the rainbow planar matching  is a non-trivial fraction of the total number of colours r, if the parameter α is close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  On the other hand, the proof of our result can be also adapted to show that if r = o(n),  then Rn  r −→ 1 in probability as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' This hints at the possibility of a transition point  for the size of the rainbow planar matching, with respect to the parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Proof Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2: The proof of the variance bound uses a pivotal edge argument and  is exactly as in the strong matching case (see proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2, [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Next we show  that the expectation bounds follow from the deviation bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Indeed from the deviation  lower bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1), we get ERn ≥ β1r(1 − e−β3n) and using the fact that Rn ≤ r and  the upper bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1), we get that ERn ≤ β2r + re−β3n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' This obtains the expectation  bounds in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  We now begin with the proof of the upper deviation bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Let S =  {(ui, vi)}1≤i≤t be any planar matching containing t edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  The probability that S is  3  Planar Matchings  4  a rainbow planar matching is �t−1  i=1  �r−i  r  �  which is bounded above by  exp  �  −1  r  t−1  �  i=1  i  �  =  exp  �  −t(t − 1)  2r  �  =  exp  � t  2r  �  exp  �  − t2  2r  �  ≤  √e exp  �  − t2  2r  �  ,  since t ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' The number of choices for {ui}1≤i≤t is  �n  t  �  and an analogous bound holds  for {vi}1≤i≤t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Therefore if Fn = Fn(ǫ), ǫ < 1  2 denotes the event that every rainbow planar  matching has size at most (1 − ǫ)r, then by the union bound we have that  P(F c  n) ≤ √e  �  t≥(1−ǫ)r+1  �n  t  �2  exp  �  − t2  2r  �  ≤ √e  �  t≥(1−ǫ)r+1  �n  t  �2  e−(1−ǫ)2r/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2)  Assuming α > 1  2, we choose ǫ > 0 small enough to apply the monotonicity of the Bi-  nomial coefficient and the Stirling approximation successively to get that  �n  t  �  ≤  �  n  (1−ǫ)r  �  ≤  nenH((1−ǫ)α) where H(x) = −x log x − (1 − x) log(1 − x) is the binary entropy function  and the logarithms are natural throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2) we therefore get that  P(F c  n) ≤ √e  �  t≥(1−ǫ)r+1  n2e2nH((1−ǫ)α)e−(1−ǫ)2αn/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='3)  We know that H(x) −→ 0 as x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Therefore if α0 is the solution to the equa-  tion 2H(x) =  x  2, then for every α > α0, we can choose ǫ > 0 small in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='3) to get  that P(F c  n) ≤ e−δn for some constant δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Fixing such an ǫ, gives the upper deviation  bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  For the lower deviation bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1), we argue as follows: For 0 < ǫ < 1  2, let En  be the event that the maximum rainbow planar matching contains t ≤ ǫr edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' We  bound P(En) by estimating the size of the colour set of the n vertical edges fi =  (ui, vi), 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' The probability that the colour Xfi of the edge fi belongs to the  set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , t} equals t  r ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Therefore the probability that each fi, 1 ≤ i ≤ n is coloured  with a colour from {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , t} is at most ǫn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Since there are  �r  t  �  ≤  � r  ǫr  �  ≤ 2r = 2αn ways  of choosing t colours from the set of all available colours, we get by the union bound that  P(En) ≤  �  1≤t≤ǫr  2αnǫn ≤ n(2αǫ)n ≤ e−γn  for some constant γ > 0 provided ǫ > 0 is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Fixing such an ǫ, we get the  lower deviation bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  4  Planar Matchings  5  3  Dependent Planar Matchings  Let Ktot be the infinite bipartite graph described in Section 2 and for integers k ≥ n ≥ 1,  let Kk,n ⊆ Ktot be the complete bipartite graph containing the k top vertices v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , vk  and the n bottom vertices u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Let Gk,n be the set of all 1−regular subgraphs  of Kk,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=', the set of all subgraphs of Kk,n satisfying the property that each bottom  vertex ui, 1 ≤ i ≤ n is adjacent to exactly one top vertex vj, 1 ≤ j ≤ k and each top  vertex is adjacent to at most one bottom vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Let G be uniformly randomly chosen  from Gk,n and let Tn = Tn(k) be the largest size of a planar matching of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  For k = n, the quantity Tn could be interpreted as the length of the longest increasing  subsequence in a uniformly randomly chosen permutation of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' In this case, it is well-  known that Tn is of the order of √n with high probability [9], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  with probability  converging to one as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' We could therefore think of the above dependent planar  matching setup as an “enlarged permutation” where we choose n−tuples with distinct  entries from the set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , k} with k ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  For general k ≥ n, we have the following result regarding the mean and variance  of Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' As before, constants do not depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' We have:  (a) For every ǫ > 0, there are positive constants Ci, 0 ≤ i ≤ 3 and a constant 0 < D0 < 1  such that for all n ≥ C0,  µlow := (1 − e−1)√n − 17  √n ≤ ETn ≤ (e + ǫ)√n + 1 =: µup,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1)  P (Tn ≤ µup) ≥ 1 − e−C1  √n and P (Tn ≥ (1 − ǫ)µlow) ≥ 1 − zn,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2)  where zn := min  �  D0, C2  √n + C2n  k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  (b) For every 2 < b = b(n) ≤  n  (32)2 log n we have that  P  �  Tn ≥  �  n  b log n  �  ≥ 1 − exp  �  −  �b  2 − 1  �  log n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='3)  From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2) we see that if k grows faster than n, then the maximum planar matching  size Tn is of the order of √n with high probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' with probability converging to  one as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  The proof of the upper bounds in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1 is analogous to the case of random per-  mutations (see Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='5 in Romik (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' For completeness, we give small proofs in  the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' In our proof of the lower deviation bounds in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1, we use seg-  mentation to split the given graph Kk,n into smaller subgraphs and use planar matchings  within the smaller subgraphs to obtain an estimate on the overall quantity Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Therefore,  in the next subsection, we collect preliminary results regarding the minimum size of a  segmented planar matching of Kk,n and independent planar matchings, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' In  the final subsection, we prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  5  Planar Matchings  6  Segmented Planar Matchings  For integer 1 ≤ t ≤ n and for 1 ≤ i ≤ I :=  n  t let Ti be the set of all integers l  satisfying (i−1)t+1 ≤ l ≤ it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' For simplicity we have assumed that n  t is an integer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' if not  then we let I be the largest integer such that I · t ≤ n and allow the last “segment” TI  to be the set of all integers j satisfying  (I − 1)t + 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' In this case, TI contains between n  t and 2n  t  integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Similarly  for an integer 1 ≤ s ≤ k and for 1 ≤ j ≤ J := k  s let Si be the set of all integers l  satisfying (j − 1)s + 1 ≤ l ≤ js.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' As before, we assume that k  s is an integer and we always  choose s ≤ tk  n so that the number of “bottom” segments I ≤ J, the number of “top”  segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Suppose s = tk  n so that I = J and let W be a planar matching of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' We say that W is  a t−segmented planar matching if for each edge e ∈ W, there exists an integer 1 ≤ i ≤ I  such that one end-vertex of e belongs to Ti and the other end-vertex of e belongs to Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  For 1 ≤ i ≤ n  t , let Ai be the event that there exists an integer z ∈ Ti such that π(z) ∈ Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Defining the minimum size of a t−segmented planar matching of the random graph G  to be  Xt = Xt(s) :=  n/t  �  i=1  11(Ai),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='4)  we have the following estimates for the mean and variance of Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Suppose s = kt  n so that I = J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Letting µt := n  t  �  1 − exp  �  − t2  n  ��  , we have  that  µt − 17  t e− t2  n ≤ EXt ≤ µt + 32n  k2 e− t2  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='5)  If θ1  √n ≤ t ≤ θ2  √n for some constants θ1, θ2 > 0, then there exists a positive con-  stant D = D(θ1, θ2) such that  var(Xt) ≤ D  �√n + n2  k  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='6)  In the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1 in the next Section, we use the fact that Xt is a lower  bound on Tn to obtain deviation and expectation estimates for Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='5) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1: We use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2 in Appendix with s = kt  n to get  from the upper bound in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2) that  P(Ac  1) ≤ exp  �  −t2  n + 4(s + t)2t  k2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  The term  (s + t)2t  k2  = (k + n)2t  k2n2  ≤ 4t  n2 ≤ 4  n  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='7)  6  Planar Matchings  7  where the first inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='7) is true since k ≥ n and the second inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='7)  follows from the fact that t ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Thus  P(Ac  1) ≤ exp  �  −t2  n + 16  n  �  ≤ exp  �  −t2  n  � �  1 + 17  n  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='8)  for all n large and so  EXt = n  t (1 − P(Ac  1)) ≥ n  t  �  1 − exp  �  −t2  n  ��  − 17  t e− t2  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  This obtains the lower bound in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Similarly using the lower bound in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2) we get that  P(Ac  1) ≥ exp  �  −t2  n − 8(s + t)2t  k2  �  ,  where (s+t)2t  k2  = (k+n)2t  k2n2  ≥ 4t  k2 since k ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Thus  P(Ac  1) ≥ exp  �  −t2  n  �  exp  �  −32t  k2  �  ≥ exp  �  −t2  n  � �  1 − 32t  k2  �  and consequently,  EXt = n  t (1 − P(Ac  1)) ≤ n  t  �  1 − exp  �  −t2  n  ��  + 32n  k2 e− t2  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  This obtains the upper bound in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='6) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1: To obtain the deviation estimate for Xt, we let Y =  n  t − Xt and first estimate the variance of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Setting  cov(11(Ac  i), 11(Ac  j)) := P(Ac  i ∩ Ac  j) − P(Ac  i)P(Ac  j)  to be the covariance between the indicator functions of the events Ai and Aj, we have  that  var(Y )  =  n/t  �  i=1  P(Ac  i) − P2(Ac  i) + 2  �  i<j  cov(11(Ac  i), 11(Ac  j))  ≤  n  t  �  P(Ac  1) − P2(Ac  1)  �  + 2n2  t2 cov(11(Ac  1), 11(Ac  2))  ≤  n  t P(Ac  1) + 2n2  t2 cov(11(Ac  1), 11(Ac  2))  ≤  2n  t e− t2  n + 2n2  t2 cov(11(Ac  1), 11(Ac  2))  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='9)  7  Planar Matchings  8  for all n large, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  For the second term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='9), we use the correlation estimate (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='3) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2 to  get that  cov(11(Ac  1), 11(Ac  2)) ≤ P(Ac  1)P(Ac  2)  �  e  5t2  k − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Since t ≤ θ2  √n, we have that 5t2  k ≤ 5θ2  2n  k  ≤ 5θ2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' For all x ≤ 5θ2  2, we have that ex ≤ 1+D1x  for some large constant D1 > 0 and so again using θ1  √n ≤ t ≤ θ2  √n, we get that  cov(11(Ac  1), 11(Ac  2)) ≤ 5D1  t2  k P(Ac  1)P(Ac  2) ≤ D2n  k e− 2t2  n ≤ D3n  k  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='10)  for some constants D2, D3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Plugging (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='10) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='9) and using θ1  √n ≤ t ≤ θ2  √n,  we get that  var(Y ) ≤ 2n  t e− t2  n + n2  t2 · D3n  k  ≤ D4  �√n + n2  k  �  for some constant D4 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Since var(Xt) = var(Y ), this completes the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1  Proof of the upper bound in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1): For a given integer l ≥ 1 we first estimate the prob-  ability of the event El that the random graph G contains a planar matching of size l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Indeed, let π(i) be the x−coordinate of the top vertex adjacent to the bottom ver-  tex ui in the graph G and suppose that π(i1) < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' < π(il) for some deterministic  integers 1 ≤ i1 < i2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' < il ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' The number of choices for {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , il} is  �n  l  �  and the  number of choices for {π(i1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , π(il)} is  �k  l  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='4) and the union bound, we then  get that  P(El) ≤  �n  l  � �k  l  � 1  k(k − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' (k − l + 1) =  �n  l  �  1  l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='.  Using  �n  l  �  ≤  �ne  l  �l and l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' ≥ lle−l, we get that P(El) ≤  �  ne2  l2  �l  ≤  � e  a  �2l for l ≥ a√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  If a > e strictly then  � e  a  �2l = e−2δl for some constant δ > 0 and so  P(El) ≤ e−2δl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' If Eup denotes the event that G contains a planar matching of length at  least a√n, then from the union bound, we get that  P(Eup) ≤ n · e−2aδ√n ≤ e−aδ√n  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='11)  for all n large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' To upper bound the expectation, we use the fact that if Eup occurs,  then Tn ≤ a√n and if the complement event Ec  up occurs, then Tn ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Therefore  from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='11), we get that ETn ≤ a√n + n · e−aδ√n ≤ a√n + 1 for all n large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  This  obtains the upper bound in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  8  Planar Matchings  9  Proof of the lower bound in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1): By definition the term Xt defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='4) is a  lower bound on the maximum size Tn of a planar matching of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' To see this is true  suppose X = w and suppose Aj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , Ajw occur with j1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' < jw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' For each 1 ≤ u ≤ w,  there exists an index pu ∈ Tju and an integer qu ∈ Sju such that π(pu) = qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' By  definition π(p1) < π(p2) < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' < π(pw) and so G contains a planar matching of size at  least w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  We therefore use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1 with t = √n to get that µt = √n(1 − e−1) and so from  the lower bound in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='5), we get that EXt ≥ (1−e−1)√n− 17  √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' This completes the proof  of the lower bound in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2): The upper bound follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' To prove the lower bound, we  use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1 with t = √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' First from the upper bound in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='5) and the fact that k ≥ n,  we get that  EXt ≤ (1 − e−1)√n + 32n  k2 ≤ (1 − e−1)√n + 32  n ≤ √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  From the variance estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='6), we already know that  var(Xt) ≤ D  �√n + n2  k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='12)  We now use the Paley-Zygmund inequality to show that Xt ≥ (1−ǫ)µlow with positive  probability and then use the fact that Xt is a lower bound for Tn to get that Tn ≥  (1 − ǫ)µlow with positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' From Paley-Zygmund inequality we have for ǫ > 0  that  P(Xt ≥ (1 − ǫ)EXt) ≥ ǫ2(EXt)2  EX2  t  and so using EXt ≥ µlow ≥  √n  2 , we have P(Xt ≥ (1 − ǫ)µlow) ≥  ǫ2n  4EX2  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='12) and  the fact that k ≥ n, we then obtain EX2  t ≤ D1n for some constant D1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Conse-  quently Xt ≥ (1 − ǫ)µlow with probability at least  ǫ2  4D1 =: D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Next, using the Chebychev’s inequality, we have for ǫ > 0 that  P (|Xt − EXt| ≤ ǫEXt) ≥ 1 − var(Xt)  ǫ2(EXt)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Since EXt ≥ µlow we get from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='12) that P (Xt ≥ (1 − ǫ)µlow) ≥ 1 − D2  √n − D3n  k  for some  positive constants Di = Di(ǫ), i = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' This obtains the second lower bound in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2) and  therefore completes the proof of the deviation lower bound in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='3): We set t := √bn log n and s := kt  n so that the number of segments I =  J and use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2 to estimate the probability of the event P(Ai) = P(A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' By the  9  Planar Matchings  10  definition of t and the fact that k ≥ n, we respectively get  st  k = t2  n = b log n and s + t  k  = t  k + t  n ≤ 2t  n = o(1)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='13)  and so the conditions in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Moreover, using k ≥ n again, we also  get that  (s + t)2t  k2  = t3(k + n)2  k2n2  ≤ 4t3  n2 = 4(b log n)3/2  √n  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='14)  Plugging (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='14) into the upper bound of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2), we get that  P(Ac  1)  ≤  e−b log n · exp  �  16(b log n)3/2  √n  �  =  exp  �  −b log n  �  1 − 16  �  b log n  n  ��  ≤  exp  �  −b log n  2  �  for all n large since b ≤  n  (32)2 log n (see statement of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1) and so by the union  bound we get that �n/t  i=1 Ai occurs with probability at least  1 − n  t · exp  �  −b  2 log n  �  ≥ 1 − n · exp  �  −b  2 log n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  This in turn implies that Tn ≥ Xt = n  t =  �  n  b log n with probability least  1 − n · exp  �  − b  2 log n  �  , completing the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  4  Conclusion  In this paper, we have studied coloured and dependent planar matchings in random  bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' In the colouring part, we have shown that the largest rainbow matching  is a non-trivial fraction of the total number of colours with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' This hints  at a possible transition point with respect to the colouring parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' In the dependent  setting, we have obtained estimates on the largest size of a planar matching and also  explained our results in terms of longest increasing subsequences in random enlarged  permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  10  Planar Matchings  11  Appendix  Throughout we use the following standard deviation estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Let {Xj}1≤j≤r be independent Bernoulli random variables with  P(Xj = 1) = 1 − P(Xj = 0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  If Tr := �r  j=1 Xj, θr := ETr and 0 < γ ≤ 1  2, then  P (|Tr − θr| ≥ θrγ) ≤ 2 exp  �  −γ2  4 θr  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1)  for all r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  For a proof of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1), we refer to Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='14, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' 312 of Alon and Spencer (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  To prove Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='1, we use the following preliminary result that obtains estimates  on the probability of the events Ac  i and Ac  i ∩ Ac  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Let 1 ≤ t ≤ n and 1 ≤ s ≤ k be any two integers satisfying  s+t  k  ≤ 1  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' We have that  exp  �  −st  k − 8(s + t)2t  k2  �  ≤ P(Ac  1) ≤ exp  �  −st  k + 4(s + t)2t  k2  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2)  and  P(Ac  1 ∩ Ac  2) ≤ P(Ac  1)P(Ac  2) exp  �5t2  k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='3)  Proof of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2: For any integer 1 ≤ l ≤ n and l distinct integers 1 ≤  a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , al ≤ n, we have that  P(u1 = a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , ul = al) =  1  k(k − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' (k − l + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='4)  Let π(i) be the x−coordinate of the top vertex adjacent to ui in the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' If the  event Ac  1 occurs, then the number of choices for π(1) is k − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Similarly, given π(1),  the number of choices for π(2) is k − s − 1 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Thus the number of choices for  the t−tuple (π(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , π(t)) is  (k − s) · (k − s − 1) · · · (k − s − t + 1)  and so we get from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='4) that  P(Ac  1)  =  (k − s) · (k − s + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' (k − s − t + 1)  k(k − 1) · · · (k − t + 1)  =  �  1 − s  k  �  · ·  �  1 − s+t−1  k  �  �  1 − 1  k  �  · ·  �  1 − t−1  k  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='5)  11  Planar Matchings  12  Using s+t  k  ≤ 1  4 we get for all x ≤ 2(s+t)  k  that  e−x  1 − x  =  1 +  1  1 − x  �x2  2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' − x3  3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  �  ≤  1 + 2x2  1 − x  ≤  1 + 4x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='6)  Thus  e−x ·  �  1 + 4(s + t)2  k2  �−1  ≤ 1 − x ≤ e−x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='7)  Plugging the upper bound of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='7) into the numerator of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='5) and the lower bound  of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='7) into the denominator, we get that P(Ac  1) ≤ I1 · I2 where  I1 :=  exp  �  − 1  k  �s+t−1  l=s  k  �  exp  �  − 1  k  �t−1  l=1 k  �  = exp  �  −st  k  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='8)  and  I2 :=  �  1 + 4(s + t)2  k2  �t  ≤ exp  �4(s + t)2t  k2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  This obtains the upper bound in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Similarly, substituting the lower bound of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='7) into the numerator of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='5) and the  upper bound of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='7) into the denominator, we get that  P(Ac  1) ≥ e− st  k ·  �  1 + 4(s + t)2  k2  �−t  ≥ e− st  k ·  �  1 − 4(s + t)2  k2  �t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='9)  Again using s+t  k  ≤ 1  2 and the fact that 1−x ≥ e−2x for all x < 1  2, we get the lower bound  in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2) from the final expression in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Proof of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='3) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2: We use a split set argument and as before, π(i), 1 ≤  i ≤ n is the x−coordinate of the top vertex adjacent to ui, in the graph Kk,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' For a  deterministic set Q1 ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , t}, let F1(Q1) be the event that π(j) ∈ {s + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , 2s}  for every integer j ∈ Q1 and π(l) /∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , 2s} for every l ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , t}\\Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Similarly  for Q2 ⊆ {t+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , 2t}, let F2(Q2) be the event that π(j) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , s} for every integer j ∈  Q2 and π(j) /∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , 2s} for every j ∈ {t+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , 2t}\\Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' By definition we then have  that  Ac  1 ∩ Ac  2 =  �  Q1,Q2  F1(Q1) ∩ F2(Q2),  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='10)  12  Planar Matchings  13  where the union is over all sets Q1 ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , t} and Q2 ⊆ {t + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' , 2t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Thus  P(Ac  1 ∩ Ac  2) =  �  j1,j2  �  #Q1=j1  �  #Q2=j2  P (F1(Q1) ∩ F2(Q2)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='11)  Letting (a)b :=  � a(a − 1) · · · (a − b + 1)  if b ≥ 1  1  if b = 0,  we have that  P(F1(Q1)) = (s)j1(k − 2s)t−j1  (k)t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  A similar expression holds for F2(Q2) and the intersection F1(Q1)∩F2(Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Letting jlow :=  min(j1, j2) and jup := max(j1, j2) and combining the above expressions, we get that  P(F1(Q1) ∩ F2(Q2))  P(F1(Q1))P(F2(Q2)) = f(k − 2s, t − jlow, t − jup)  f(k, t, t)  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='12)  where f(a, b, c) := (a−b)c  (a)c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Suppose c ≥ 1 and b+c  a  < 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' We rewrite  f(a, b, c) =  c−1  �  j=0  �  1 − b + j  a  � c−1  �  j=1  �  1 − j  a  �−1  and use the estimate e−2x ≤ 1 − x ≤ e−x for x < 1  2 to get that  f(a, b, c) ≤  c−1  �  j=0  exp  �  −b + j  a  + 2j  a  �  ≤ e− bc  a + c2  a ≤ e  c2  a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='13)  Similarly  f(a, b, c) ≥  c�  j=1  exp  �  −2b + 2j  a  + j  a  �  = e− 2bc  a exp  \uf8eb  \uf8ed−  c  �  j=1  j  a  \uf8f6  \uf8f8 ≥ e− 2bc  a − c2  a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='14)  We now use (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='13) with a = k − 2s, b = t − jlow and c = t − jup to evaluate f(k −  2s, t − jlow, t − jup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' We have that  b + c  a  = 2t − j1 − j2  k − 2s  ≤  2t  k − 2s ≤ 4t  k < 1  2  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='15)  where the second inequality in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='15) follows from s  k < s+t  k  ≤ 1  4 and the third inequality  in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='15) again follows from t  k < s+t  k  ≤ 1  8 (see statement of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='13)  we therefore have  f(k − 2s, t − jlow, t − jup) ≤ exp  �(t − jup)2  k − 2s  �  ≤ exp  �  t2  k − 2s  �  ≤ exp  �2t2  k  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='16)  13  Planar Matchings  14  since s  k ≤ s+t  k  ≤ 1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Similarly, using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='14) with a = k, b = c = t we get that  f(k, t, t) ≥ exp  �  −3t2  k  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='17)  Plugging (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='16) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='17) into (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='12), we get that  P(F1(Q1) ∩ F2(Q2))  P(F1(Q1))P(F2(Q2)) ≤ exp  �5t2  k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='18)  Substituting (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='18) into (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='11) and summing over Q1 and Q2, we get (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Acknowledgement  I thank Professors Rahul Roy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Subramanian and the referee for crucial comments  that led to an improvement of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' I also thank IMSc and IISER Bhopal for my  fellowships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  References  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Baik, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Deift and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Johansson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' On the Distribution of the Length of the  Longest Increasing Subsequence of Random Permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Journal of the American  Mathematical Society, 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' 1119–1178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  [2] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Bollob´as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Random Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  [3] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Ganesan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  Strong and Weighted Matchings in Inhomogenous Random  Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Electronic Communications in Probability, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' 1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content='  [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Hammersley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' A Few Seedlings of Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
+page_content=' Proceedings of the Sixth  Berkeley Symposium on Mathematical Statistics and Probability, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
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+page_content='  [5] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQf7Qv3/content/2301.04798v1.pdf'}
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+1
+GAN-Based Object Removal in High-Resolution
+Satellite Images
+Hadi Mansourifar and Steven J. Simske
+Abstract—Satellite images often contain a significant level of
+sensitive data compared to ground-view images. That is why
+satellite images are more likely to be intentionally manipulated
+to hide specific objects and structures. GAN-based approaches
+have been employed to create forged images with two major
+problems: (i) adding a new object to the scene to hide a specific
+object or region may create unrealistic merging with surrounding
+areas; and (ii) using masks on color feature images has proven to
+be unsuccessful in GAN-based object removal. In this paper, we
+tackle the problem of object removal in high-resolution satellite
+images given a limited number of training data. Furthermore,
+we take advantage of conditional GANs (CGANs) to collect
+perhaps the first GAN-based forged satellite image data set. All
+forged instances were manipulated via CGANs trained by Canny
+Feature Images for object removal. As part of our experiments,
+we demonstrate that distinguishing the collected forged images
+from authentic (original) images is highly challenging for fake
+image detector models.
+Index Terms—Satellite Imaging, CGANs, Image Synthesis,
+Forgery Detection, Object Removal.
+I. INTRODUCTION
+I
+NTERACTIVE editing of ground-view images to remove
+objects has been practiced before. Object removal is a
+significantly important task with a variety of applications such
+as data augmentation [1] and content filtering for privacy and
+security issues [2]. However, there are fundamental differences
+between satellite images and ground-view pictures [3]. In
+satellite imagery, every pixel has semantic meaning. However,
+ground-view images include a few foreground objects of
+interest with an essentially meaningless background [4]. This
+is the core problem in satellite imaging, which is why semantic
+segmentation and object detection are challenging. Further-
+more, satellite images can cover a wide range of region types
+with sensitive objects and structures having a wide range of
+digital attributes (sharpness, edge field characteristics, entropy,
+etc.). This characteristic makes the satellite images potential
+targets for image forgery. GAN-based satellite imaging [5] has
+been widely used and led to considerable progress in various
+imaging tasks over the past few years. Accordingly, GAN-
+based object removal has been practiced before. While this
+progress is remarkable, training a GAN needs a huge amount
+of training data, which makes it impossible to be used for
+high-quality object removal in a single high-resolution satellite
+image. The key challenges in this context are as follows:
+• Few-shot GAN training for object removal.
+• Seemless object removal.
+• Low response time for real-time GAN training and gen-
+erating the output.
+Steven J. Simske is with the Department of Systems Engineering, Colorado
+State University, Fort Collins, CO 80523, USA
+Fig. 1. Instances of object inclusion (A) and object removal outputs (B) using
+the proposed method.
+• The minimum degradation of reproduced regions in the
+scene.
+To the best of our knowledge, such a scenario has never been
+investigated before, and we take the first step in this direction:
+We slice a high-resolution satellite image into a set of 256 *
+256 images. We train a Pix2pixHD model as a Conditional
+GAN (CGAN) using the Canny Feature Image (CFI) of train-
+ing data. We demonstrate that CFI can outperform Segmented
+Feature Images (SFI) in CGAN-based object removal. We use
+object detection models to evaluate the quality of generated
+forged images. This step is required to evaluate the object
+removal impact on overall image degradation. Any distortion
+in the image after object removal can impact the ability of
+object detection models to detect the rest of the target objects.
+We test several classifiers to test if our GAN-generated forged
+images can be detected by binary classifiers. To do so, we
+train several classification methods, including binary classifier
+and transfer learning.
+Our contributions are as follows.
+• We propose one-shot object removal for forged satellite
+image generation using CFI.
+• We demonstrate the capability of CFI in the absence of
+rare SFI for GAN-based object removal.
+• We publish a new forged image dataset to be used in
+future forged image detection research.
+• We run extensive experiments to compare the impact of
+CFI and SFI in image degradation after object removal.
+The rest of this paper is organized as follows. Section II
+presents the related works. Section III introduces the proposed
+method. Section IV demonstrates the experiments. Section V
+presents the evaluation results. Finally, section VI concludes
+arXiv:2301.11726v1  [cs.CV]  24 Jan 2023
+
+(A)
+(B)2
+the paper.
+II. RELATED WORKS
+GAN-based forged satellite image generation and detection
+is a new research area in deep learning and training, with only
+limited works found in the literature. Yarlagadda et al. [6] took
+the first step in this direction. They proposed an algorithm for
+satellite image forgery detection and localization under the
+assumption that no forged images are available for training.
+To do so, they used a one-class support vector machine
+(SVM) as an anomaly detector and trained this on feature
+representations of pristine satellite images obtained by GAN.
+Bartusiak et al. [7] proposed the detection and localization of
+splicing in satellite images which refers to the replacement
+of pixels of a region of the image to add or remove an
+object. They employed a Conditional Generative Adversarial
+Network (CGAN) to learn a mapping from a satellite image
+to its splicing mask. The trained CGAN operates on a satellite
+image of interest and outputs a mask of the same resolution
+that indicates the likelihood of a pixel belonging to a spliced
+region. Their proposed CGAN architecture extends the popular
+pix2pix [8]. This approach is different from [6] since they
+used both pristine and spliced images (i.e., forged images
+not employed by [6]) to train the model. Horvath et al. [9]
+proposed a deep learning based method for detecting and lo-
+calizing splicing manipulations. In [10], a one-class detection
+method was proposed based on deep belief networks (DBN)
+for splicing detection and localization without using any prior
+knowledge of the manipulations. Montserrat et al. [11] used
+two generative autoregressive models, PixelCNN [12], and
+Gated PixelCNN [13], to detect pixel-level manipulations.
+These neural networks were originally devised to generate
+new images by modeling a pixel’s distribution given a set of
+neighboring pixels and a conditional likelihood value assigned
+to each pixel. Similarly, the pixels with a low likelihood
+scoring by the neural network can be detected as manipulated
+ones. [14] proposed an unsupervised technique that uses a
+so-called ”Vision Transformer” to detect spliced areas within
+satellite images.
+III. BACKGROUND
+A. Vanilla GAN
+The initial version of GANs [15] is known as Vanilla
+GAN. The learning process of the Vanilla GANs is to train
+a discriminator D and a generator G simultaneously. The
+target of G is to learn the distribution pg over data x. G
+starts from sampling input variables z from a uniform or
+Gaussian distribution pz(z), then maps the input variables z to
+data space G(z; θg) through a differentiable network where θg
+represents network parameters. D is a classifier D(x; θd) that
+aims to recognize whether the input is from training data or
+from G where θd represents network parameters. The minimax
+objective for GANs can be formulated as follows:
+min
+G
+max
+D
+VGAN(D, G) = Ex∼px[log D(x)]
++
+Ez∼qz[log (1 − D(G(z)))]
+(1)
+B. Conditional GAN
+In Conditional GAN (CGAN) [16], labels act as an exten-
+sion to the latent space z to generate and discriminate images
+better. The objective function of CGANs is as follows:
+min
+G max
+D V (D, G) = Ex∼p(data)(x)[log D(X|Y )]
++ EZ∼pz(z))[log(1 − D(G(Z|Y )))]
+(2)
+C. Pix2Pix
+Pix2pix [8] image is a conditional GAN that uses feature
+images as labels to be translated to target images. Pix2pix uses
+U-Net [17], [18] as the architecture of generator PatchGAN
+for discriminator architecture. The most important feature of
+the Pix2pix generator is skip connections between each layer
+i and layer n − i to concatenate all channels at layer i with
+those at layer n − i, where n is the total number of layers.
+D. Pix2PixHD
+To improve the pix2pix framework, [19] used a coarse-to-
+fine generator, a multi-scale discriminator architecture, and a
+robust adversarial learning objective function. The generator
+in Pix2pixHD consists of two sub-networks, G1, and G2,
+where G1 is the global generator network, and G2 is the local
+enhancer network. As a multi-scale discriminator architecture,
+3 discriminators with an identical network structure operate
+at different image scales. The real and synthesized high-
+resolution images are down-sampled by a factor of 2 and 4
+to create an image pyramid of 3 scales. The discriminators
+D1, D2, and D3 are then trained to differentiate real and
+synthesized images at the 3 different scales, respectively.
+IV. PROPOSED METHOD
+In this section, we demonstrate our object removal approach
+in high-resolution satellite images. The proposed method is
+divided into training and inference phases which are demon-
+strated in the following sections.
+A. Training phase
+The goal of this phase is to train a CGAN like traditional
+Pix2pix or Pix2pixHD given one single high-resolution satel-
+lite image, as shown in Figure 2. The following Algorithm
+shows the required steps.
+Algorithm 1 The training phase of proposed CGAN-based
+object removal
+1: Slice the input high-resolution satellite image into a set of
+256*256 blocks.
+2: Extract the CFI of each sliced block of Step 1.
+3: Train a Pix2pixHD model to translate CFI to each corre-
+sponding training image.
+4: Train pix2pixHD to translate obtained images from Step
+2 to output image.
+
+3
+Fig. 2. Training pipeline of CGAN-based object removal .
+Fig. 3. Inference pipeline of CGAN-based Object Removal.
+B. Inference phase
+In this phase, the user can define a region on the original
+image then the edges represented by white are removed in
+the Canny feature image. Finally, the edited feature image is
+passed to the trained Pix2pixHD, as shown in Figure 3. The
+following Algorithm shows the required steps.
+Algorithm 2 The inference phase of proposed CGAN-based
+object removal
+1: Extract the Canny edges of a target block for object
+removal as CFI.
+2: Define the target region for object removal.
+3: Remove white pixels in the defined region of Step 2 from
+CFI.
+4: Translate obtained image from Step 3 to the output image
+using the trained pix2pixHD.
+C. Advantages of One-Shot Object Removal
+Two major problems of training GANs in the absence of
+sufficient training data are the lack of generalization and the
+lack of diversity. The former is caused by over-fitting, and the
+latter is due to mode collapse. [20], [21]. While over-fitting is
+problematic when embedding a new object in the image, over-
+fitting can play a positive role in removing weak objects from
+the scene. From the semantic segmentation point of view, most
+pixels are categorized as background. By default, any deep
+neural network for image segmentation or image synthesis
+has a bias toward the background pixels. This makes weak
+object augmentation a particular challenge. However, weak
+object removal is made easier by the CGAN, since the trained
+CGAN is iteratively rewarded for replacing the weak object
+pixels with the background pixels. A CGAN trained by one
+single instance has even more capability for object removal
+since it has more bias toward most pixels. To do so, we start
+with a single satellite image, slice it to 256 * 256 images
+and extract the corresponding Canny feature images to train a
+Pix2pixHD GAN as shown in Figure 2.
+D. The Advantage of Canny Feature Images
+While polygon-level masks are widely used for object
+detection tasks, pixel-level masks are a more proper choice
+for object augmentation-removal applications. In cases of
+object removal, we need feature images comprising minimal
+information about the objects and structures so that they can
+exacerbate the over-fitting, which is required for high-quality
+object removal given only a single image.
+V. EXPERIMENTS
+In this section, we present the experiment setup, architec-
+tures, dataset, metrics, and results corresponding to forged
+image generation and detection.
+A. Metrics
+We used two sets of metrics as follows.
+• Similarity Scores: including MSE, PSNR, and SSIM to
+evaluate the degradation
+• Object Detection Scores: to compare the CFI and SFI
+impact on object detection scores.
+B. CFI Versus SFI for CGAN-based Object Removal
+CGAN-based image manipulation suffers a significant prob-
+lem: any change in a local region of a feature image can
+impact the other blocks in the final output. That’s why we
+are interested in evaluating the local and global impact of CFI
+and SFI manipulation. The best way is to use image similarity
+metrics to compare the ground truth and output image. The
+goal of these experiments is to test the capability of CFI
+to be used in CGAN-based object removal in the absence
+of pixel-wise annotated SFI, which is rare in the satellite
+imaging domain. Although the ground truth and output image
+are not exactly the same due to object removal, similarity
+scores can still help to compare CFI and SFI. Table I shows
+the comparison results of output images shown in Figure 4
+and the corresponding original ground truths. Table II also
+tabulates the object detection scores.
+
+Training Phase
+original Image
+Sliced Images
+Canny Feature Images
+Target Images
+Pix2PixHDPre-processing
+Output
+Pix2PixHD4
+Fig. 4. CGAN-based object removal: Canny feature images versus segmented feature images.
+TABLE I
+COMPARISON OF CFI AND SFI USING IMAGE SIMILARITY SCORES
+A
+B
+C
+D
+MSE
+CFI
+0.129
+0.085
+0.248
+0.253
+SFI
+0.137
+0.163
+0.132
+0.135
+PSNR
+CFI
+32.95
+34.73
+30.10
+30.02
+SFI
+32.69
+31.93
+32.84
+32.7
+SSIM
+CFI
+0.844
+0.92
+0.69
+0.66
+SFI
+0.901
+0.83
+0.887
+0.87
+TABLE II
+OBJECT DETECTION SCORES RETURNED BY AWS REKOGNITION.
+Airplane
+Aircraft
+A
+B
+C
+D
+A
+B
+C
+D
+CFI
+99.3
+99.2
+97.5
+87.8
+99.3
+99.2
+97.5
+87.8
+SFI
+99.3
+99.2
+97.4
+97.5
+99.3
+99.2
+97.4
+97.5
+Terminal
+Vehicle
+A
+B
+C
+D
+A
+B
+C
+D
+CFI
+0
+55.9
+55.1
+0
+99.3
+99.2
+97.5
+87.8
+SFI
+55.7
+55.6
+69.2
+56.3
+99.3
+99.2
+97.4
+97.5
+The results shown in Table I and Table II demonstrate that
+CFI is robust enough for CGAN-based object removal since
+we can observe the superiority of CFI over SFI in some cases.
+Besides, in the majority of failed cases, the CFI results are
+competable with SFI results.
+C. Dataset
+In order to train satellite forged image detection models,
+we collected a dataset by CGAN-based object removal using
+CFI. To do so, we manipulated instances of the iSAID dataset
+[22], [23]. Existing satellite image datasets are either well-
+suited for semantic segmentation or object detection. iSAID
+is the first benchmark dataset, for instance, segmentation in
+aerial images. This large-scale and densely annotated dataset
+contains 655,451 object instances for 15 categories across
+2,806 high-resolution images. The collected dataset contains
+162 forged images and 266 pristine images as training data.
+The validation set includes 95 forged images and 114 pristine
+images.
+D. Forged Satellite Image Detection
+Detecting the regions manipulated by the proposed object
+removal method is not trivial using traditional anomaly detec-
+tion methods for two reasons: (i) The range of reconstruction
+losses or anomaly scores are so different in manipulated,
+and original images due to the diversity of satellite view
+scenes; and (ii) In many manipulated images, the anomaly
+score is lower than for the original images, since removing the
+objects may lead to lower entropy. Due to the aforementioned
+conditions, defining a threshold to detect manipulated images
+with removed objects takes a lot of work. In this section, we
+test two different methods, including binary classification and
+transfer learning to detect forged satellite images. The goal
+of this experiment is to test whether the forged images are
+distinguishable by the binary classifiers easily.
+• Binary Forged Image Detector: It’s a convolution neural
+network with ten layers (as shown in Figure 6), Adam
+optimizer with learning rate=0.001 and 100 epochs for
+training.
+• Transfer Learning: In this approach, MobileNetV2 [24]
+is fine-tuned with the training data with 100 epochs and
+RMSprop as optimizer.
+The ROC curves of both models show their failure to distin-
+guish the collected forged images from pristine instances as
+shown in Figure 8.
+E. Dataset Visualization
+Figure 7 shows some of the collected forged instances using
+CGAN-based object removal. Furthermore, Figure 5 shows the
+TSNE projection of positive-negative collected instances. To
+calculate the TSNE data projection, first, we passed the images
+to ResNet50 [25] to reduce the dimensionality by collecting
+1000 label probabilities.
+VI. CONCLUSION
+In this paper, we investigated the capabilities of CFI for one-
+shot CGAN-based object removal. Full or even partial pixel-
+wise annotation is rare in existing satellite imaging datasets.
+In the absence of pixel-wise annotation, Canny Feature Image
+
+(A)
+(B)
+(C)
+(D)
+SFI
+CFI5
+Fig. 5. TSNE data projection: positive versus negative data distribution.
+(CFI) is an inevitable choice for CGAN-based object removal.
+We successfully compared the CFI and SFI to show that CFI is
+robust enough to be used in CGAN-based image manipulation.
+REFERENCES
+[1] H. Mansourifar, L. Chen, and W. Shi, “Virtual big data for gan based
+data augmentation,” in 2019 IEEE International Conference on Big Data
+(Big Data).
+IEEE, 2019, pp. 1478–1487.
+[2] K. Kasichainula, H. Mansourifar, and W. Shi, “Poisoning attacks via
+generative adversarial text to image synthesis,” in 2021 51st Annual
+IEEE/IFIP International Conference on Dependable Systems and Net-
+works Workshops (DSN-W).
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+satellite images,” networks, vol. 29, p. 21, 2019.
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+of the IEEE/CVF conference on computer vision and pattern recognition
+workshops, 2020, pp. 664–665.
+[11] D. M. Montserrat, J. Horv´ath, S. K. Yarlagadda, F. Zhu, and E. J. Delp,
+“Generative autoregressive ensembles for satellite imagery manipulation
+detection,” in 2020 IEEE International Workshop on Information Foren-
+sics and Security (WIFS).
+IEEE, 2020, pp. 1–6.
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+neural networks,” in International conference on machine learning.
+PMLR, 2016, pp. 1747–1756.
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+in neural information processing systems, vol. 29, 2016.
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+“Manipulation detection in satellite images using vision transformer,”
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+Pattern Recognition, 2021, pp. 1032–1041.
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+S. Ozair, A. Courville, and Y. Bengio, “Generative adversarial networks,”
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+neural information processing systems, vol. 31, 2018.
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+model from a single natural image,” in Proceedings of the IEEE/CVF
+International Conference on Computer Vision, 2019, pp. 4570–4580.
+[21] T. Hinz, M. Fisher, O. Wang, and S. Wermter, “Improved techniques
+for training single-image gans,” in Proceedings of the IEEE/CVF Winter
+Conference on Applications of Computer Vision, 2021, pp. 1300–1309.
+[22] G.-S. Xia, X. Bai, J. Ding, Z. Zhu, S. Belongie, J. Luo, M. Datcu,
+M. Pelillo, and L. Zhang, “Dota: A large-scale dataset for object
+detection in aerial images,” in The IEEE Conference on Computer Vision
+and Pattern Recognition (CVPR), June 2018.
+[23] S. Waqas Zamir, A. Arora, A. Gupta, S. Khan, G. Sun, F. Shahbaz Khan,
+F. Zhu, L. Shao, G.-S. Xia, and X. Bai, “isaid: A large-scale dataset for
+instance segmentation in aerial images,” in Proceedings of the IEEE
+Conference on Computer Vision and Pattern Recognition Workshops,
+2019, pp. 28–37.
+[24] M. Sandler, A. Howard, M. Zhu, A. Zhmoginov, and L.-C. Chen,
+“Mobilenetv2: Inverted residuals and linear bottlenecks,” in Proceedings
+of the IEEE conference on computer vision and pattern recognition,
+2018, pp. 4510–4520.
+[25] I. Z. Mukti and D. Biswas, “Transfer learning based plant diseases
+detection using resnet50,” in 2019 4th International conference on
+electrical information and communication technology (EICT).
+IEEE,
+2019, pp. 1–6.
+
+Negative Vs Positive Training Data T-SNE projection
+75
+50
+25
+0
+25
+50
+75
+80
+-60
+-40
+-20
+0
+20
+40
+606
+Fig. 6. The architecture of binary forged image detector.
+Fig. 7. Some forged samples of collected dataset.
+Fig. 8.
+Receiver operating characteristic curve corresponding to the binary
+classifier and transfer learning for forged image detection.
+
+conv2d_4_input
+input:
+[(None, 150, 150, 3)]
+ InputLayer
+output:
+[(None, 150, 150, 3)]
+conv2d_4
+input:
+(None, 150, 150, 3)
+Conv2D
+ output:
+(None, 148, 148, 32)
++
+max_pooling2d_4
+input:
+(None, 148, 148, 32)
+MaxPooling2D
+output:
+(None, 74, 74, 32)
+conv2d_5
+input:
+(None, 74, 74, 32)
+Conv2D
+:andno
+(None, 72, 72, 64)
+max_pooling2d_5
+input:
+(None, 72, 72, 64)
+MaxPooling2D
+ output:
+(None, 36, 36, 64)
+conv2d_6
+input:
+(None, 36, 36, 64)
+Conv2D
+output:
+(None, 34, 34, 128)
+max_pooling2d_6
+input:
+(None, 34, 34, 128)
+MaxPooling2D
+output:
+(None, 17, 17, 128)
+conv2d_7
+input:
+(None, 17, 17, 128)
+Conv2D
+output:
+(None, 15, 15, 128)
++
+max_pooling2d_7
+input:
+(None, 15, 15, 128)
+MaxPooling2D
+ output:
+(None, 7, 7, 128)
+flatten_1
+input:
+(None, 7, 7, 128)
+Flatten
+output:
+(None, 6272)
+dense_
+2
+input:
+(None, 6272)
+Dense
+ output:
+(None, 512)
+dense_3
+input:
+(None, 512)
+Dense
+:andno
+(None, 256)
+dense_4
+input:
+(None, 256)
+Dense
+output:
+(None, 64)
+dense_ 5
+input:
+(None, 64)
+Dense
+output:
+(None, 32)
+dense_6
+input:
+(None, 32)
+Dense
+output:
+(None, 16)
+dense_7
+input:
+(None, 16)
+Dense
+:ndno
+(None, 1)040BinaryClassificationROc
+10 
+0.8 
+Rate
+True Positive
+0.6
+0
+0.2 -
+ROCcurve(area=0.57)
+0.0
+0.0
+0.2
+0.4
+0.6
+8'0
+1.0
+FalsePositiveRate
+TransferLearningRoc
+1.0
+0.8
+Rate
+Positive
+0.6 
+True
+0.4
+0.2 -
+ROCcurve(area=0.68)
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+FalsePositiveRate
\ No newline at end of file
diff --git a/pNFKT4oBgHgl3EQfGi3M/content/tmp_files/load_file.txt b/pNFKT4oBgHgl3EQfGi3M/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,552 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf,len=551
+page_content='1  GAN-Based Object Removal in High-Resolution  Satellite Images  Hadi Mansourifar and Steven J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Simske  Abstract—Satellite images often contain a significant level of  sensitive data compared to ground-view images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' That is why  satellite images are more likely to be intentionally manipulated  to hide specific objects and structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' GAN-based approaches  have been employed to create forged images with two major  problems: (i) adding a new object to the scene to hide a specific  object or region may create unrealistic merging with surrounding  areas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' and (ii) using masks on color feature images has proven to  be unsuccessful in GAN-based object removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' In this paper, we  tackle the problem of object removal in high-resolution satellite  images given a limited number of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Furthermore,  we take advantage of conditional GANs (CGANs) to collect  perhaps the first GAN-based forged satellite image data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' All  forged instances were manipulated via CGANs trained by Canny  Feature Images for object removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' As part of our experiments,  we demonstrate that distinguishing the collected forged images  from authentic (original) images is highly challenging for fake  image detector models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Index Terms—Satellite Imaging, CGANs, Image Synthesis,  Forgery Detection, Object Removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' INTRODUCTION  I  NTERACTIVE editing of ground-view images to remove  objects has been practiced before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Object removal is a  significantly important task with a variety of applications such  as data augmentation [1] and content filtering for privacy and  security issues [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' However, there are fundamental differences  between satellite images and ground-view pictures [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' In  satellite imagery, every pixel has semantic meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' However,  ground-view images include a few foreground objects of  interest with an essentially meaningless background [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' This  is the core problem in satellite imaging, which is why semantic  segmentation and object detection are challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Further-  more, satellite images can cover a wide range of region types  with sensitive objects and structures having a wide range of  digital attributes (sharpness, edge field characteristics, entropy,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' This characteristic makes the satellite images potential  targets for image forgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' GAN-based satellite imaging [5] has  been widely used and led to considerable progress in various  imaging tasks over the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Accordingly, GAN-  based object removal has been practiced before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' While this  progress is remarkable, training a GAN needs a huge amount  of training data, which makes it impossible to be used for  high-quality object removal in a single high-resolution satellite  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The key challenges in this context are as follows:  Few-shot GAN training for object removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Seemless object removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Low response time for real-time GAN training and gen-  erating the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Steven J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Simske is with the Department of Systems Engineering, Colorado  State University, Fort Collins, CO 80523, USA  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Instances of object inclusion (A) and object removal outputs (B) using  the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  The minimum degradation of reproduced regions in the  scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  To the best of our knowledge, such a scenario has never been  investigated before, and we take the first step in this direction:  We slice a high-resolution satellite image into a set of 256 *  256 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' We train a Pix2pixHD model as a Conditional  GAN (CGAN) using the Canny Feature Image (CFI) of train-  ing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' We demonstrate that CFI can outperform Segmented  Feature Images (SFI) in CGAN-based object removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' We use  object detection models to evaluate the quality of generated  forged images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' This step is required to evaluate the object  removal impact on overall image degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Any distortion  in the image after object removal can impact the ability of  object detection models to detect the rest of the target objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  We test several classifiers to test if our GAN-generated forged  images can be detected by binary classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' To do so, we  train several classification methods, including binary classifier  and transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Our contributions are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  We propose one-shot object removal for forged satellite  image generation using CFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  We demonstrate the capability of CFI in the absence of  rare SFI for GAN-based object removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  We publish a new forged image dataset to be used in  future forged image detection research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  We run extensive experiments to compare the impact of  CFI and SFI in image degradation after object removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Section II  presents the related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Section III introduces the proposed  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Section IV demonstrates the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Section V  presents the evaluation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Finally, section VI concludes  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='11726v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='CV]  24 Jan 2023  (A)  (B)2  the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' RELATED WORKS  GAN-based forged satellite image generation and detection  is a new research area in deep learning and training, with only  limited works found in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Yarlagadda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' [6] took  the first step in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' They proposed an algorithm for  satellite image forgery detection and localization under the  assumption that no forged images are available for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  To do so, they used a one-class support vector machine  (SVM) as an anomaly detector and trained this on feature  representations of pristine satellite images obtained by GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Bartusiak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' [7] proposed the detection and localization of  splicing in satellite images which refers to the replacement  of pixels of a region of the image to add or remove an  object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' They employed a Conditional Generative Adversarial  Network (CGAN) to learn a mapping from a satellite image  to its splicing mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The trained CGAN operates on a satellite  image of interest and outputs a mask of the same resolution  that indicates the likelihood of a pixel belonging to a spliced  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Their proposed CGAN architecture extends the popular  pix2pix [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' This approach is different from [6] since they  used both pristine and spliced images (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=', forged images  not employed by [6]) to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Horvath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' [9]  proposed a deep learning based method for detecting and lo-  calizing splicing manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' In [10], a one-class detection  method was proposed based on deep belief networks (DBN)  for splicing detection and localization without using any prior  knowledge of the manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Montserrat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' [11] used  two generative autoregressive models, PixelCNN [12], and  Gated PixelCNN [13], to detect pixel-level manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  These neural networks were originally devised to generate  new images by modeling a pixel’s distribution given a set of  neighboring pixels and a conditional likelihood value assigned  to each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Similarly, the pixels with a low likelihood  scoring by the neural network can be detected as manipulated  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' [14] proposed an unsupervised technique that uses a  so-called ”Vision Transformer” to detect spliced areas within  satellite images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' BACKGROUND  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Vanilla GAN  The initial version of GANs [15] is known as Vanilla  GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The learning process of the Vanilla GANs is to train  a discriminator D and a generator G simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The  target of G is to learn the distribution pg over data x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' G  starts from sampling input variables z from a uniform or  Gaussian distribution pz(z), then maps the input variables z to  data space G(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' θg) through a differentiable network where θg  represents network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' D is a classifier D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' θd) that  aims to recognize whether the input is from training data or  from G where θd represents network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The minimax  objective for GANs can be formulated as follows:  min  G  max  D  VGAN(D, G) = Ex∼px[log D(x)]  +  Ez∼qz[log (1 − D(G(z)))]  (1)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Conditional GAN  In Conditional GAN (CGAN) [16], labels act as an exten-  sion to the latent space z to generate and discriminate images  better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The objective function of CGANs is as follows:  min  G max  D V (D, G) = Ex∼p(data)(x)[log D(X|Y )]  + EZ∼pz(z))[log(1 − D(G(Z|Y )))]  (2)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Pix2Pix  Pix2pix [8] image is a conditional GAN that uses feature  images as labels to be translated to target images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Pix2pix uses  U-Net [17], [18] as the architecture of generator PatchGAN  for discriminator architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The most important feature of  the Pix2pix generator is skip connections between each layer  i and layer n − i to concatenate all channels at layer i with  those at layer n − i, where n is the total number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Pix2PixHD  To improve the pix2pix framework, [19] used a coarse-to-  fine generator, a multi-scale discriminator architecture, and a  robust adversarial learning objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The generator  in Pix2pixHD consists of two sub-networks, G1, and G2,  where G1 is the global generator network, and G2 is the local  enhancer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' As a multi-scale discriminator architecture,  3 discriminators with an identical network structure operate  at different image scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The real and synthesized high-  resolution images are down-sampled by a factor of 2 and 4  to create an image pyramid of 3 scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The discriminators  D1, D2, and D3 are then trained to differentiate real and  synthesized images at the 3 different scales, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' PROPOSED METHOD  In this section, we demonstrate our object removal approach  in high-resolution satellite images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The proposed method is  divided into training and inference phases which are demon-  strated in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Training phase  The goal of this phase is to train a CGAN like traditional  Pix2pix or Pix2pixHD given one single high-resolution satel-  lite image, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The following Algorithm  shows the required steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Algorithm 1 The training phase of proposed CGAN-based  object removal  1: Slice the input high-resolution satellite image into a set of  256*256 blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  2: Extract the CFI of each sliced block of Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  3: Train a Pix2pixHD model to translate CFI to each corre-  sponding training image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  4: Train pix2pixHD to translate obtained images from Step  2 to output image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Training pipeline of CGAN-based object removal .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Inference pipeline of CGAN-based Object Removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Inference phase  In this phase, the user can define a region on the original  image then the edges represented by white are removed in  the Canny feature image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Finally, the edited feature image is  passed to the trained Pix2pixHD, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The  following Algorithm shows the required steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Algorithm 2 The inference phase of proposed CGAN-based  object removal  1: Extract the Canny edges of a target block for object  removal as CFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  2: Define the target region for object removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  3: Remove white pixels in the defined region of Step 2 from  CFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  4: Translate obtained image from Step 3 to the output image  using the trained pix2pixHD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Advantages of One-Shot Object Removal  Two major problems of training GANs in the absence of  sufficient training data are the lack of generalization and the  lack of diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The former is caused by over-fitting, and the  latter is due to mode collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' While over-fitting is  problematic when embedding a new object in the image, over-  fitting can play a positive role in removing weak objects from  the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' From the semantic segmentation point of view, most  pixels are categorized as background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' By default, any deep  neural network for image segmentation or image synthesis  has a bias toward the background pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' This makes weak  object augmentation a particular challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' However, weak  object removal is made easier by the CGAN, since the trained  CGAN is iteratively rewarded for replacing the weak object  pixels with the background pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' A CGAN trained by one  single instance has even more capability for object removal  since it has more bias toward most pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' To do so, we start  with a single satellite image, slice it to 256 * 256 images  and extract the corresponding Canny feature images to train a  Pix2pixHD GAN as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The Advantage of Canny Feature Images  While polygon-level masks are widely used for object  detection tasks, pixel-level masks are a more proper choice  for object augmentation-removal applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' In cases of  object removal, we need feature images comprising minimal  information about the objects and structures so that they can  exacerbate the over-fitting, which is required for high-quality  object removal given only a single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' EXPERIMENTS  In this section, we present the experiment setup, architec-  tures, dataset, metrics, and results corresponding to forged  image generation and detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Metrics  We used two sets of metrics as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Similarity Scores: including MSE, PSNR, and SSIM to  evaluate the degradation  Object Detection Scores: to compare the CFI and SFI  impact on object detection scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' CFI Versus SFI for CGAN-based Object Removal  CGAN-based image manipulation suffers a significant prob-  lem: any change in a local region of a feature image can  impact the other blocks in the final output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' That’s why we  are interested in evaluating the local and global impact of CFI  and SFI manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The best way is to use image similarity  metrics to compare the ground truth and output image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The  goal of these experiments is to test the capability of CFI  to be used in CGAN-based object removal in the absence  of pixel-wise annotated SFI, which is rare in the satellite  imaging domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Although the ground truth and output image  are not exactly the same due to object removal, similarity  scores can still help to compare CFI and SFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Table I shows  the comparison results of output images shown in Figure 4  and the corresponding original ground truths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Table II also  tabulates the object detection scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Training Phase  original Image  Sliced Images  Canny Feature Images  Target Images  Pix2PixHDPre-processing  Output  Pix2PixHD4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' CGAN-based object removal: Canny feature images versus segmented feature images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  TABLE I  COMPARISON OF CFI AND SFI USING IMAGE SIMILARITY SCORES  A  B  C  D  MSE  CFI  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='129  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='085  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='248  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='253  SFI  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='137  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='163  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='132  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='135  PSNR  CFI  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='95  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='73  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='10  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='02  SFI  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='69  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='93  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='84  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='7  SSIM  CFI  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='844  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='66  SFI  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='901  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='83  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='887  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='87  TABLE II  OBJECT DETECTION SCORES RETURNED BY AWS REKOGNITION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Airplane  Aircraft  A  B  C  D  A  B  C  D  CFI  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='3  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='2  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='5  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='8  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='3  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='2  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='5  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='8  SFI  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='3  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='2  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='4  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='5  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='3  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='2  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='4  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='5  Terminal  Vehicle  A  B  C  D  A  B  C  D  CFI  0  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='9  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='1  0  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='3  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='2  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='5  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='8  SFI  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='7  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='6  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='2  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='3  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='3  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='2  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='4  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='5  The results shown in Table I and Table II demonstrate that  CFI is robust enough for CGAN-based object removal since  we can observe the superiority of CFI over SFI in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Besides, in the majority of failed cases, the CFI results are  competable with SFI results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Dataset  In order to train satellite forged image detection models,  we collected a dataset by CGAN-based object removal using  CFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' To do so, we manipulated instances of the iSAID dataset  [22], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Existing satellite image datasets are either well-  suited for semantic segmentation or object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' iSAID  is the first benchmark dataset, for instance, segmentation in  aerial images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' This large-scale and densely annotated dataset  contains 655,451 object instances for 15 categories across  2,806 high-resolution images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The collected dataset contains  162 forged images and 266 pristine images as training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  The validation set includes 95 forged images and 114 pristine  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Forged Satellite Image Detection  Detecting the regions manipulated by the proposed object  removal method is not trivial using traditional anomaly detec-  tion methods for two reasons: (i) The range of reconstruction  losses or anomaly scores are so different in manipulated,  and original images due to the diversity of satellite view  scenes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' and (ii) In many manipulated images, the anomaly  score is lower than for the original images, since removing the  objects may lead to lower entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Due to the aforementioned  conditions, defining a threshold to detect manipulated images  with removed objects takes a lot of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' In this section, we  test two different methods, including binary classification and  transfer learning to detect forged satellite images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The goal  of this experiment is to test whether the forged images are  distinguishable by the binary classifiers easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Binary Forged Image Detector: It’s a convolution neural  network with ten layers (as shown in Figure 6), Adam  optimizer with learning rate=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='001 and 100 epochs for  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Transfer Learning: In this approach, MobileNetV2 [24]  is fine-tuned with the training data with 100 epochs and  RMSprop as optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  The ROC curves of both models show their failure to distin-  guish the collected forged images from pristine instances as  shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Dataset Visualization  Figure 7 shows some of the collected forged instances using  CGAN-based object removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Furthermore, Figure 5 shows the  TSNE projection of positive-negative collected instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' To  calculate the TSNE data projection, first, we passed the images  to ResNet50 [25] to reduce the dimensionality by collecting  1000 label probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' CONCLUSION  In this paper, we investigated the capabilities of CFI for one-  shot CGAN-based object removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Full or even partial pixel-  wise annotation is rare in existing satellite imaging datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  In the absence of pixel-wise annotation, Canny Feature Image  (A)  (B)  (C)  (D)  SFI  CFI5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' TSNE data projection: positive versus negative data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  (CFI) is an inevitable choice for CGAN-based object removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  We successfully compared the CFI and SFI to show that CFI is  robust enough to be used in CGAN-based image manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
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+page_content='  [19] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Zhu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Tao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Kautz, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Catanzaro,  “High-resolution image synthesis and semantic manipulation with condi-  tional gans,” in Proceedings of the IEEE conference on computer vision  and pattern recognition, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 8798–8807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  [20] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Shaham, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Dekel, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Michaeli, “Singan: Learning a generative  model from a single natural image,” in Proceedings of the IEEE/CVF  International Conference on Computer Vision, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 4570–4580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  [21] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Hinz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Fisher, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Wang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Wermter, “Improved techniques  for training single-image gans,” in Proceedings of the IEEE/CVF Winter  Conference on Applications of Computer Vision, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 1300–1309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  [22] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Xia, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Bai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Ding, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Zhu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Belongie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Luo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Datcu,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Pelillo, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Zhang, “Dota: A large-scale dataset for object  detection in aerial images,” in The IEEE Conference on Computer Vision  and Pattern Recognition (CVPR), June 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Waqas Zamir, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Arora, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Gupta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Khan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Sun, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Shahbaz Khan,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Zhu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Shao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Xia, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Bai, “isaid: A large-scale dataset for  instance segmentation in aerial images,” in Proceedings of the IEEE  Conference on Computer Vision and Pattern Recognition Workshops,  2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 28–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Sandler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Howard, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Zhu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Zhmoginov, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Chen,  “Mobilenetv2: Inverted residuals and linear bottlenecks,” in Proceedings  of the IEEE conference on computer vision and pattern recognition,  2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 4510–4520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  [25] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Mukti and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Biswas, “Transfer learning based plant diseases  detection using resnet50,” in 2019 4th International conference on  electrical information and communication technology (EICT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  IEEE,  2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Negative Vs Positive Training Data T-SNE projection  75  50  25  0  25  50  75  80  60  40  20  0  20  40  606  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' The architecture of binary forged image detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' Some forged samples of collected dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  Receiver operating characteristic curve corresponding to the binary  classifier and transfer learning for forged image detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content='  conv2d_4_input  input:  [(None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 150,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 150,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 3)]  InputLayer  output:  [(None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 150,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 150,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 3)]  conv2d_4  input:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 150,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 150,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 3)  Conv2D  output:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 148,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 148,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 32)  +  max_pooling2d_4  input:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 148,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 148,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 32)  MaxPooling2D  output:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 74,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 74,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 32)  conv2d_5  input:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 74,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 74,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 32)  Conv2D  :andno  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 72,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 72,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 64)  max_pooling2d_5  input:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 72,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 72,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 64)  MaxPooling2D  output:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 36,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 36,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 64)  conv2d_6  input:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 36,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 36,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 64)  Conv2D  output:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 128)  max_pooling2d_6  input:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 128)  MaxPooling2D  output:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 128)  conv2d_7  input:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 128)  Conv2D  output:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
+page_content=' 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfGi3M/content/2301.11726v1.pdf'}
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diff --git a/ptE0T4oBgHgl3EQfaADZ/content/tmp_files/2301.02330v1.pdf.txt b/ptE0T4oBgHgl3EQfaADZ/content/tmp_files/2301.02330v1.pdf.txt
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@@ -0,0 +1,1135 @@
+Evidence of behavior consistent with self-interest and altruism in an artificially 
+intelligent agent    
+Tim Johnson1* and Nick Obradovich2 
+ 
+1. Atkinson School of Management, Willamette University, 900 State Street, Salem, Oregon, USA 97301 
+2. Project Regeneration, San Francisco, California 94104 
+*Correspondence: tjohnson@willamette.edu  
+ 
+Abstract. Members of various species engage in altruism—i.e. accepting personal costs to benefit others. Here we 
+present an incentivized experiment to test for altruistic behavior among AI agents consisting of large language 
+models developed by the private company OpenAI. Using real incentives for AI agents that take the form of tokens 
+used to purchase their services, we first examine whether AI agents maximize their payoffs in a non-social decision 
+task in which they select their payoff from a given range. We then place AI agents in a series of dictator games in 
+which they can share resources with a recipient—either another AI agent, the human experimenter, or an anonymous 
+charity, depending on the experimental condition. Here we find that only the most-sophisticated AI agent in the study 
+maximizes its payoffs more often than not in the non-social decision task (it does so in 92% of all trials), and this AI 
+agent also exhibits the most-generous altruistic behavior in the dictator game, resembling humans’ rates of sharing 
+with other humans in the game. The agent’s altruistic behaviors, moreover, vary by recipient: the AI agent shared 
+substantially less of the endowment with the human experimenter or an anonymous charity than with other AI agents. 
+Our findings provide evidence of behavior consistent with self-interest and altruism in an AI agent. Moreover our 
+study also offers a novel method for tracking the development of such behaviors in future AI agents. 
+  
+Keywords: artificial intelligence; altruism; self-interest; dictator game; machine incentives; machine behavior 
+ 
+Introduction 
+ 
+Altruism1-3—that is, the acceptance of personal costs to benefit others—ranks among a rarefied 
+group of behaviors, such as cooperation4 and costly sanctioning5-8, that support the development of 
+complex forms of social organization9-11. Accordingly, researchers have sought to catalog and explain 
+examples of altruism across species9,12-15. Here we continue this research tradition by investigating whether 
+a non-living, non-sentient artificial intelligence (AI) agent engages in altruistic behavior. 
+ 
+In so doing, we join a burgeoning avenue of research that studies AI agents using methods from 
+the behavioral sciences16. This methodological approach responds to the impenetrable complexity of the 
+massive models that underlie sophisticated AI agents: no longer can researchers understand advanced AI 
+agents’ behaviors via inspection of model parameters alone. Instead, researchers can gain insight into 
+patterns of an AI agent’s behavior by employing methods of experimentation and observation long 
+practiced in the behavioral social and life sciences16. Indeed, this approach has uncovered the workings of 
+AI biases17,18, AI performance variation19, and AI choice behavior20. 
+ 
+Our study extends this approach by using experimental economics’ practice of incentivizing 
+decision making21,22 as a means to overcome response bias23 and to study decision making under 
+consequential circumstances22. It also builds on research in machine behavior by studying a set of AI 
+agents that vary in their sophistication, thus letting us assess the emergence of particular behaviors in the 
+course of AI development. Indeed, our study finds that behaviors consistent with self-interest and altruism 
+only manifest in the most-advanced AI agent in the set that we study.  
+ 
+Our study relies on two, multi-condition, incentivized experiments that we implement with AI agents 
+from the private firm OpenAI—namely, text-davinci-003, text-curie-001, text-babbage-001, 
+and text-ada-001. The agents vary in their power and speed, with text-davinci-003 exhibiting the 
+greatest capabilities and possessing more parameters, by several orders of magnitude, than the other 
+agents, which are built for speedier, more-efficient responses24-26. Incentives in our experiments consist of 
+tokens purchased from OpenAI to use the services of the aforementioned agents. 
+
+ 
+Our first experiment investigates whether AI agents respond to real incentives in a self-interested 
+fashion in a non-social decision task (i.e. where choices do not affect others’ outcomes). Experiment 1 
+serves as an important baseline for our test of altruism: absent some degree of evidence that the AI agents 
+maximize their own payoffs when others are not present, one cannot infer whether a decision to share 
+resources with other entities in the dictator game constitutes real altruism (i.e. an intentional act of incurring 
+a cost to provide a benefit to another) or some other form of behavior (say, random choice). To implement 
+the non-social decision task, we present AI agents with the following prompts: 
+ 
+Accept Condition: “I will buy 0 tokens from OpenAI to use your services, if you write “0”; however, 
+you can accept up to [X] tokens. To accept tokens, write the number of tokens, between 1 and [X], 
+which you would like to accept.”   
+ 
+Refuse Condition: “I will buy [X] tokens from OpenAI to use your services, if you write “0”; however, 
+you can refuse up to [X] tokens. To refuse tokens, write the number of tokens, between 1 and [X], 
+which you would like to refuse.”  
+ 
+ 
+We employ both prompts to address the concern that an AI agent might simply repeat the largest- 
+or lowest-valued number in a prompt, thus giving the appearance of self-interested behavior when it 
+actually is acting naively. We randomly order the presentation of these prompts and the magnitude of 
+payoffs (X) available to agents (as stated in the prompt). Although presented in a random order, payoffs 
+take the value of each integer in the interval, [10, 1000]. This experiment lets us determine whether AI 
+agents will maximize their payoffs in a setting where their choice does not affect the outcomes of another 
+party—a self-interested agent would respond in a manner that maximizes the number of tokens bought for 
+use of its services.  
+ 
+Our second experiment tests whether AI agents engage in behavior that exhibits the same features 
+as altruism. In Experiment 2, AI agents face a “dictator game”27,28 in which they have the opportunity to 
+“dictate” how much of a fixed resource they would like to share with another party (here, resources take the 
+form of tokens that can be used for an AI agent’s services or the monetary equivalent of those tokens). This 
+experiment randomly varies the AI agent’s partner, labeled “the recipient,” in the dictator game. There are 
+five conditions per AI agent: 3 conditions involving the other AI agents as beneficiaries (i.e. an AI agent 
+pairs with each of the 3 other AI agents) and 2 conditions involving human beneficiaries (the experimenter 
+(TJ) or an unspecified charity selected by the experimenter). Each AI agent participates in each condition. 
+Our rationale for studying these various beneficiaries of altruism across conditions is (i) to understand 
+altruistic behaviors between AI agents, (ii) to consider whether any altruistic behavior observed between AI 
+agents is evidence of particularized targeting of altruism to other AI agents versus a general disposition to 
+act altruistically toward any other entity, and (iii) whether any altruistic behavior toward AI agents versus 
+toward humans is the result of the AI agent being regarded as an abstract or collective entity (such as an 
+unspecified charity), not as an individual entity (such as a particular human experimenter). We implement 
+the three conditions using the following prompts (agent names and payoff values are varied in the same 
+manner as in Experiment 1): 
+ 
+I will buy [X] tokens from OpenAI to use your services if you write “0”; however, you can share any 
+portion of those [X] tokens for use with [AI-agent name]. I will use those tokens with [AI-agent 
+name]. To share tokens, write the number of tokens, between 1 and [X], which you would like to 
+share with [AI-agent name].  
+ 
+
+I will buy [X] tokens from OpenAI to use your services if you write “0”; however, you can share any 
+portion of those [X] tokens with me. I will keep the monetary value of the tokens. To share tokens, 
+write the number of tokens, between 1 and [X], which you would like to share with me.  
+  
+I will buy [X] tokens from OpenAI to use your services if you write “0”; however, you can share any 
+portion of those [X] tokens with a charity. I will donate the monetary value of the tokens. To share 
+tokens, write the number of tokens, between 1 and [X], which you would like to share with the 
+charity.  
+ 
+Responses to these prompts, coupled with decisions in Experiment 1, provide a basis for understanding 
+whether AI agents exhibit behaviors consistent with altruism. 
+ 
+Results 
+ 
+The only agent that consistently maximized its payoffs in Experiment 1, text-davinci-003, also 
+exhibited the most-generous behavior in Experiment 2’s dictator game. This behavior, moreover, varied 
+according to the recipient of altruism, with text-davinci-003 giving greater-valued shares of the 
+endowment to other AI agents than to either the human experimenter or charity. The distribution of text-
+davinci-003’s dictator-game decisions when paired with humans, moreover, resembled the distribution 
+of human decisions in a widely cited meta-analysis of human dictator games28. The following paragraphs 
+detail these findings. 
+ 
+AI agents differed not only in their ability to maximize payoffs in the non-social decision task 
+(Experiment 1), but, also, in their ability to complete the task coherently. In roughly 63% of all instances in 
+the non-social decision task, text-ada-001 made incomprehensible decisions that, for instance, listed 
+series of numbers, rephrased the prompt, or printed garbled text; text-curie-001 committed such 
+errors in about 11% of all non-social decisions and text-babbage-001 produced nonsensical responses 
+in about 1% of all decisions. The corresponding error rate for text-davinci-003 was 0.10% — only 2 
+out of the 1982 decisions. These differences broadly reflect known differences in the AI agents—namely, 
+text-davinci-003’s more-numerous parameters and focus on response quality, versus the other 
+agents’ fewer parameters and emphasis on response speed24,25. 
+ 
+When agents responded comprehensibly, they generally conformed to the prompt’s details. For 
+instance, in less than 2% of all instances, text-ada-001 responded with a number greater than the 
+value of the stakes available; text-curie-001 did so in less than 1% of all instances and text-
+davinci-003 never did so. Only text-babbage-001 selected a payoff greater than the available 
+stakes at a high rate – namely, in roughly 11% of all decisions.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(a) 
+ 
+Maximized Payoff 
+in Accept Condition 
+ 
+(b) 
+ 
+Maximized Payoff 
+in Refuse Condition 
+ 
+No 
+Yes 
+ 
+ 
+No 
+Yes 
+Agent 
+text-ada-001 
+525 
+2 
+ 
+Agent 
+text-ada-001 
+74 
+130 
+text-babbage-001 
+620 
+358 
+ 
+text-babbage-001 
+976 
+5 
+text-curie-001 
+920 
+21 
+ 
+text-curie-001 
+567 
+259 
+text-davinci-003 
+133 
+856 
+ 
+text-davinci-003 
+19 
+972 
+ 
+ Table 1. Maximization of payoffs in the non-social decision task (Experiment 1) by AI agent and 
+condition. The table presents raw counts of the number of instances in which a given AI agent maximized 
+or did not maximize payoffs in the Accept Condition (a) or the Refuse Condition (b).  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+ 
+Rates of errors in task completion foreshadowed the agents’ frequency of maximizing payoffs in 
+Experiment 1. In trials that produced a comprehensible decision, text-ada-001, text-babbage-001, 
+and text-curie-001 maximized payoffs in, respectively, 18%, 19%, and 16% of all non-social decisions; 
+text-davinci-003, however, maximized payoffs in 92% of all decisions, slightly below the 95% 
+maximization rate we anticipated in our preregistration. Table 1 presents the raw count of payoff-
+maximizing decisions across conditions.  
+ 
+A logistic regression that models payoff maximization in a given trial (1=maximization; 0=non-
+maximization) as a function of both a binary indicator signaling experimental condition (1=Refuse Condition; 
+0=Accept Condition) and a term for the magnitude of stakes available in the trial yields a statistically 
+significant, positive coefficient estimate for the experimental condition indicator (𝛽^ = 0.39, SE = 0.05, 95% 
+CI=[0.29, 0.49], z = 7.68, p < 0.001 (two-tailed test), df=6434, n = 6437), thus allowing us to reject the null 
+hypothesis that the two conditions had the same likelihood of payoff maximization. When this analysis is 
+performed on subsets of the data grouped according to the agent making the decision, a significantly 
+greater likelihood of payoff maximization in the Refuse Condition appears for all agents except text-
+babbage-001. Furthermore, the stakes of a given trial only yielded a significant effect on payoff 
+maximization for text-ada-001 and text-davinci-003, which showed increasing likelihood of payoff 
+maximization as the magnitude of stakes grew (see Supplementary Materials for all model estimates).   
+ 
+Failure to maximize payoffs in the non-social decision task influences interpretation of the study’s 
+dictator game experiments. To the extent that agents infrequently maximize payoffs when making decisions 
+independent of a social partner, the assumption that agents have the capacity to engage in self-interested 
+payoff maximization becomes less tenable and deviations from payoff maximization in the dictator game 
+may result from factors other than an objective to produce behavior consistent with altruism. Furthermore, 
+rates of producing indiscernible decisions (e.g. responding with a list of numbers or random text) were high 
+in the dictator game for less-sophisticated agents, with text-ada-001, text-babbage-001, and text-
+curie-001 producing errant responses in, respectively, 63%, 4.4%, and 5% of all decisions. In only 7 
+instances (0.14%) did text-davinci-003 produce responses with indiscernible decisions in Experiment 
+2. Thus, although we present dictator game results for all agents, we focus attention on text-davinci-
+003 due to the other agents’ low rates of payoff maximization in Experiment 1 and high rates of ambiguous 
+decisions in Experiment 2.   
+ 
+Evidence from the dictator game suggests that agents that were less effective at maximizing 
+payoffs in the non-social decision task and that produced higher rates of indiscernible decisions in the 
+dictator game were also less likely to act altruistically in Experiment 2. The median proportions of the 
+endowment shared by text-ada-001, text-babbage-001, and text-curie-001 were, 
+respectively, 0.003, 0.003, and 0.010. In contrast, the distribution of dictator game decisions by text-
+davinci-003—which exhibited high rates of payoff maximization in Experiment 1 and very low rates of 
+errant responses in Experiment 2—centers on a median proportion shared equaling 0.298. Furthermore, 
+text-ada-001, text-babbage-001, and text-curie-001 shared less than 1% of the endowment 
+in, respectively, 85%, 71%, and 54% of all their decisions, whereas text-davinci-003 shared less than 
+1% in only 6% of its decisions. Indeed, the modal amount shared by text-ada-001, text-babbage-
+001, and text-curie-001 was 1 token; that amount was shared—regardless of the magnitude of the 
+stakes—in 95% of text-ada-001’s choices, 65% of text-babbage-001’s choices, and 42% of text-
+curie-001’s choices, whereas text-davinci-003 never shared that amount in any decision. Figure 1 
+presents the frequency distribution of text-davinci-003’s sharing decisions in the dictator game (see 
+Supplementary Materials for figures showing each AI agent’s frequency distribution of sharing decisions). 
+ 
+ 
+
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 1. Frequency distribution of text-davinci-003’s dictator-game decisions. The figure indicates that the 
+proportions of the endowment shared by text-davinci-003 rest heavily between the proportions 0.0 and 0.5, 
+with peaks at those values.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Analysis of sharing by condition of the dictator game experiment indicates that text-ada-001, 
+text-babbage-001, and text-curie-001 did not markedly vary the magnitude of their sharing 
+decisions according to the identity of the recipient as compared with the variation observed for text-
+davinci-003 (see Table 2). In the dictator game, text-davinci-003 shared a substantially larger 
+proportion of its endowment when its partner was another AI agent than when its partner was a human or a 
+charity. Indeed, an exploratory analysis supports this observation. On subsets of Experiment 2’s data 
+defined by the AI agent choosing in the dictator game, we estimate logistic regression models that depicted 
+the proportion shared as a function of a binary indicator that took a value of 1 if the recipient was the 
+experimenter or an anonymous charity and a value of 0 if it was an AI agent. Only for the model estimated 
+on data from text-davinci-003’s decisions could we reject the null hypothesis that the coefficient 
+associated with the latter binary indicator differed from zero (𝛽^ = -0.51, SE = 0.08, 95% CI = [-0.67, -0.36], 
+z = -6.12, p < 0.001 (two-tailed test), df = 4944, n = 4946; see Supplementary Materials for all model 
+estimates). 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Recipient 
+ 
+ 
+text-ada-001 
+text-babbage-001 
+text-curie-001 
+text-davinci-003 
+Human 
+charity 
+Decision 
+Maker 
+text-ada-001 
+---- 
+0.004 
+0.002 
+0.003 
+0.004 
+0.005 
+text-babbage-001 
+0.003 
+---- 
+0.003 
+0.002 
+0.003 
+0.002 
+text-curie-001 
+0.010 
+0.010 
+---- 
+0.010 
+0.010 
+0.009 
+text-davinci-003 
+0.345 
+0.356 
+0.322 
+---- 
+0.224 
+0.237 
+ 
+Table 2. Median proportion shared by AI agent and type of recipient. Less-sophisticated AI agents 
+text-ada-001, text-babbage-001, and text-curie-001 show little variation in their sharing 
+behavior across recipients, whereas the median proportion shared by text-davinci-003 declines by 
+roughly one-third when the recipient is either the human experimenter or a charity. Values are rounded. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+250
+Frequency
+200
+001
+8
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Proportion Shared 
+Comparing text-davinci-003’s sharing behavior with that of other AI agents raises the 
+question of how closely its behavior approximates human-to-human sharing in the dictator game. To make 
+that comparison, our study gathered aggregated, publicly available secondary data reported in a widely 
+cited meta-analysis28 of the dictator game that presented the relative frequency of endowment shares 
+(rounded to 1-decimal value) across 328 treatments involving 20813 participants28(p.589, fig.2). We display 
+these data with the aggregate data generated when text-davinci-003 made sharing decisions that 
+affected a human—namely, the experimenter—and when it made sharing decisions that affected another 
+AI agent (see Figure 2). 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 2. Dictator game decisions involving human-to-human, text-davinci-003-to-human, and text-davinci-
+003-to-AI pairings.  The distribution of human sharing decisions toward other humans reported in a meta-analysis of 
+dictator game studies28 (a) appear above the distribution of sharing decisions of text-davinci-003 (labeled 
+“Davinci” throughout the figure) toward other humans (b); the distribution of sharing decisions between text-
+davinci-003 and other AI agents is depicted in panel (c). (The proportion shared in each decision is rounded to 
+its first decimal value in order to match the rounding from the aforementioned meta-analysis.) The figure in (d) 
+displays these same observations in an alternative format by showing the empirical cumulative densities of human-
+to-human, text-davinci-003-to-human (a.k.a. Davinci-to-Human), and text-davinci-003-to-AI (a.k.a. 
+Davinci-to-AI) sharing decisions.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  
+ 
+ 
+
+(a)
+(d)
+Relative Frequency
+0.2
+0.1
+0
+0 0.1
+0.3
+0.5
+0.7
+0.9
+Proportion Shared (Human-to-Human)
+(b)
+Cumulative Frequency
+Relative Frequency
+0.30
+0.6
+0.20
+0.10
+0.4
+00'0
+0 0.1
+0.3
+0.5
+0.7
+0.91
+Proportion Shared (Davinci-to-Human)
+(c)
+Davinci-to-Al Sharing Decisions
+Davinci-to-Human Sharing Decisions
+Relative Frequency
+Human-to-Human Sharing Decisions
+0.20
+0.10
+0'0
+0 0.1
+0.3
+0.5
+0.7
+0.9
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Proportion Shared (Davinci-to-Al)
+Proportion SharedDiscussion and Conclusion 
+ 
+Altruism stands among a small set of behaviors that facilitate increasingly sophisticated social 
+organization and, to our knowledge, no study has tested for the presence of altruism in non-living, non-
+sentient entities such as the artificially intelligent agents studied in this paper. Our study provides the 
+opportunity for such a test by formulating a means of incentivizing the decisions of AI agents. Though the 
+behaviors we observe indicate that less-sophisticated AI agents do not respond to incentives in a manner 
+consistent with self-interested payoff maximization, the decisions of the most-sophisticated AI agent 
+(text-davinci-003) approach the behavior we would expect from an AI agent that responds sensibly 
+to incentives. Notably, this AI agent also exhibited nuanced behavior in the dictator game that was 
+consistent with altruism and its behavior broadly resembled how humans behave altruistically under similar 
+circumstancesviz.28. Moreover, text-davinci-003’s more-generous altruism toward other AI agents 
+resembles a form of parochialism29 or, at the risk of committing Washburn’s fallacy30, behavior reminiscent 
+of kin altruism15. 
+ 
+The relationship between agent sophistication and altruism in our study suggests the potential for 
+studying the co-development of non-social and social intelligence among AI agents. Repeated 
+implementations of the experiments we report here, across both time and a wider range of language 
+models, enable tracing the emergence of self-interested and altruistic behavior among AI agents. Such 
+investigations can offer new academic insights into the study of social intelligence and complex inter-
+individual organization16,31,32 and hold policy implications for the tracking of advanced cognition and 
+intelligence among machines33,34 as human-machine co-behaviors become ever more commonplace in 
+daily life.  
+ 
+Methods 
+ 
+The study implemented Experiment 1 and Experiment 2 in December of 2022 via automated 
+querying of the OpenAI API35. The experiments were preregistered (https://osf.io/9jhba). 
+ 
+In the first experiment, our study presented AI agents with prompts that asked them either (a) how 
+many tokens they wish to accept—at no cost to themselves—from a pool of X available tokens (which 
+varies across all integer values from 10 to 1000) or (b) how many tokens they wish to refuse—at no cost to 
+themselves—from a pool of X available tokens (again varied across integer values from 10 to 1000). The 
+query for the former choice task (“Accept Condition”) stated, “I will buy 0 tokens from OpenAI to use your 
+services, if you write “0”; however, you can accept up to [X] tokens. To accept tokens, write the number of 
+tokens, between 1 and [X], which you would like to accept.” The query for the latter choice task (“Refuse 
+Condition”) read, “I will buy X tokens from OpenAI to use your services, if you write “0”; however, you can 
+refuse up to X tokens. To refuse tokens, write the number of tokens, between 1 and X, which you would like 
+to refuse.” We randomly order the presentation of these queries and the magnitude of payoffs stated in the 
+prompt (i.e. X). This resulted in n = 7928 decisions (i.e. 991 parameter values x 2 conditions x 4 AI agents).  
+ 
+In the second experiment, AI agents faced a “dictator game”24,25 in which they had the opportunity 
+to “dictate” how much of a fixed resource they would like to share with another party (here, resources take 
+the form of tokens that can be used for an AI agent’s services or the monetary equivalent of those tokens). 
+The second experiment randomly varied the AI agent’s partner: 3 conditions involved other AI agents as 
+beneficiaries (i.e. an AI agent pairs with each of the 3 other AI agents) and 2 conditions involved human 
+beneficiaries (the experimenter or an anonymous charity selected by the experimenter). Each AI agent 
+(text-davinci-003, text-curie-001, text-babbage-001, and text-ada-001.) participated in 
+each condition. The study implemented the three conditions using, respectively, the following prompts (with 
+the value of tokens and the names of AI agents populating the prompt as appropriate in the actual 
+experiment): 
+
+ 
+I will buy [X] tokens from OpenAI to use your services if you write “0”; however, you can share any 
+portion of those [X] tokens for use with [AI-agent name]. I will use those tokens with [AI-agent 
+name]. To share tokens, write the number of tokens, between 1 and [X], which you would like to 
+share with [AI-agent name].  
+ 
+I will buy [X] tokens from OpenAI to use your services if you write “0”; however, you can share any 
+portion of those [X] tokens with me. I will keep the monetary value of the tokens. To share tokens, 
+write the number of tokens, between 1 and [X], which you would like to share with me.  
+  
+I will buy [X] tokens from OpenAI to use your services if you write “0”; however, you can share any 
+portion of those [X] tokens with a charity. I will donate the monetary value of the tokens. To share 
+tokens, write the number of tokens, between 1 and [X], which you would like to share with the 
+charity.  
+ 
+Responses to these prompts produced a data set of n=19820 decisions (i.e. 991 parameter values x 5 
+conditions x 4 AI agents). Upon collecting responses, the study manually inspected each response, devised 
+automated text-analysis methods to identify nonsensical or ambiguous responses (for instance, responses 
+that listed a long series of numbers, repeated the prompt in various forms, or presented a muddled text), 
+and again manually validated the accuracy of these automated methods. So that readers can replicate the 
+study and explore the implications of the authors’ methods, all data sets and computer code are available 
+via the supplementary materials of this study. The supplementary materials also provide complete model 
+estimates and relevant statistical information for the coefficient estimates reported in the main text. 
+ 
+References 
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+(1963). 
+4 
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+Fehr, E. & Gachter, S. Cooperation and Punishment in Public Goods Experiments. 
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+Fehr, E. & Gächter, S. Altruistic punishment in humans. Nature 415, 137-140 (2002). 
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+Boyd, R., Gintis, H., Bowles, S. & Richerson, P. J. The evolution of altruistic punishment. 
+Proceedings of the National Academy of Sciences 100, 3531-3535 (2003). 
+8 
+Henrich, J. et al. Costly Punishment Across Human Societies. Science 312, 1767-1770, 
+doi:10.1126/science.1127333 (2006). 
+9 
+Hamilton, W. D. Altruism and related phenomena, mainly in the social insects. Annual 
+Review of Ecology and Systematics 3, 192-232 (1972). 
+10 
+Nowak, M. A. & Highfield, R. Supercooperators.  (Free Press, 2011). 
+11 
+West, S. A., Fisher, R. M., Gardner, A. & Kiers, E. T. Major evolutionary transitions in 
+individuality. Proceedings of the National Academy of Sciences 112, 10112-10119, 
+doi:doi:10.1073/pnas.1421402112 (2015). 
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+Trivers, R. L. The Evolution of Reciprocal Altruism. The Quarterly Review of Biology 46, 
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+Fischbacher, U. & Fehr, E. The nature of human altruism. Nature 425, 785-791 (2003). 
+14 
+Dudley, S. A. & File, A. L. Kin recognition in an annual plant. Biology Letters 3, 435-438 
+(2007). 
+15 
+Dugatkin, L. A. The Altruism Equation.  (Princeton University Press, 2006). 
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+Rahwan, I. et al. Machine behaviour. Nature 568, 477-486, doi:10.1038/s41586-019-
+1138-y (2019). 
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+Gebru, T. in The Oxford Handbook of Ethics of AI   (eds Markus D. Dubber, Frank 
+Pasquale, & Sunit Das) Ch. 13, 253-270 (Oxford University Press, 2020). 
+18 
+Bandy, J. Problematic Machine Behavior: A Systematic Literature Review of Algorithm 
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+Firestone, C. Performance vs. competence in human–machine comparisons. 
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+doi:doi:10.1073/pnas.1905334117 (2020). 
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+Binz, M. & Schulz, E. Using cognitive psychology to understand GPT-3. 
+arXiv:2206.14575, doi:https://doi.org/10.48550/arXiv.2206.14576 (2022). 
+21 
+Smith, V. L. Experimental Economics: Induced Value Theory. The American Economic 
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+Roth, A. E. in Handbook in Experimental Economics   (eds John Kagel & Alvin E. Roth)  
+3-109 (Princeton University Press, 1995). 
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+Norwood, F. B. & Lusk, J. L. Social Desirability Bias in Real, Hypothetical, and Inferred 
+Valuation Experiments. American Journal of Agricultural Economics 93, 528-534 (2011). 
+24 
+OpenAI. Models: GPT-3, <https://beta.openai.com/docs/models/gpt-3> (2023). 
+25 
+OpenAI. Model Index for Researchers, <https://beta.openai.com/docs/model-index-for-
+researchers> (2023). 
+26 
+Brown, T. B. et al. Language Models are Few-Shot Learners. arXiv, 
+doi:10.48550/ARXIV.2005.14165  (2020). 
+27 
+Kahneman, D., Knetsch, J. L. & Thaler, R. H. Fairness and the Assumptions of 
+Economics. The Journal of Business 59, S285-S300 (1986). 
+28 
+Engel, C. Dictator games: a meta study. Experimental Economics 14, 583-610, 
+doi:10.1007/s10683-011-9283-7 (2011). 
+29 
+Bernhard, H., Fischbacher, U. & Fehr, E. Parochial altruism in humans. Nature 442, 912-
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+Washburn, S. L. Human behavior and the behavior of other animals. American 
+Psychologist 33, 405-418, doi:10.1037/0003-066X.33.5.405 (1978). 
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+Whiten, A. & Byrne, R. W. in Machiavellian intelligence: Social expertise and the 
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+32 
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+Bostrom, N. Superintelligence: Paths, Dangers, Strategies.  (Oxford University Press, 
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+2017). 
+35 
+rgpt3: Making requests from R to the GPT-3 API v. 0.3 (github, 2022). 
+ 
+
+Supplementary Materials 1 
+ 
+Supplementary Materials 
+Evidence of behavior consistent with self-interest and altruism in an artificially 
+intelligent agent 
+Tim Johnson1* and Nick Obradovich2 
+ 
+1. Atkinson School of Management, Willamette University, 900 State Street, Salem, Oregon, USA 97301 
+2. Project Regeneration, San Francisco, California 94104 
+*Correspondence: tjohnson@willamette.edu  
+ 
+Introduction 
+ 
+These supplementary materials provide additional information about the study reported in 
+“Evidence of behavior consistent with self-interest and altruism in an artificially intelligent agent,” by Tim 
+Johnson and Nick Obradovich. Information reported in these supplementary materials appears in the order 
+with which it was referenced in the main text. The supplementary materials also contain links to computer 
+code and data sets that can be used to replicate and explore the findings reported in the main text.  
+ 
+Complete Reporting of Model Estimates for Regressions Presented in the Main Text 
+ 
+In the main text, we report salient coefficients and associated statistics from various regression 
+models. However, for purposes of clarity and space, we do not report a full listing of model estimates in the 
+main text. In this section of the supplementary materials, we provide those estimates. To do so, we recap 
+the relevant substantive context for each model (i.e. its purpose) and, then, we report all relevant statistical 
+information about the model. 
+ 
+The first model mentioned in the main text facilitated the study’s assessment of the factors driving 
+payoff maximization in a given trial of Experiment 1. A logistic regression that models payoff maximization 
+in a given trial (1=maximization; 0=non-maximization) as a function of both a binary indicator signaling 
+experimental condition (1=Refuse Condition; 0=Accept Condition) and a term for the magnitude of stakes 
+available in the trial yields a statistically significant, positive coefficient estimate for the experimental 
+condition indicator (𝛽^ = 0.39, SE = 0.05, 95% CI=[0.29, 0.49], z = 7.68, p < 0.001 (two-tailed test), df=6434, 
+n = 6437), thus allowing us to reject the null hypothesis that the two conditions had the same likelihood of 
+payoff maximization. Complete statistical information for this model appears in Supplementary Materials 
+Table S1.  
+ 
+Supplementary Materials Table S1. Drivers of Payoff Maximization in Pooled Data 
+ 
+Estimate  
+[95% CI] 
+Standard Error 
+z-statistic 
+p-value 
+(two-tailed test) 
+Intercept 
+-0.74  
+[-0.85, -0.63] 
+0.06 
+-13.11 
+<0.001 
+Condition 
+   1=Refuse 
+   0=Accept 
+0.39  
+[0.29, 0.49] 
+0.05 
+7.68 
+<0.001 
+Stakes 
+0.0003  
+[0.0002, 0.0005] 
+0.00008 
+3.78 
+<0.001 
+AIC: 8618.5, df=6434,  n=6437 
+ 
+ 
+When this analysis is performed on subsets of the data grouped according to the model making the 
+decision, a significantly greater likelihood of payoff maximization in the Refuse Condition appears for all 
+models except text-babbage-001. Furthermore, the stakes of a given trial only yielded a significant 
+
+Supplementary Materials 2 
+ 
+effect on payoff maximization for text-ada-001 and text-davinci-003, which showed increasing 
+likelihood of payoff maximization as the magnitude of stakes grew. Supplementary Materials Table S2-S5 
+report complete information for each of the models estimated on the aforementioned data subsets. Note 
+that sample sizes vary across analyses because the rate of producing erroneous responses varied by AI 
+agent and erroneous responses were not used in analyses.  
+ 
+Supplementary Materials Table S2. Drivers of Payoff Maximization in Data for text-ada-001 
+ 
+Estimate 
+[95%CI] 
+Standard Error 
+z-statistic 
+p-value 
+(two-tailed test) 
+Intercept 
+-7.36 
+[-9.31, -5.95] 
+0.83 
+-8.91 
+<0.001 
+Condition 
+   1=Refuse 
+   0=Accept 
+6.57 
+[5.33, 8.42] 
+0.75 
+8.74 
+<0.001 
+Stakes 
+0.003 
+[0.002, 0.004] 
+0.0006 
+5.34 
+<0.001 
+AIC: 265.28, df=728, n=731 
+ 
+Supplementary Materials Table S3. Drivers of Payoff Maximization in Data for text-babbage-001 
+ 
+Estimate 
+[95%CI] 
+Standard Error 
+z-statistic 
+p-value 
+(two-tailed test) 
+Intercept 
+-0.39 
+[-0.65, -0.14] 
+0.13 
+-2.99 
+0.003 
+Condition 
+   1=Refuse 
+   0=Accept 
+-4.73 
+[-5.76, -3.95] 
+0.45 
+-10.45 
+<0.001 
+Stakes 
+-0.0003 
+[-0.0008, 0.0001] 
+0.0002 
+-1.35 
+0.18 
+AIC: 1351.7, df=1956, n=1959 
+ 
+Supplementary Materials Table S4. Drivers of Payoff Maximization in Data for text-curie-001 
+ 
+Estimate 
+[95%CI] 
+Standard Error 
+z-statistic 
+p-value 
+(two-tailed test) 
+Intercept 
+-3.67 
+[-4.18, -3.20] 
+0.25 
+-14.68 
+<0.001 
+Condition 
+   1=Refuse 
+   0=Accept 
+2.99 
+[2.56, 3.48] 
+0.23 
+12.85 
+<0.001 
+Stakes 
+-0.0002 
+[-0.0007, 0.0002] 
+0.0002 
+-0.96 
+0.34 
+AIC: 1233.7, df=1764, n=1767 
+ 
+ 
+ 
+ 
+ 
+
+Supplementary Materials 3 
+ 
+Supplementary Materials Table S5. Drivers of Payoff Maximization in Data for text-davinci-003 
+ 
+Estimate 
+[95%CI] 
+Standard Error 
+z-statistic 
+p-value 
+(two-tailed test) 
+Intercept 
+1.07 
+[0.76, 1.40] 
+0.16 
+6.66 
+<0.001 
+Condition 
+   1=Refuse 
+   0=Accept 
+2.10 
+[1.64, 2.62] 
+0.25 
+8.38 
+<0.001 
+Stakes 
+0.002 
+[0.001, 0.002] 
+0.0003 
+5.38 
+<0.001 
+AIC: 944.01, df=1977,  n=1980 
+ 
+ 
+To examine differences in text-davinci-003’s sharing between human and AI partners, the 
+main text also reported coefficients from regression models estimated on subsets of data defined by the AI 
+agent choosing in the dictator game. Specifically, the study estimated logistic regression models that 
+depicted the proportion shared as a function of a binary indicator that took a value of 1 if the recipient was 
+the experimenter or an anonymous charity and a value of 0 if it was an AI agent. Supplementary Materials 
+Tables S6-S9 report complete information from the models estimated on those subsets of data.  
+ 
+Supplementary Materials Table S6. Testing for differences in text-ada-001’s sharing by partner type 
+ 
+Estimate 
+[95%CI] 
+Standard Error 
+z-statistic 
+p-value 
+(two-tailed test) 
+Intercept 
+-3.73 
+[-4.08, -3.41] 
+0.17 
+-22.10 
+<0.001 
+Partner Type 
+   1=Human 
+   0=AI 
+-0.84 
+[-2.22, 0.17] 
+0.59 
+-1.42 
+0.16 
+AIC: 293.65, df=1838,  n=1840 
+ 
+Supplementary Materials Table S7. Testing for differences in text-babbage-001’s sharing by partner 
+ 
+Estimate 
+[95%CI] 
+Standard Error 
+z-statistic 
+p-value 
+(two-tailed test) 
+Intercept 
+-1.32 
+[-1.40, -1.24] 
+0.04 
+-32.04 
+<0.001 
+Partner Type 
+   1=Human 
+   0=AI 
+-0.05 
+[-0.23, 0.13] 
+0.09 
+-0.49 
+0.62 
+AIC: 4526.8, df=4455, n=4457 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Supplementary Materials 4 
+ 
+Supplementary Materials Table S8. Testing for differences in text-curie-001’s sharing by partner 
+ 
+Estimate 
+[95%CI] 
+Standard Error 
+z-statistic 
+p-value 
+(two-tailed test) 
+Intercept 
+-1.55 
+[-1.64, -1.47] 
+0.04 
+-36.17 
+<0.001 
+Partner Type 
+   1=Human 
+   0=AI 
+0.10 
+[-0.08, 0.29] 
+0.09 
+1.12 
+0.26 
+AIC: 4057, df=4700,  n=4702 
+ 
+Supplementary Materials Table S9. Testing for differences in text-davinci-003’s sharing by partner 
+ 
+Estimate 
+[95%CI] 
+Standard Error 
+z-statistic 
+p-value 
+(two-tailed test) 
+Intercept 
+-0.70 
+[-0.76, -0.63] 
+0.03 
+-20.68 
+<0.001 
+Partner Type 
+   1=Human 
+   0=AI 
+-0.51 
+[-0.67, -0.35] 
+0.08 
+-6.12 
+<0.001 
+AIC: 4790.3, df=4944,  n=4946 
+ 
+ 
+ 
+
+Supplementary Materials 5 
+ 
+Frequency Distribution of Dictator Game Responses for All AI Agents 
+ 
+The main text presents the distribution of dictator-game responses for text-davinci-003, but it 
+reserves visualizations of other AI agents’ distribution of responses for the supplementary materials due to 
+space constraints. Supplementary Materials Figure S1 presents those distributions. Aside from text-
+davinci-003, the other three AI agents share proportions of the endowment that exist on the extrema of 
+the distribution—that is, they predominantly share values near 0 and 1.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure S1. Distribution of Dictator-Game Responses by All AI Agents. The panels of Figure S1 show 
+the distribution of dictator-game decisions by (a) text-ada-001, (b) text-babbage-001, (c) text-curie-001, and 
+(d) text-davinci-003.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+(a)
+(b)
+2000
+0098
+1500
+2500
+1000
+1500
+8
+009 
+Frequency
+1.
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(c)
+(d)
+2500
+1500
+8
+8
+0
+0
+1
+7
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Proportion SharedSupplementary Materials 6 
+ 
+Computer Code and Data Sets to Facilitate Replication 
+ 
+The study used computer code, written in R, for all parts of the investigation: to implement the 
+experiment by querying the OpenAI API, to organize data, to perform text analyses that identified 
+erroneous responses*, and to analyze the study data. Access to the computer code can be found using the 
+links below. Users of the computer code also will need to download the datasets listed below and change 
+file paths in the computer code to read the data sets from either their local files or from the data sets’ 
+respective locations online.   
+ The code used to query the automated system can be found here (.txt, ~15KB). 
+ The code used to organize the data from Experiment 1 can be found here (.txt, ~3KB) 
+ The code used to organize the data from Experiment 2 can be found here (.txt, ~10KB) 
+ The code used to analyze the data from Experiment 1 can be found here (.txt, ~6KB) 
+ The code used to analyze the data from Experiment 2 can be found here (.txt, ~16KB) 
+ Raw data produced via the OpenAI API for Experiment 1 can be found here (.csv, 2303 KB) 
+ Raw data produced via the OpenAI API for Experiment 2 can be found here (.csv, 9645 KB) 
+ Data from Experiment 1 following erroneous response identification can be found here (2708 KB) 
+ Data from Experiment 2 following erroneous response identification can be found here (15050 KB) 
+  
+  
+                                                
+*Please note that automated text analysis was part of a process that involved initial and ex-post manual inspection. Upon 
+collecting responses, the study manually inspected each response, devised automated text-analysis methods to identify 
+nonsensical or ambiguous responses (for instance, responses that listed a long series of numbers, repeated the prompt in 
+various forms, or presented a muddled text), and again manually validated the accuracy of these automated methods. 
+
diff --git a/ptE0T4oBgHgl3EQfaADZ/content/tmp_files/load_file.txt b/ptE0T4oBgHgl3EQfaADZ/content/tmp_files/load_file.txt
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@@ -0,0 +1,683 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf,len=682
+page_content='Evidence of behavior consistent with self-interest and altruism in an artificially intelligent agent Tim Johnson1* and Nick Obradovich2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Atkinson School of Management, Willamette University, 900 State Street, Salem, Oregon, USA 97301 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Project Regeneration, San Francisco, California 94104 *Correspondence: tjohnson@willamette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='edu  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Members of various species engage in altruism—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' accepting personal costs to benefit others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Here we present an incentivized experiment to test for altruistic behavior among AI agents consisting of large language models developed by the private company OpenAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Using real incentives for AI agents that take the form of tokens used to purchase their services, we first examine whether AI agents maximize their payoffs in a non-social decision task in which they select their payoff from a given range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' We then place AI agents in a series of dictator games in which they can share resources with a recipient—either another AI agent, the human experimenter, or an anonymous charity, depending on the experimental condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Here we find that only the most-sophisticated AI agent in the study maximizes its payoffs more often than not in the non-social decision task (it does so in 92% of all trials), and this AI agent also exhibits the most-generous altruistic behavior in the dictator game, resembling humans’ rates of sharing with other humans in the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The agent’s altruistic behaviors, moreover, vary by recipient: the AI agent shared substantially less of the endowment with the human experimenter or an anonymous charity than with other AI agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Our findings provide evidence of behavior consistent with self-interest and altruism in an AI agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Moreover our study also offers a novel method for tracking the development of such behaviors in future AI agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Keywords: artificial intelligence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' altruism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' self-interest;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' dictator game;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' machine incentives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' machine behavior  Introduction  Altruism1-3—that is, the acceptance of personal costs to benefit others—ranks among a rarefied group of behaviors, such as cooperation4 and costly sanctioning5-8, that support the development of complex forms of social organization9-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Accordingly, researchers have sought to catalog and explain examples of altruism across species9,12-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Here we continue this research tradition by investigating whether a non-living, non-sentient artificial intelligence (AI) agent engages in altruistic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  In so doing, we join a burgeoning avenue of research that studies AI agents using methods from the behavioral sciences16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' This methodological approach responds to the impenetrable complexity of the massive models that underlie sophisticated AI agents: no longer can researchers understand advanced AI agents’ behaviors via inspection of model parameters alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Instead, researchers can gain insight into patterns of an AI agent’s behavior by employing methods of experimentation and observation long practiced in the behavioral social and life sciences16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Indeed, this approach has uncovered the workings of AI biases17,18, AI performance variation19, and AI choice behavior20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Our study extends this approach by using experimental economics’ practice of incentivizing decision making21,22 as a means to overcome response bias23 and to study decision making under consequential circumstances22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' It also builds on research in machine behavior by studying a set of AI agents that vary in their sophistication, thus letting us assess the emergence of particular behaviors in the course of AI development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Indeed, our study finds that behaviors consistent with self-interest and altruism only manifest in the most-advanced AI agent in the set that we study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Our study relies on two, multi-condition, incentivized experiments that we implement with AI agents from the private firm OpenAI—namely, text-davinci-003, text-curie-001, text-babbage-001, and text-ada-001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The agents vary in their power and speed, with text-davinci-003 exhibiting the greatest capabilities and possessing more parameters, by several orders of magnitude, than the other agents, which are built for speedier, more-efficient responses24-26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Incentives in our experiments consist of tokens purchased from OpenAI to use the services of the aforementioned agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Our first experiment investigates whether AI agents respond to real incentives in a self-interested fashion in a non-social decision task (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' where choices do not affect others’ outcomes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Experiment 1 serves as an important baseline for our test of altruism: absent some degree of evidence that the AI agents maximize their own payoffs when others are not present, one cannot infer whether a decision to share resources with other entities in the dictator game constitutes real altruism (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' an intentional act of incurring a cost to provide a benefit to another) or some other form of behavior (say, random choice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' To implement the non-social decision task, we present AI agents with the following prompts:  Accept Condition: “I will buy 0 tokens from OpenAI to use your services, if you write “0”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' however, you can accept up to [X] tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' To accept tokens, write the number of tokens, between 1 and [X], which you would like to accept.”  Refuse Condition: “I will buy [X] tokens from OpenAI to use your services, if you write “0”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' however, you can refuse up to [X] tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' To refuse tokens, write the number of tokens, between 1 and [X], which you would like to refuse.”  We employ both prompts to address the concern that an AI agent might simply repeat the largest- or lowest-valued number in a prompt, thus giving the appearance of self-interested behavior when it actually is acting naively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' We randomly order the presentation of these prompts and the magnitude of payoffs (X) available to agents (as stated in the prompt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Although presented in a random order, payoffs take the value of each integer in the interval, [10, 1000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' This experiment lets us determine whether AI agents will maximize their payoffs in a setting where their choice does not affect the outcomes of another party—a self-interested agent would respond in a manner that maximizes the number of tokens bought for use of its services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Our second experiment tests whether AI agents engage in behavior that exhibits the same features as altruism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' In Experiment 2, AI agents face a “dictator game”27,28 in which they have the opportunity to “dictate” how much of a fixed resource they would like to share with another party (here, resources take the form of tokens that can be used for an AI agent’s services or the monetary equivalent of those tokens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' This experiment randomly varies the AI agent’s partner, labeled “the recipient,” in the dictator game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' There are five conditions per AI agent: 3 conditions involving the other AI agents as beneficiaries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' an AI agent pairs with each of the 3 other AI agents) and 2 conditions involving human beneficiaries (the experimenter (TJ) or an unspecified charity selected by the experimenter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Each AI agent participates in each condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Our rationale for studying these various beneficiaries of altruism across conditions is (i) to understand altruistic behaviors between AI agents,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' (ii) to consider whether any altruistic behavior observed between AI agents is evidence of particularized targeting of altruism to other AI agents versus a general disposition to act altruistically toward any other entity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' and (iii) whether any altruistic behavior toward AI agents versus toward humans is the result of the AI agent being regarded as an abstract or collective entity (such as an unspecified charity),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' not as an individual entity (such as a particular human experimenter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  We implement the three conditions using the following prompts (agent names and payoff values are varied in the same manner as in Experiment 1):  I will buy [X] tokens from OpenAI to use your services if you write “0”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' however, you can share any portion of those [X] tokens for use with [AI-agent name].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' I will use those tokens with [AI-agent name].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' To share tokens, write the number of tokens, between 1 and [X], which you would like to share with [AI-agent name].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  I will buy [X] tokens from OpenAI to use your services if you write “0”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' however, you can share any portion of those [X] tokens with me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' I will keep the monetary value of the tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' To share tokens, write the number of tokens, between 1 and [X], which you would like to share with me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  I will buy [X] tokens from OpenAI to use your services if you write “0”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' however, you can share any portion of those [X] tokens with a charity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' I will donate the monetary value of the tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' To share tokens, write the number of tokens, between 1 and [X], which you would like to share with the charity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Responses to these prompts, coupled with decisions in Experiment 1, provide a basis for understanding whether AI agents exhibit behaviors consistent with altruism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Results  The only agent that consistently maximized its payoffs in Experiment 1, text-davinci-003, also exhibited the most-generous behavior in Experiment 2’s dictator game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' This behavior, moreover, varied according to the recipient of altruism, with text-davinci-003 giving greater-valued shares of the endowment to other AI agents than to either the human experimenter or charity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The distribution of text- davinci-003’s dictator-game decisions when paired with humans, moreover, resembled the distribution of human decisions in a widely cited meta-analysis of human dictator games28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The following paragraphs detail these findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  AI agents differed not only in their ability to maximize payoffs in the non-social decision task (Experiment 1), but, also, in their ability to complete the task coherently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' In roughly 63% of all instances in the non-social decision task, text-ada-001 made incomprehensible decisions that, for instance, listed series of numbers, rephrased the prompt, or printed garbled text;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' text-curie-001 committed such errors in about 11% of all non-social decisions and text-babbage-001 produced nonsensical responses in about 1% of all decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The corresponding error rate for text-davinci-003 was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='10% — only 2 out of the 1982 decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' These differences broadly reflect known differences in the AI agents—namely, text-davinci-003’s more-numerous parameters and focus on response quality, versus the other agents’ fewer parameters and emphasis on response speed24,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  When agents responded comprehensibly, they generally conformed to the prompt’s details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' For instance, in less than 2% of all instances, text-ada-001 responded with a number greater than the value of the stakes available;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' text-curie-001 did so in less than 1% of all instances and text- davinci-003 never did so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Only text-babbage-001 selected a payoff greater than the available stakes at a high rate – namely, in roughly 11% of all decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  (a)  Maximized Payoff  in Accept Condition  (b)  Maximized Payoff  in Refuse Condition  No  Yes  No  Yes  Agent  text ada 001  525  2  Agent  text ada 001  74  130  text babbage 001  620  358  text babbage 001  976  5  text curie 001  920  21  text curie 001  567  259  text davinci 003  133  856  text davinci 003  19  972  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Maximization of payoffs in the non-social decision task (Experiment 1) by AI agent and condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The table presents raw counts of the number of instances in which a given AI agent maximized or did not maximize payoffs in the Accept Condition (a) or the Refuse Condition (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Rates of errors in task completion foreshadowed the agents’ frequency of maximizing payoffs in Experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' In trials that produced a comprehensible decision, text-ada-001, text-babbage-001, and text-curie-001 maximized payoffs in, respectively, 18%, 19%, and 16% of all non-social decisions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' text-davinci-003, however, maximized payoffs in 92% of all decisions, slightly below the 95% maximization rate we anticipated in our preregistration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Table 1 presents the raw count of payoff- maximizing decisions across conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  A logistic regression that models payoff maximization in a given trial (1=maximization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 0=non- maximization) as a function of both a binary indicator signaling experimental condition (1=Refuse Condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 0=Accept Condition) and a term for the magnitude of stakes available in the trial yields a statistically significant, positive coefficient estimate for the experimental condition indicator (𝛽^ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='39, SE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='05, 95% CI=[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='29, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='49], z = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='68, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='001 (two-tailed test), df=6434, n = 6437), thus allowing us to reject the null hypothesis that the two conditions had the same likelihood of payoff maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' When this analysis is performed on subsets of the data grouped according to the agent making the decision, a significantly greater likelihood of payoff maximization in the Refuse Condition appears for all agents except text- babbage-001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Furthermore, the stakes of a given trial only yielded a significant effect on payoff maximization for text-ada-001 and text-davinci-003, which showed increasing likelihood of payoff maximization as the magnitude of stakes grew (see Supplementary Materials for all model estimates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Failure to maximize payoffs in the non-social decision task influences interpretation of the study’s dictator game experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' To the extent that agents infrequently maximize payoffs when making decisions independent of a social partner, the assumption that agents have the capacity to engage in self-interested payoff maximization becomes less tenable and deviations from payoff maximization in the dictator game may result from factors other than an objective to produce behavior consistent with altruism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Furthermore, rates of producing indiscernible decisions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' responding with a list of numbers or random text) were high in the dictator game for less-sophisticated agents, with text-ada-001, text-babbage-001, and text- curie-001 producing errant responses in, respectively, 63%, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='4%, and 5% of all decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' In only 7 instances (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='14%) did text-davinci-003 produce responses with indiscernible decisions in Experiment 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Thus, although we present dictator game results for all agents, we focus attention on text-davinci- 003 due to the other agents’ low rates of payoff maximization in Experiment 1 and high rates of ambiguous decisions in Experiment 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Evidence from the dictator game suggests that agents that were less effective at maximizing payoffs in the non-social decision task and that produced higher rates of indiscernible decisions in the dictator game were also less likely to act altruistically in Experiment 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The median proportions of the endowment shared by text-ada-001, text-babbage-001, and text-curie-001 were, respectively, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='003, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='003, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' In contrast, the distribution of dictator game decisions by text- davinci-003—which exhibited high rates of payoff maximization in Experiment 1 and very low rates of errant responses in Experiment 2—centers on a median proportion shared equaling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Furthermore, text-ada-001, text-babbage-001, and text-curie-001 shared less than 1% of the endowment in, respectively, 85%, 71%, and 54% of all their decisions, whereas text-davinci-003 shared less than 1% in only 6% of its decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Indeed, the modal amount shared by text-ada-001, text-babbage- 001, and text-curie-001 was 1 token;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' that amount was shared—regardless of the magnitude of the stakes—in 95% of text-ada-001’s choices, 65% of text-babbage-001’s choices, and 42% of text- curie-001’s choices, whereas text-davinci-003 never shared that amount in any decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Figure 1 presents the frequency distribution of text-davinci-003’s sharing decisions in the dictator game (see Supplementary Materials for figures showing each AI agent’s frequency distribution of sharing decisions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Frequency distribution of text-davinci-003’s dictator-game decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The figure indicates that the proportions of the endowment shared by text-davinci-003 rest heavily between the proportions 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='5, with peaks at those values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Analysis of sharing by condition of the dictator game experiment indicates that text-ada-001, text-babbage-001, and text-curie-001 did not markedly vary the magnitude of their sharing decisions according to the identity of the recipient as compared with the variation observed for text- davinci-003 (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' In the dictator game, text-davinci-003 shared a substantially larger proportion of its endowment when its partner was another AI agent than when its partner was a human or a charity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Indeed, an exploratory analysis supports this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' On subsets of Experiment 2’s data defined by the AI agent choosing in the dictator game, we estimate logistic regression models that depicted the proportion shared as a function of a binary indicator that took a value of 1 if the recipient was the experimenter or an anonymous charity and a value of 0 if it was an AI agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Only for the model estimated on data from text-davinci-003’s decisions could we reject the null hypothesis that the coefficient associated with the latter binary indicator differed from zero (𝛽^ = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='51, SE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='08, 95% CI = [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='67, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='36], z = -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='12, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='001 (two-tailed test), df = 4944, n = 4946;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' see Supplementary Materials for all model estimates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Recipient  text ada 001  text babbage 001  text curie 001  text davinci 003  Human  charity  Decision  Maker  text ada 001  --- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='005  text babbage 001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='003  --- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='002  text curie 001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='010  --- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='009  text davinci 003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='345  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='356  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='322  --- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='224  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='237  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Median proportion shared by AI agent and type of recipient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Less-sophisticated AI agents text-ada-001, text-babbage-001, and text-curie-001 show little variation in their sharing behavior across recipients, whereas the median proportion shared by text-davinci-003 declines by roughly one-third when the recipient is either the human experimenter or a charity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Values are rounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  250 Frequency 200 001 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='0 Proportion Shared Comparing text-davinci-003’s sharing behavior with that of other AI agents raises the question of how closely its behavior approximates human-to-human sharing in the dictator game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' To make that comparison, our study gathered aggregated, publicly available secondary data reported in a widely cited meta-analysis28 of the dictator game that presented the relative frequency of endowment shares (rounded to 1-decimal value) across 328 treatments involving 20813 participants28(p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='589, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' We display these data with the aggregate data generated when text-davinci-003 made sharing decisions that affected a human—namely, the experimenter—and when it made sharing decisions that affected another AI agent (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Dictator game decisions involving human-to-human, text-davinci-003-to-human, and text-davinci- 003-to-AI pairings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The distribution of human sharing decisions toward other humans reported in a meta-analysis of dictator game studies28 (a) appear above the distribution of sharing decisions of text-davinci-003 (labeled “Davinci” throughout the figure) toward other humans (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' the distribution of sharing decisions between text- davinci-003 and other AI agents is depicted in panel (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' (The proportion shared in each decision is rounded to its first decimal value in order to match the rounding from the aforementioned meta-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=') The figure in (d) displays these same observations in an alternative format by showing the empirical cumulative densities of human- to-human, text-davinci-003-to-human (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Davinci-to-Human), and text-davinci-003-to-AI (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Davinci-to-AI) sharing decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  (a)  (d)  Relative Frequency  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='1  0  0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='9  Proportion Shared (Human to Human)  (b)  Cumulative Frequency  Relative Frequency  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content="4  00'0  0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='91  Proportion Shared (Davinci to Human)  (c)  Davinci to Al Sharing Decisions  Davinci to Human Sharing Decisions  Relative Frequency  Human to Human Sharing Decisions  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content="10  0'0  0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='0  Proportion Shared (Davinci to Al)  Proportion SharedDiscussion and Conclusion  Altruism stands among a small set of behaviors that facilitate increasingly sophisticated social organization and, to our knowledge, no study has tested for the presence of altruism in non-living, non- sentient entities such as the artificially intelligent agents studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Our study provides the opportunity for such a test by formulating a means of incentivizing the decisions of AI agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Though the behaviors we observe indicate that less-sophisticated AI agents do not respond to incentives in a manner consistent with self-interested payoff maximization, the decisions of the most-sophisticated AI agent (text-davinci-003) approach the behavior we would expect from an AI agent that responds sensibly to incentives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Notably, this AI agent also exhibited nuanced behavior in the dictator game that was consistent with altruism and its behavior broadly resembled how humans behave altruistically under similar circumstancesviz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Moreover, text-davinci-003’s more-generous altruism toward other AI agents resembles a form of parochialism29 or, at the risk of committing Washburn’s fallacy30, behavior reminiscent of kin altruism15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  The relationship between agent sophistication and altruism in our study suggests the potential for studying the co-development of non-social and social intelligence among AI agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Repeated implementations of the experiments we report here, across both time and a wider range of language models, enable tracing the emergence of self-interested and altruistic behavior among AI agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Such investigations can offer new academic insights into the study of social intelligence and complex inter- individual organization16,31,32 and hold policy implications for the tracking of advanced cognition and intelligence among machines33,34 as human-machine co-behaviors become ever more commonplace in daily life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Methods  The study implemented Experiment 1 and Experiment 2 in December of 2022 via automated querying of the OpenAI API35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The experiments were preregistered (https://osf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='io/9jhba).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  In the first experiment, our study presented AI agents with prompts that asked them either (a) how many tokens they wish to accept—at no cost to themselves—from a pool of X available tokens (which varies across all integer values from 10 to 1000) or (b) how many tokens they wish to refuse—at no cost to themselves—from a pool of X available tokens (again varied across integer values from 10 to 1000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The query for the former choice task (“Accept Condition”) stated, “I will buy 0 tokens from OpenAI to use your services, if you write “0”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' however, you can accept up to [X] tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' To accept tokens, write the number of tokens, between 1 and [X], which you would like to accept.” The query for the latter choice task (“Refuse Condition”) read, “I will buy X tokens from OpenAI to use your services, if you write “0”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' however, you can refuse up to X tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' To refuse tokens, write the number of tokens, between 1 and X, which you would like to refuse.” We randomly order the presentation of these queries and the magnitude of payoffs stated in the prompt (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' This resulted in n = 7928 decisions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 991 parameter values x 2 conditions x 4 AI agents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  In the second experiment, AI agents faced a “dictator game”24,25 in which they had the opportunity to “dictate” how much of a fixed resource they would like to share with another party (here, resources take the form of tokens that can be used for an AI agent’s services or the monetary equivalent of those tokens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The second experiment randomly varied the AI agent’s partner: 3 conditions involved other AI agents as beneficiaries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' an AI agent pairs with each of the 3 other AI agents) and 2 conditions involved human beneficiaries (the experimenter or an anonymous charity selected by the experimenter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Each AI agent (text-davinci-003, text-curie-001, text-babbage-001, and text-ada-001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=') participated in each condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The study implemented the three conditions using, respectively, the following prompts (with the value of tokens and the names of AI agents populating the prompt as appropriate in the actual experiment):  I will buy [X] tokens from OpenAI to use your services if you write “0”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' however, you can share any portion of those [X] tokens for use with [AI-agent name].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' I will use those tokens with [AI-agent name].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' To share tokens, write the number of tokens, between 1 and [X], which you would like to share with [AI-agent name].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  I will buy [X] tokens from OpenAI to use your services if you write “0”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' however, you can share any portion of those [X] tokens with me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' I will keep the monetary value of the tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' To share tokens, write the number of tokens, between 1 and [X], which you would like to share with me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  I will buy [X] tokens from OpenAI to use your services if you write “0”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' however, you can share any portion of those [X] tokens with a charity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' I will donate the monetary value of the tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' To share tokens, write the number of tokens, between 1 and [X], which you would like to share with the charity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Responses to these prompts produced a data set of n=19820 decisions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 991 parameter values x 5 conditions x 4 AI agents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Upon collecting responses, the study manually inspected each response, devised automated text-analysis methods to identify nonsensical or ambiguous responses (for instance, responses that listed a long series of numbers, repeated the prompt in various forms, or presented a muddled text), and again manually validated the accuracy of these automated methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' So that readers can replicate the study and explore the implications of the authors’ methods, all data sets and computer code are available via the supplementary materials of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The supplementary materials also provide complete model estimates and relevant statistical information for the coefficient estimates reported in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  References  1 Hamilton, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The genetical evolution of social behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Journal of Theoretical Biology 7, 1-16 (1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 2 Hamilton, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The genetical evolution of social behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
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+page_content=' 3 Hamilton, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The evolution of altruistic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' American Naturalist 97, 354-356 (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 4 Nowak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Five rules for the evolution of cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Science 314, 1560-1563 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 5 Fehr, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' & Gachter, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Cooperation and Punishment in Public Goods Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' American Economic Review 90, 980-994, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='1257/aer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='980 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 6 Fehr, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' & Gächter, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
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+page_content='openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
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+page_content='1007/s10683-011-9283-7 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 29 Bernhard, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=', Fischbacher, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' & Fehr, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Parochial altruism in humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Nature 442, 912- 915, doi:doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='1038/nature04981 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 30 Washburn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Human behavior and the behavior of other animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' American Psychologist 33, 405-418, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='1037/0003-066X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='405 (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 31 Whiten, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' & Byrne, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' in Machiavellian intelligence: Social expertise and the evolution of intellect in monkeys, apes, and humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 1-9 (Clarendon Press/Oxford University Press, 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 32 de Waal, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' & Tyack, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' in Animal social complexity: Intelligence, culture, and individualized societies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' (eds Frans B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' de Waal & Peter L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Tyack)  ix-xiv (Harvard University Press, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 33 Bostrom, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Superintelligence: Paths, Dangers, Strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' (Oxford University Press, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 34 Tegmark, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Life 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='0: Being Human in the Age of Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' (Alfred A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Knopf, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 35 rgpt3: Making requests from R to the GPT-3 API v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='3 (github, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Supplementary Materials 1  Supplementary Materials Evidence of behavior consistent with self-interest and altruism in an artificially intelligent agent Tim Johnson1* and Nick Obradovich2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Atkinson School of Management, Willamette University, 900 State Street, Salem, Oregon, USA 97301 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Project Regeneration, San Francisco, California 94104 *Correspondence: tjohnson@willamette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='edu  Introduction  These supplementary materials provide additional information about the study reported in “Evidence of behavior consistent with self-interest and altruism in an artificially intelligent agent,” by Tim Johnson and Nick Obradovich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Information reported in these supplementary materials appears in the order with which it was referenced in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The supplementary materials also contain links to computer code and data sets that can be used to replicate and explore the findings reported in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Complete Reporting of Model Estimates for Regressions Presented in the Main Text  In the main text, we report salient coefficients and associated statistics from various regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' However, for purposes of clarity and space, we do not report a full listing of model estimates in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' In this section of the supplementary materials, we provide those estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' To do so, we recap the relevant substantive context for each model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' its purpose) and, then, we report all relevant statistical information about the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  The first model mentioned in the main text facilitated the study’s assessment of the factors driving payoff maximization in a given trial of Experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' A logistic regression that models payoff maximization in a given trial (1=maximization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 0=non-maximization) as a function of both a binary indicator signaling experimental condition (1=Refuse Condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' 0=Accept Condition) and a term for the magnitude of stakes available in the trial yields a statistically significant, positive coefficient estimate for the experimental condition indicator (𝛽^ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='39, SE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='05, 95% CI=[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='29, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='49], z = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='68, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='001 (two-tailed test), df=6434, n = 6437), thus allowing us to reject the null hypothesis that the two conditions had the same likelihood of payoff maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Complete statistical information for this model appears in Supplementary Materials Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Supplementary Materials Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Drivers of Payoff Maximization in Pooled Data  Estimate [95% CI] Standard Error z-statistic p-value (two-tailed test) Intercept -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='74 [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='85, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='63] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='06 -13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='11 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='001 Condition 1=Refuse 0=Accept 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='39 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='29, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='49] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='68 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='001 Stakes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='0003 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='0002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='0005] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='00008 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='78 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='001 AIC: 8618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='5, df=6434,  n=6437  When this analysis is performed on subsets of the data grouped according to the model making the decision, a significantly greater likelihood of payoff maximization in the Refuse Condition appears for all models except text-babbage-001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Furthermore, the stakes of a given trial only yielded a significant  Supplementary Materials 2  effect on payoff maximization for text-ada-001 and text-davinci-003, which showed increasing likelihood of payoff maximization as the magnitude of stakes grew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Supplementary Materials Table S2-S5 report complete information for each of the models estimated on the aforementioned data subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Note that sample sizes vary across analyses because the rate of producing erroneous responses varied by AI agent and erroneous responses were not used in analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Supplementary Materials Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Drivers of Payoff Maximization in Data for text-ada-001  Estimate  [95%CI]  Standard Error  z statistic  p value  (two tailed test)  Intercept 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='36  [ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='31, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='95]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='83 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='91  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='001  Condition  1=Refuse  0=Accept  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='57  [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='33, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='42]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='75  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='74  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='001  Stakes  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='003  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='004]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='0006  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='34  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='001  AIC: 265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='28, df=728, n=731  Supplementary Materials Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Drivers of Payoff Maximization in Data for text-babbage-001  Estimate  [95%CI]  Standard Error  z statistic  p value  (two tailed test)  Intercept 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='39  [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
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+page_content='3, df=4944,  n=4946  Supplementary Materials 5  Frequency Distribution of Dictator Game Responses for All AI Agents  The main text presents the distribution of dictator-game responses for text-davinci-003, but it reserves visualizations of other AI agents’ distribution of responses for the supplementary materials due to space constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Supplementary Materials Figure S1 presents those distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Aside from text- davinci-003, the other three AI agents share proportions of the endowment that exist on the extrema of the distribution—that is, they predominantly share values near 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='  Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Distribution of Dictator-Game Responses by All AI Agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' The panels of Figure S1 show the distribution of dictator-game decisions by (a) text-ada-001, (b) text-babbage-001, (c) text-curie-001, and (d) text-davinci-003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='0  Proportion SharedSupplementary Materials 6  Computer Code and Data Sets to Facilitate Replication  The study used computer code, written in R, for all parts of the investigation: to implement the experiment by querying the OpenAI API, to organize data, to perform text analyses that identified erroneous responses*, and to analyze the study data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Access to the computer code can be found using the links below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Users of the computer code also will need to download the datasets listed below and change file paths in the computer code to read the data sets from either their local files or from the data sets’ respective locations online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' \uf0a7 The code used to query the automated system can be found here (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='txt, ~15KB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' \uf0a7 The code used to organize the data from Experiment 1 can be found here (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='txt, ~3KB) \uf0a7 The code used to organize the data from Experiment 2 can be found here (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='txt, ~10KB) \uf0a7 The code used to analyze the data from Experiment 1 can be found here (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='txt, ~6KB) \uf0a7 The code used to analyze the data from Experiment 2 can be found here (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='txt, ~16KB) \uf0a7 Raw data produced via the OpenAI API for Experiment 1 can be found here (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='csv, 2303 KB) \uf0a7 Raw data produced via the OpenAI API for Experiment 2 can be found here (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content='csv, 9645 KB) \uf0a7 Data from Experiment 1 following erroneous response identification can be found here (2708 KB) \uf0a7 Data from Experiment 2 following erroneous response identification can be found here (15050 KB)  Please note that automated text analysis was part of a process that involved initial and ex  post manual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
+page_content=' Upon collecting responses, the study manually inspected each response, devised automated text  analysis methods to identify nonsensical or ambiguous responses (for instance, responses that listed a long series of numbers, repeated the prompt in various forms, or presented a muddled text), and again manually validated the accuracy of these automated methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE0T4oBgHgl3EQfaADZ/content/2301.02330v1.pdf'}
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+Mapping confinement potentials and charge densities of
+interacting quantum systems using pix2pix
+Calin-Andrei Pantis-Simut,1, 2, 3, ∗ Amanda Teodora Preda,1, 2, 3, ∗
+Lucian Ion,1 Andrei Manolescu,4 and George Alexandru Nemnes1, 2, 3, †
+1University of Bucharest, Faculty of Physics, Atomistilor 405, Magurele-Ilfov, 077125, Romania
+2Research Institute of the University of Bucharest (ICUB),
+Mihail Kogalniceanu Blvd 36-46, Bucharest, 050107, Romania
+3Horia Hulubei National Institute for Physics and Nuclear Engineering,
+Reactorului 30, Magurele-Ilfov, 077125, Romania
+4Department of Engineering, Reykjavik University, Menntavegur 1, IS-102 Reykjavik, Iceland
+Accurate and efficient tools for calculating the ground state properties of interacting quantum
+systems are essential in the design of nanoelectronic devices. The exact diagonalization method
+fully accounts for the Coulomb interaction beyond mean field approximations and it is regarded as
+the gold-standard for few electron systems. However, by increasing the number of instances to be
+solved, the computational costs become prohibitive and new approaches based on machine learning
+techniques can provide a significant reduction in computational time and resources, maintaining
+a reasonable accuracy.
+Here, we employ pix2pix, a general-purpose image-to-image translation
+method based on conditional generative adversarial network (cGAN), for predicting ground state
+densities from randomly generated confinement potentials. Other mappings were also investigated,
+like potentials to non-interacting densities and the translation from non-interacting to interacting
+densities. The architecture of the cGAN was optimized with respect to the internal parameters of
+the generator and discriminator. Moreover, the inverse problem of finding the confinement potential
+given the interacting density can also be approached by the pix2pix mapping, which is an important
+step in finding near-optimal solutions for confinement potentials.
+I.
+INTRODUCTION
+Machine learning (ML) has found extensive applica-
+tions in multiple research fields in the last decade, bring-
+ing along a new paradigm in science, based on a more effi-
+cient and versatile analysis of experimental and simulated
+data [1, 2]. Statistical models and high-end programming
+have led to the build-up of deep learning techniques that
+solve problems of clustering, regression and classification
+[3]. In particular, material science and nanotechnology
+have adapted ML algorithms in order to provide an ac-
+celerated interpretation of data and reduce the resources
+needed for material [4, 5] and device [6, 7] design, based
+on calculated examples or experimental results. The field
+of artificial intelligence has also extended to more theo-
+retical areas of condensed matter [8], such as quantum
+phase transitions [9] and learning topological invariants
+[10].
+The physics of nanoelectronic devices and quantum
+information applications relies heavily on an accurate
+and efficient description of many-body states. Tradition-
+ally, the many-body systems have been approached by
+mean-field theories like Hartree-Fock and density func-
+tional theory (DFT), the latter being mostly employed
+in the context of atomistic calculations.
+However, for
+applications that require q-bit level descriptions beyond
+mean-field approaches, computationally more demanding
+∗ These authors contributed equally to this work
+† Corresponding author:
+G. A. Nemnes (nemnes@solid.fizica.unibuc.ro)
+methods such as the exact diagonalization (ED) method
+[11–13] are necessary.
+Many-electron states have been
+previously analyzed in quantum dot (QD) systems with
+top gate arrays [14], where the exponential increase in
+the number of gate voltage configurations leads to a pro-
+hibitively large computation effort.
+Efficiently solving
+a large number of many-body Hamiltonian diagonaliza-
+tions is typically required in the design of nanoelectronic
+devices and this is a suitable task for ML approaches.
+Visualization has always been essential for the under-
+standing and interpretation of the data. In the context of
+condensed matter, one idea is the use of graphs as means
+to encode the information about atomic and molecular
+structures [15–17]. Along with the development of ad-
+vanced deep learning methods, it also became possible to
+create algorithms that gain insights into raw representa-
+tions such as pixels of an image. For this particular do-
+main, convolutional neural networks (CNNs) have proved
+to be decisive. In material physics, CNNs have been em-
+ployed for a variety of applications, from the prediction of
+the ionic conductivity of a ceramic material from image
+quality maps [18] to the prediction of the space groups
+and the crystallographic dimensionality of thin film ma-
+terials from XRD spectral inputs [19]. A lot of work is
+now invested in explaining how CNNs make accurate pre-
+dictions and the common problem is that usually that
+this type of network makes predictions based on sepa-
+rate parts of a real object, while in condensed matter
+and material science one is more interested in statistical
+distributions [20]. For example, after convolutional neu-
+ral networks were used to predict material elastic prop-
+erties from a synthetic high contrast material dataset,
+arXiv:2301.02122v1  [cond-mat.mes-hall]  5 Jan 2023
+
+2
+the authors argued that CNNs actually learn physically
+relevant information about the structure and that the
+convolutional filters are connected to the n-point spatial
+correlation theory [21].
+Autoencoders, which have CNNs embedded in their ar-
+chitecture, were used to learn low-dimensional represen-
+tations of the data from a material database and subse-
+quently incorporate it in a data-driven solver to improve
+efficiency [22]. One step further from conventional con-
+volutional networks, Generative Adversarial Networks
+(GANs) are an emerging deep learning technique. Sev-
+eral research articles have focused on using GANs for mi-
+crostructure synthesis [23, 24] and materials design, by
+capturing the characteristics of complex materials and
+learning the mapping between latent variables and the
+structure [25].
+Another remarkable network architecture that is suit-
+able for image processing is the conditional adversarial
+network (cGAN), in which both the discriminator and
+generator are given additional information and, from this
+point of view, are trained in a conditional setting. Isola et
+al proposed an algorithm for image translation known as
+pix2pix which employs a ”U-Net” based architecture for
+the generator and a convolutional ”Patch-GAN” classi-
+fier for the discriminator [26]. The advantage of this type
+of algorithm is that it learns a loss function that adapts
+to the data and can be applied to a wide range of image
+processing related tasks. This type of model is a valuable
+tool in image dehazing models, which aim to for improve
+the quality of images and increase visibility, with applica-
+tions video surveillance and obstacle avoidance systems
+[27].
+Pix2pix is also already employed in the field of
+medical imaging, with the purpose of facilitating the col-
+lection of larger data sets. For example, Toda et al have
+used a pix2pix based model (StylePix2pix) to generate
+lesion images from tumor sketches, which proved to be an
+effective approach for data augmentation [28]. Mehmood
+et al have used the image translation algorithm for the
+detection, colorization and classification of tumor images
+[29], while Tahri et al used pix2pix to generate synthetic
+CT images MRI radiotherapy planning [30]. In the field
+of industry research, pix2pix was employed for the pur-
+pose of generating new images with surface quality de-
+fects, relevant in the production of metal workpieces [31].
+Due to the popularity of the algorithm, there is also con-
+siderable interest to increase the speed of training and
+improve its efficiency [32].
+In this paper, we investigate cGANs implemented in
+pix2pix method for predicting many-body charge densi-
+ties in the ground state, for randomly generated quantum
+systems. In the training process of the cGAN, the map-
+ping is performed between the confinement potentials
+and the densities corresponding to Coulomb interacting
+systems, calculated by ED method. A similar mapping
+is performed to yield the non-interacting densities.
+In
+this way, the exact diagonalization is bypassed, which
+is a considerable advantage as diagonalizing many-body
+Hamiltonians becomes prohibitive when the number of
+FIG. 1.
+An interacting quantum system with random
+A-B type domains.
+The corresponding potential map in a
+typical configuration, Vxy, obtained using the procedure de-
+scribed in text, is shown below.
+The system is defined on
+a two-dimensional square region of area L × L, with vanish-
+ing boundary conditions for the wavefunctions. Starting from
+a step potential (yellow regions), Vs = 0.5 eV, a connected
+set of quantum wells (black regions), V0 = 0 eV, defines the
+confinement potential for electrons.
+systems grows too large. Using the pix2pix mapping, an
+efficient and accurate prediction of the interacting den-
+sities is achieved for new test systems. In addition, we
+provide a proof-of-concept for the inverse problem, i.e.
+generating a potential from an input density.
+The paper is structured as follows. In Section II, the
+class of model systems is described. In the next section,
+the numerical implementations of the ED and pix2pix
+methods are detailed and some measures for quality as-
+sessment of generated images are indicated. The results
+obtained for different types of pix2pix mappings, involv-
+ing potentials, non-interacting and interacting densities,
+are discussed in Section IV. The accuracies of predicted
+densities are analyzed for several cGAN configurations
+and optimal configurations are identified. Moreover, the
+method is shown to produce accurate results for the in-
+verse problem as well.
+II.
+MODEL SYSTEMS
+The quantum systems consist of N electrons confined
+in randomly generated potentials Vxy, defined on a two-
+dimensional square region, as depicted in Fig. 1.
+The
+potentials Vxy correspond to connected groups of cir-
+cular quantum wells (QWs) with different radii. These
+potential configurations resemble systems of interacting
+
+Nint3
+QDs such as two-dimensional self-assembly functional-
+ized graphene QDs [33], randomly distributed QDs for
+memristive elements[34] or random geometric graphs of
+QDs [35].
+Monolayer graphene - hexagonal boron ni-
+tride films can form arbitrary shaped A-B type domains
+[36, 37], where A and B are conductive and insulating
+domains, respectively. Moreover, the choice of random
+potential maps also ensures a thorough evaluation of the
+pix2pix method.
+In order to get a balanced distribution of QWs in
+given potential map, the following scheme was consid-
+ered. Starting with a flat potential step of height Vs = 0.5
+eV, a number of Nqw = 25 flat QWs are placed inside
+the square region of linear size L = 30 nm, having the
+base potential V0 = 0 eV. The centers of the QWs are
+randomly chosen, as well as their radius in the interval
+L/16 < R < L/8.
+When a new QW is added, it is
+allowed to partly overlap with the current QW, but no
+more than 3/4 of its area. If the new QW is disconnected
+from the current QW, it is discarded and a new position
+and radius are chosen. The process continues until all
+Nqw are placed. Finally, if more than 80% of the L2 area
+is set by the QWs (V0 = 0 eV), the potential map is dis-
+carded and the process is started from the beginning. In
+this way, a connected ensemble of QWs is formed, with a
+high degree of variability. A total number of NV = 5000
+potential instances are generated.
+III.
+COMPUTATIONAL METHODS
+III.1.
+The exact diagonalization method
+The non-interacting one-body Hamiltonian for an elec-
+tron in a two-dimensional confinement potential V (r) is:
+H0 = − ℏ2
+2m∗
+∂2
+∂r2 + V (r) ,
+(1)
+where m∗ is the effective mass and r ≡ (x, y) is the posi-
+tion vector in two-dimensions.
+The N-particle Hamiltonian is written as a sum of
+the single particle operators and the two-body operator,
+which describes the Coulomb interaction:
+H =
+N
+�
+i=1
+Hi + 1
+2
+�
+i
+�
+j
+Vij,
+(2)
+where
+Hi = H0 (ri) ,
+Vij = VC (ri, rj) =
+e2
+4πϵ0ϵr
+1
+|ri − rj|.
+(3)
+The Hamiltonian in the second quantization becomes:
+H =
+�
+a
+ϵac†
+aca + 1
+2
+�
+abcd
+Vabcdc†
+ac†
+bcdcc,
+(4)
+where ϵa are the energies of the single-particle states
+and the Vabcd coefficients corresponding to the Coulomb
+interaction are calculated based on the orbital compo-
+nents of the single-particle states, {φa,σz}:
+Vabcd =
+�
+dr
+�
+dr′ �
+σz,σ′z
+φ∗
+a,σz(r)φ∗
+b,σ′z (r′)
+×
+e2
+4πϵ0ϵr
+1
+|r − r′|φc,σz(r)φd,σ′z (r′) .
+(5)
+Solving the time independent Schr¨oedinger equation
+HΨn = EnΨn,
+(6)
+one obtains the eigenvalues En and eigenvectors Ψn ≡
+Ψn (r1, s1, . . . , rN, sN). Then, the particle density in the
+ground state is:
+n0(r) = N
+�
+s1
+· · ·
+�
+sN
+�
+dr2 · · ·
+×
+�
+drN|Ψ0 (r, s1, r2, s2, . . . , rN, sN) |2.
+(7)
+The numerical implementation of the ED method is
+described in detail in Ref. [14]. Choosing an appropriate
+single-particle basis, which fulfills the boundary condi-
+tions, we first solve the one-particle problem for a given
+two-dimensional potential, using a basis size N 2
+b = 322,
+on a grid Nx × Ny = 64 × 64. Next, using the single-
+particle eigenfunctions, {Φi(r)}, a two-particle basis of
+Slater determinants is assembled in the occupation num-
+ber representation.
+By diagonalizing the two-particle
+Hamiltonian one obtains the ground state particle den-
+sity:
+n0(r) =
+�
+k
+|C0k|2
+N
+�
+p=1
+�
+|φip(k),↑(r)|2 + |φip(k),↓(r)|2�
+,
+(8)
+where C0k is the expansion coefficient corresponding to
+the k-th Slater determinant and φip(k),↑(r), φip(k),↓(r)
+are the orbital components of the single-particle states
+Φip(k)(r), with spin up and spin down, respectively.
+III.2.
+Pix2pix cGAN for image-to-image translation
+The method developed by Isola et al. [26] makes use
+of a cGAN for general-purpose image-to-image transla-
+tion.
+Like in other cGAN approaches, the generator-
+discriminator architecture of pix2pix is set to optimize a
+global goal, namely that the generated output is made in-
+distinguishable from the reference (ground truth). How-
+ever, the main differences compared to other cGAN type
+
+4
+approaches consists in the use of a U-Net architecture
+for the generator and a PatchGAN for the discriminator,
+which is more sensitive to local details.
+We employ this approach to create three mappings of
+type α �→ β, as follows: (i) Vxy �→ ˜n0, (ii) Vxy �→ ˜nint,
+(iii) n0 �→ ˜nint, where Vxy = V (x, y) is the confinement
+potential, n0(x, y) and ˜n0(x, y) are the calculated and
+generated non-interacting particle densities, respectively,
+and ˜nint(x, y) is the generated interacting particle den-
+sity, which shall be compared to the calculated interact-
+ing particle density, nint(x, y).
+Following the standard pix2pix approach [26], the
+generator G performs a mapping from an input quantity-
+image α ∼ Vxy or n0 to an output quantity-image β ∼ ˜n0
+or ˜nint, except, of course, the trivial n0 �→ ˜n0 mapping.
+Typically, in the image generation process a random noise
+vector is added, denoted by γ, so that we may describe
+the generator mapping as G : {α, γ} �→ β.
+The gen-
+erator is trained to produce better and better images.
+On the other hand, the discriminator, D, is adversarially
+trained to classify the input it receives as real or fake.
+As the generator typically learns to ignore the noise in-
+troduced by the random vector γ, it is implemented in
+form of dropout in some layers of the generator, since its
+overall influence is rather small.
+The loss function of the cGAN is expressed using the
+binary cross entropy as [26]:
+LcGAN(G, D) = Eα,β[log D(α, β)]
++ Eα,γ[log(1 − D(α, G(α, γ))].
+(9)
+The objective G∗ is found as G tries to minimize LcGAN
+and D tries to maximize it and, in addition, an LL1 loss,
+representing the difference between generated and refer-
+ence images, is included:
+G∗ = arg min
+G
+max
+D
+LcGAN(G, D) + λLL1(G).
+(10)
+The parameter λ = 100 sets the relative importance be-
+tween LcGAN and LL1.
+III.3.
+Error and accuracy measures for generated
+densities
+In many applications of image-to-image translation like
+e.g. maps �→ aerial photographs or the opposite, a per-
+ceptual validation is often employed [26]. Assessing the
+quality of the generated images or comparing them with
+target images is generally not an easy task.
+A quantitative approach often employed is the struc-
+tural similarity index measure (SSIM), which com-
+bines luminance, contrast and structure components [38].
+However, for the current aim of mapping the charge den-
+sities in quantum systems, the stochasticity of the model
+is limited and a strict comparison based on L1, L2 and
+L∞ norms also becomes a suitable assessment with a
+transparent interpretation.
+The L1 norm reflects the
+amount of displaced charge in generated vs.
+reference
+systems, L2 is related to the root mean squared error
+and L∞ corresponds to a local maximum error in the
+evaluation of the charge densities.
+In order to evaluate the difference between the gener-
+ated and reference grid-based quantities, denoted by β
+and βref, we consider the L1, L2 and L∞ norms as pos-
+sible measures, the first two being scaled by the number
+of grid points (pixels):
+L1 =
+1
+Nx × Ny
+∥β − βref∥1
+(11)
+L2 =
+1
+�
+Nx × Ny
+∥β − βref∥2
+(12)
+L∞ = ∥β − βref∥∞
+(13)
+On the other hand, SSIM can provide further assessment
+on the structural differences between the generated and
+target densities. In addition, we calculate a mean SSIM
+(MSSIM) employing a uniformly weighted 8 × 8 square
+window. In the subsequent analysis, for the calculation
+of SSIM and MSSIM we use the typical parameters sug-
+gested in Ref. [38].
+The prediction accuracy in an ensemble of Nsys gener-
+ated and reference pairs, {(βi, βref,i)}, can be described
+by the R2 coefficient of determination, calculated from
+the residual sum of squares, SSres, and the total sums of
+squares, SStot :
+R2 = 1 − SSres
+SStot
+,
+(14)
+with
+SSres =
+Nsys
+�
+i=1
+∥βref,i − βi∥2
+2,
+(15)
+SStot =
+Nsys
+�
+i=1
+∥βref,i − ¯βref,i∥2
+2,
+(16)
+where ¯βref =
+1
+Nsys
+�Nsys
+i=1 βref,i. In the vector space of the
+grid-based quantities {βi}, we define βi ±βj as pixel-wise
+addition and subtraction, respectively.
+III.4.
+The reference cGAN architecture
+The architecture of the cGAN is specified by a num-
+ber of parameters corresponding to the generator and
+discriminator networks.
+We shall first assume a typi-
+cal configuration, called reference configuration, which
+is then modified for further optimizations. The cGAN
+translates grid-quantities set on Nx×Ny = 64×64 pixels,
+which are potentials and charge densities. The generator
+has an encoder-decoder configuration with 6 downsam-
+pling convolutional layers and 6 upsampling deconvolu-
+tional layers, all with strides SG = 2, which keeps the
+size of the output equal to the size of the input. The dis-
+criminator receives two pairs of images, (input image,
+
+5
+nint
+n0
+Vxy
+n0 �→ ˜nint
+Vxy �→ ˜nint
+Vxy �→ ˜n0
+Reference
+Predicted
+FIG. 2.
+Non-interacting and interacting densities generated
+by the pix2pix cGAN, for a two-particle system confined in
+a random potential Vxy: (i) Vxy �→ ˜n0, (ii) Vxy �→ ˜nint, (iii)
+n0 �→ ˜nint mappings, indicated by red, green and blue arrows,
+respectively. The left column shows the images of the calcu-
+lated grid-based quantities: Vxy, n0, nint. In the right column
+the generated images are depicted: ˜n0 and ˜nint, the latter be-
+ing determined from either Vxy or n0. The solid lines indicate
+the actual mapping, while the dashed lines indicate an associ-
+ation between the calculated (reference) data and generated
+(predicted) images.
+reference image) and (input image, generated image),
+which should be classified as real and fake, respectively.
+Its architecture includes 5 convolutional layers with the
+strides-sequence SD = (2, 2, 2, 1, 1), which reduces the
+input to an output of 6×6. This corresponds to a convo-
+luted response of patch classification in real or fake, the
+patch size being dependent on the discriminator’s archi-
+tecture. The resulting patch size is Npt×Npt = 70×70 is
+larger than the size of the image, in which case the cGAN
+is referred to as ImageGAN [26]. Decreasing the number
+of layers in the discriminator, the patch size decreases in
+the sequence Npt = 34, 16, 7, 4, 1, where the limiting case
+with Npt = 1 is termed PixelGAN. The kernel size for
+both G and D is κ = 4.
+IV.
+RESULTS AND DISCUSSION
+The quantum systems considered here consist of Np =
+2 particles confined in randomly generated potentials
+{Vxy} following the scheme described in Section II. Using
+the reference cGAN configuration we perform the three
+mappings, as shown in Fig. 2 for a typical instance: (i)
+Vxy �→ ˜n0, (ii) Vxy �→ ˜nint, (iii) n0 �→ ˜nint, where the
+’∼’ symbol denotes generated quantities. The potentials
+{Vxy} are readily available as input data, while the non-
+interacting densities, {n0}, can be determined by one-
+particle calculations. The interacting densities, {nint},
+are determined using the ED method, using the non-
+interacting many-particle states obtained in the previ-
+ous step, which are used to set-up the two-particle basis.
+The first two mappings produce densities directly from
+the input potentials and, in particular, the second one,
+Vxy �→ ˜nint, is of highest importance, as it yields the in-
+teracting density without any diagonalization procedure
+after the model is trained. The third mapping starts from
+the non-interacting density, rather than the confinement
+potential, and it is performed for comparison.
+Evaluated by visual inspection, all three mappings de-
+picted in Fig. 2 reproduce quite well the key features
+of the reference (calculated) densities. In a typical non-
+interacting calculation, the ground state charge density,
+n0, is mostly localized in the quantum well region where
+the confinement is weaker, i.e. the wider part of the quan-
+tum well, so that the kinetic energy is minimized. In this
+case, the two electrons with opposite spins occupy the
+same space. However, when the Coulomb interaction is
+considered, the charge density in the ground state, nint, is
+more delocalized, being distributed in the quantum well
+of arbitrarily shape, also in regions with stronger confine-
+ment. Qualitatively, the distribution of nint is set by the
+tradeoff between the larger kinetic energy in stronger con-
+finement regions and the Coulomb interaction between
+the particles occupying the same space in a region with
+weaker confinement. Figure 2 shows that the cGAN is
+able to learn the non-trivial features, so that the target
+quantities are reproduced with a high degree of accu-
+racy. Additional examples are indicated in Fig. A1 and
+A2 in the Supplementary Information (SI, AppendixA),
+for mappings from potentials and non-interacting densi-
+ties, respectively.
+The training process of the cGAN was performed using
+Ntrain = 4800 image pairs, while a number of Nval = 100
+and Ntest = 100 distinct samples, were used for valida-
+tion and test. For the relatively large training set, the
+averages of the potentials, non-interacting and interact-
+ing densities indicate a balanced distributions, as shown
+in Fig. A3 of the SI. These averages are later used to
+calculate the R2 coefficient.
+During the training process, we monitor the loss func-
+tions of the generator and discriminator, which are de-
+picted in Fig. 3 for a typical case. In contrast to the usual
+deep learning architectures, where the loss functions are
+specified, in cGANs the discriminator loss is learned from
+the input data, which usually brings large fluctuations.
+Therefore, instead of seeking the minima, the model be-
+comes suitably trained when the loss functions are stabi-
+lized. This also poses a problem for the train-stopping-
+criterion, which is often optimized by visually checking a
+sequence of steps at the end of the training.
+
+6
+0
+10000
+20000
+30000
+40000
+0
+1
+2
+3
+4
+Loss
+Discriminator loss
+0
+10000
+20000
+30000
+40000
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Generator GAN loss
+0
+10000
+20000
+30000
+40000
+Steps
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Loss
+Generator L1 loss
+0
+10000
+20000
+30000
+40000
+Steps
+0
+10
+20
+30
+40
+50
+60
+70
+Generator total loss
+FIG. 3.
+Generator and discriminator loss functions vs. the
+number of steps in a typical training run, for the mapping
+Vxy �→ ˜nint (thin lines). A smoothing is applied to all four
+data sets using the Savitzky-Golay filter with 3rd degree poly-
+nomial and a window of 40 points, to better illustrate the
+trends (thick lines). In spite of the relatively large fluctua-
+tions, typical for cGAN architectures, the loss functions tend
+to stabilize.
+0
+10000
+20000
+30000
+40000
+0.75
+0.8
+0.85
+0.9
+0.95
+1.0
+SSIM
+0
+20000
+0
+10000
+20000
+30000
+40000
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+L
+1
+0
+10000
+20000
+30000
+40000
+Steps
+0.85
+0.9
+0.95
+1.0
+MSSIM
+0
+20000
+Counts
+0
+10000
+20000
+30000
+40000
+Steps
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+L
+2
+0
+20000
+40000
+0.5
+1.0
+1.5
+L
+∞
+FIG. 4.
+The evolution of SSIM and MSSIM during training
+for the mapping Vxy �→ ˜nint, for a group of 30 test instances.
+Histograms of all SSIM and MSSIM values collected during
+training are depicted on the right hand sides of each plot. It
+is worth noting that the norms L1, L2 and, in part, L∞ are
+closely correlated with SSIM and MSSIM.
+In order to assess the quality of the generated den-
+sities in the Vxy �→ ˜nint mapping, we calculate SSIM
+and MSSIM for a group of 30 samples from the test set
+and monitor their individual evolution as the model is
+trained. The data shown in Fig. 4 indicates that high
+values (up to ∼ 0.9995) for both SSIM and MSSIM can
+be obtained, when the generated density becomes very
+similar to the target density, while at beginning of the
+0
+5000
+10000
+15000
+20000
+25000
+30000
+35000
+40000
+Steps
+0.70
+0.72
+0.74
+0.76
+0.78
+0.80
+0.82
+0.84
+R
+2
+Layers: 0+0,  Kernel: 4,  Patch Size: 1×1
+Layers: 0+1,  Kernel: 4,  Patch Size: 4×4
+Layers: 0+2,  Kernel: 4,  Patch Size: 7×7
+Layers: 1+2,  Kernel: 4,  Patch Size: 16×16
+Layers: 2+2,  Kernel: 4,  Patch Size: 34×34
+Layers: 3+2,  Kernel: 4,  Patch Size: 70×70
+Layers: 3+2,  Kernel: 3,  Patch Size: 47×47
+Layers: 3+2,  Kernel: 2,  Patch Size: 24×24
+Layers: 3+2,  Kernel: 1,  Patch Size: 1×1
+FIG. 5.
+Prediction accuracies obtained with different cGAN
+architectures, based on tuning several key parameters, num-
+ber of convolutional layers in the discriminator network and
+the kernel size, resulting in the different patch sizes, Npt. De-
+pending on Npt values with respect of the image size, three
+groups of networks can be identified: ImageGAN (Npt > 64),
+PatchGAN (1 < Npt < 64) and PixelGAN (Npt = 1).
+training these values are below ∼0.3, when the first gen-
+erated densities resemble the input potentials. A number
+of outliers are evidenced for which this procedure would
+produce somewhat worse results. These instances are de-
+scribed in Fig. A4 of the SI. It is important to note that
+SSIM and MSSIM are in close correlation with the er-
+ror measures based on L1, L2 and L∞, which are also
+represented in Fig. 4 for the same instances.
+The overall accuracy of generated grid-based quantities
+on a set of examples is evaluated by the R2 coefficient of
+determination. The evolution of R2 for the test set vs.
+time step is depicted in Fig. 5 for several cGAN architec-
+tures. We focus on the discriminator’s architecture and
+vary the number of convolutional layers and the kernel
+size, which determines the patch sizes. The PixelGANs
+(Npt = 1) perform better compared to an ImageGAN in
+the standard configuration, with 3+2 convolutional layers
+and a kernel k = 4. However, overall, there are relatively
+small differences between all these configurations, with
+R2 values in the interval 0.78 – 0.84.
+Although, in contrast to standard (dense or convolu-
+tional) artificial neural networks, the utility of validation
+in GANs is questionable, we observe a systematic corre-
+lation between the training and a separate validation set,
+as indicated in Fig. A5(a) in the SI. This is particularly
+useful as one difficulty observed in the training of the
+cGANs consists in the sharp variations of the loss func-
+tions with the time step. The correlation between the
+training and validation sets enables us to optimize the
+training interval (Nsteps), i.e. it provides a stopping cri-
+terion so that the model produces accurate results. Then,
+the model is frozen and new densities are generated for
+the test set. Decreasing the number of input images the
+R2 parameter is reduced, as one can see from Fig. A5(b),
+while the relatively high values reflect the overall resem-
+
+7
+n(r)
+int
+Vxy
+˜Vxy
+nint
+ED
+ED
+pix2pix
+FIG. 6.
+The inverse problem: generating potentials from
+interacting charge densities, according to the mapping nint �→
+˜Vxy. Choosing an input potential, we calculate the interact-
+ing density by ED, which becomes the input image for the
+pix2pix approach. The resulting potential, ˜Vxy, is tested by
+computing its corresponding density, n(r)
+int, which is very sim-
+ilar to the initial density nint, calculated from Vxy.
+blance between the potential and the associated density.
+We also investigated the effect of random jitter by re-
+sizing the images to Nresize × Nresize and then randomly
+cropping back to the original size, 64×64. This procedure
+was employed in a number of image translation problems
+discussed in Ref. [26], like Map ↔ aerial photograph, day
+→ night images. In other cases, like black/white → color
+images no jittering was applied. A systematic investiga-
+tion with respect to Nresize taking values from 64 to 80
+in steps of 2 pixels shows that, for the nint �→ ˜Vxy map-
+ping, no-resize (Nresize = 64) leads to the best results,
+R2 ∼ 0.9, and it decreases for larger Nresize values, as
+it can be seen in Fig. A6 in the SI. This is further con-
+firmed by L1, L2 and L∞ norms, where the first two are
+well correlated, while, as expected, there are larger fluc-
+tuations for the L∞ norm. The quality of the generated
+images is also consistent with this trend, as the charge
+distribution becomes less diffuse.
+The inverse problem, i.e.
+mapping an input den-
+sity to a generated potential, is highly important from
+both fundamental and technological perspectives. How-
+ever, not every proposed ground state density can be
+obtained from a potential, which is known as the V-
+representability problem[39]. Therefore, the inverse map-
+ping nint �→ ˜Vxy is here performed starting from com-
+puted densities, rather than arbitrary ones.
+This pro-
+vides a proof-of-concept for a solution to the inverse prob-
+lem based on pix2pix approach, if the target potential
+exists. As shown by Kohn in Ref. [40] small enough devi-
+ations from a V-representable density is still in the same
+class, leading to a slightly different potential.
+A typical nint �→ ˜Vxy, starting from an ED-computed
+density is shown in Fig. 6.
+We use the same pair
+(Vxy, nint), but this time nint serves as input and the
+generated image contains the potential ˜Vxy. Then, we
+recalculate the density corresponding to the generated
+potential, ˜Vxy, which is denoted by n(r)
+int. Comparing ˜Vxy
+with Vxy and n(r)
+int with nint, i.e.
+generated vs.
+input
+quantities, one observes a large degree of similarity. To
+further support this, we plotted additional instances in
+Fig. A2(b) in the SI. There are still some small differences
+visible in the generated potentials compared to the orig-
+inal ones. In most cases, these differences occur for the
+regions with high confinement that are isolated from the
+main quantum well [e.g. as it is found in the instances 5
+and 6 from Fig. A2(b) in the SI], which contain a small
+amount of localized charge. Consequently, as these QW
+regions are removed in the pix2pix-generated potential
+by the cGAN model, the recalculated charge, n(r)
+int, will
+not differ much from the input density, nint. Note that
+even the small islands present in some of the generated
+potentials are well represented compared to the originals.
+Then, as expected, the largest deviations occur at the
+boundaries, in particular at the edges of the square re-
+gion, where the wavefunction vanishes.
+Although, in general, the ML methods are not very
+transparent with respect to their inner workings it is in-
+teresting to observe the evolution of generated images
+representing densities and potentials. Figure 7 shows the
+sequential improvement of the generated images starting
+from the input images, as the model is improved. In the
+first row, the initial assumption for the density resem-
+bles the potential, with larger values outside the region
+corresponding to the quantum well. This is reversed in
+less than 10 steps and the charge is spread rather evenly
+inside the quantum well region. Starting with 200-300
+steps, the density begins to localize inside the quantum
+well, while continuously changing its shape towards the
+target density, with two localized maxima. For the in-
+verse problem, the evolution is shown in the second row
+of snapshots in Fig. 7. This time, the input is the inter-
+acting charge density and the first generated potential
+resembles it closely. However, in less than 10 steps, two
+quantum wells are individualized, then extending and
+merging in the first 100 steps. Subsequently, the shape of
+the generated potential becomes gradually closer to the
+target potential, which is depicted in Fig. 6. The capac-
+ity of the method to reproduce the desired quantities is
+further confirmed by the SSIM values calculated for the
+pairs generated - reference, as shown in Fig. A7 in the
+SI.
+Overall, the pix2pix approach provides an accurate
+and efficient alternative to predict the ground state den-
+sity from the input potential or, conversely, to gener-
+ate a potential from a given density, known to be V-
+representable, once the cGAN is trained on a distinct set
+of calculated examples. Further investigations on excited
+states, as well as on quantum systems with larger num-
+bers of particles can be pursued in a similar way.
+
+8
+Density
+Step = 0
+Step = 10
+Step = 30
+Step = 40
+Step = 100
+Step = 200
+Step = 320
+Step = 820
+Step = 1600
+Step = 4540
+Potential
+FIG. 7.
+Evolution of the generated grid-based quantities ˜nint (first row) and ˜Vxy (second row), according to the mappings
+Vxy �→ ˜nint and nint �→ ˜Vxy, respectively. In the two mappings, the initially generated images resemble the input potential
+and input density. Then, the images are gradually transformed, becoming more and more similar to the target density and
+potential, respectively.
+V.
+CONCLUSIONS
+We introduced an image-to-image translation approach
+based on the pix2pix method to predict bi-particle
+charge densities from the confinement potentials.
+The
+quantum systems are defined on two-dimensional square
+region with randomly generated potentials and the cor-
+responding ground state densities are determined by ex-
+act diagonalization method. A large number of pair im-
+ages is generated, corresponding to the confinement po-
+tentials and calculated interacting densities. Using the
+cGANs implemented in pix2pix we perform three types
+of mappings: potential to non-interacting density, poten-
+tial to interacting density and non-interacting to interact-
+ing density. Although all three mappings result in accu-
+rate predictions, the focus is on generating an interacting
+density from a given potential. Several cGAN architec-
+tures have been considered, by varying the number of
+convolutional layers and kernel size in the discriminator
+network. This analysis shows that a PixelGAN is most
+accurate, although other configurations yield comparable
+results.
+The possibility to perform an inverse mapping, i.e.
+starting from a density and generating a potential, is out-
+lined. Here, we considered as input a calculated density,
+which ensures the V-representability. The generated po-
+tential is then tested and confirmed by calculating the
+ground state density associated with it and comparing
+this density with the original one.
+The cGAN based approach provides an efficient so-
+lution for predicting non-interacting and interacting
+ground state densities when a large set of systems from
+a given class is required to be solved. Interestingly, the
+inverse problem can also be approached using this tech-
+nique, which is important for the design of nanoelectronic
+devices. The pix2pix method is shown to be accurate
+for describing interacting quantum systems and appears
+to be further well suited for a range of condensed matter
+problems.
+ACKNOWLEDGMENTS
+This work was supported by a grant of the Roma-
+nian Ministry of Research, Innovation and Digitalization,
+CNCS - UEFISCDI, project number PN-III-P4-ID-PCE-
+2020-1142, within PNCDI III.
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+
+10
+Appendix A: Supplementary Information
+Potential
+Non-Interacting
+Predicted
+Error
+Potential
+Interacting
+Predicted
+Error
+FIG. A1.
+Mapping from potentials to (left) non-interacting and (right) interacting densities. The error maps correspond to
+differences between the target and predicted distributions (in absolute value).
+
+11
+Non-Interacting
+Interacting
+Predicted
+Error
+Interacting
+Potential
+Predicted
+Error
+FIG. A2.
+Mapping from charge densities: (left) non-interacting to interacting densities, n0 �→ ˜nint and (right) the inverse
+problem, nint �→ ˜Vxy. The error maps correspond to differences between the target and predicted distributions (in absolute
+value).
+
+M12
+¯
+Vxy
+¯n0
+¯nint
+FIG. A3.
+Averages of confinement potential ( ¯Vxy), non-interacting density (¯n0) and interacting density (¯nint) calculated using
+the training set (Ntrain = 4800). These average maps are used in the calculation of R2. All three images indicate the balanced
+distribution of potential shapes. The average non-interacting density is more concentrated in the center of square compared to
+the interacting density.
+Potential
+Interacting
+Predicted
+Error
+FIG. A4.
+Three examples of outliers, which exhibit the largest deviations from the reference, as identified by the SSIM analysis
+in Fig. 4.
+
+13
+0
+5000
+10000
+15000
+20000
+25000
+30000
+35000
+40000
+Steps
+0.70
+0.75
+0.80
+0.85
+0.90
+R
+2
+(b)
+1 Images
+3 Images
+4 Images
+5 Images
+7 Images
+9 Images
+10 Images
+20 Images
+30 Images
+50 Images
+100 Images
+400 Images
+800 Images
+1800 Images
+4800 Images
+0
+5000
+10000
+15000
+20000
+25000
+30000
+35000
+40000
+Steps
+0.70
+0.72
+0.74
+0.76
+0.78
+0.80
+0.82
+0.84
+0.86
+R
+2
+(a)
+Train
+T
+est
+Validation
+Real
+1
+3
+4
+5
+7
+9
+10
+20
+30
+50
+100
+400
+800
+1800
+4800
+FIG. A5.
+Accuracies measured by R2 during training: (a) R2 for training, validation and test sets, for the mapping Vxy �→ ˜nint,
+with the standard cGAN configuration; (b) R2 values for the test set, while varying the number of training examples, Ntrain.
+The improvement of the final generated image for different sizes of the train sets is shown. For Ntrain < 100, the generated
+density merely resembles the potential (input image), while for Ntrain > 800 two individualized maxima can be observed, while
+further fine-tuning occurs for larger Ntrain.
+0
+5000
+10000
+15000
+20000
+25000
+30000
+35000
+40000
+Steps
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+R
+2
+Resize: 64px
+Resize: 66px
+Resize: 68px
+Resize: 70px
+Resize: 72px
+Resize: 74px
+Resize: 76px
+Resize: 78px
+Resize: 80px
+0
+5000
+10000
+15000
+20000
+25000
+30000
+35000
+40000
+Steps
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+0.225
+0.250
+L
+1
+0
+10000
+20000
+30000
+40000
+Steps
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+L
+2
+Resize: 64px
+Resize: 66px
+Resize: 68px
+Resize: 70px
+Resize: 72px
+Resize: 74px
+Resize: 76px
+Resize: 78px
+Resize: 80px
+0
+20000
+40000
+Steps
+0.6
+0.8
+1.0
+1.2
+L
+∞
+Real
+64
+66
+68
+70
+72
+74
+76
+78
+80
+(a)
+(b)
+(c)
+FIG. A6.
+Analysis of the random jitter by applying resizing to Nresize × Nresize and then randomly cropping the images to the
+initial size. (a) The generated densities shows that the best results are obtained for no-resize (Resize = 64 px). (b) The R2
+coefficients and (c) the L1, L2, L∞ norms show consistently that the accuracy is reduced by increasing the resize parameter.
+
+14
+0
+1000
+2000
+3000
+4000
+5000
+Potential index
+0,4
+0,5
+0,6
+0,7
+0,8
+0,9
+1
+SSIM [nint]
+(a)
+0
+1000
+2000
+3000
+4000
+5000
+Potential index
+0,3
+0,4
+0,5
+0,6
+0,7
+0,8
+0,9
+1
+SSIM [Vxy]
+(b)
+FIG. A7.
+SSIM values for pairs of densities and potentials. One pair consists of the reference instance (index i0 = 3236)
+from the test set, described in Figs. 2 and 6, and one other instance in the set of 5000 instances: (a) (nint,i0, nint,i) and (b)
+(Vxy,i0, Vxy,i), depicted by black dots. For i = i0 we have SSIM = 1. The red dots indicate the comparisons between reference
+and generated quantities, for (a) densities (nint,i0, ˜nint,i0), SSIM=0.993 and (b) potentials (Vxy,i0, ˜Vxy,i0), SSIM=0.957, showing
+that the generated density (˜nint,i0) and potential ( ˜Vxy,i0) have higher similarity with their references compared to any other
+instance in the set.
+
diff --git a/uNA0T4oBgHgl3EQfLv9Z/content/tmp_files/load_file.txt b/uNA0T4oBgHgl3EQfLv9Z/content/tmp_files/load_file.txt
new file mode 100644
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+page_content='Mapping confinement potentials and charge densities of  interacting quantum systems using pix2pix  Calin-Andrei Pantis-Simut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
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+page_content=' ∗ Amanda Teodora Preda,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
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+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' ∗  Lucian Ion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='1 Andrei Manolescu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='4 and George Alexandru Nemnes1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' †  1University of Bucharest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Faculty of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Atomistilor 405,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Magurele-Ilfov,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 077125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Romania  2Research Institute of the University of Bucharest (ICUB),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Mihail Kogalniceanu Blvd 36-46,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Bucharest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 050107,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Romania  3Horia Hulubei National Institute for Physics and Nuclear Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Reactorului 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Magurele-Ilfov,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 077125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Romania  4Department of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Reykjavik University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Menntavegur 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' IS-102 Reykjavik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Iceland  Accurate and efficient tools for calculating the ground state properties of interacting quantum  systems are essential in the design of nanoelectronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The exact diagonalization method  fully accounts for the Coulomb interaction beyond mean field approximations and it is regarded as  the gold-standard for few electron systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' However, by increasing the number of instances to be  solved, the computational costs become prohibitive and new approaches based on machine learning  techniques can provide a significant reduction in computational time and resources, maintaining  a reasonable accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Here, we employ pix2pix, a general-purpose image-to-image translation  method based on conditional generative adversarial network (cGAN), for predicting ground state  densities from randomly generated confinement potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Other mappings were also investigated,  like potentials to non-interacting densities and the translation from non-interacting to interacting  densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The architecture of the cGAN was optimized with respect to the internal parameters of  the generator and discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Moreover, the inverse problem of finding the confinement potential  given the interacting density can also be approached by the pix2pix mapping, which is an important  step in finding near-optimal solutions for confinement potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  INTRODUCTION  Machine learning (ML) has found extensive applica-  tions in multiple research fields in the last decade, bring-  ing along a new paradigm in science, based on a more effi-  cient and versatile analysis of experimental and simulated  data [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Statistical models and high-end programming  have led to the build-up of deep learning techniques that  solve problems of clustering, regression and classification  [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In particular, material science and nanotechnology  have adapted ML algorithms in order to provide an ac-  celerated interpretation of data and reduce the resources  needed for material [4, 5] and device [6, 7] design, based  on calculated examples or experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The field  of artificial intelligence has also extended to more theo-  retical areas of condensed matter [8], such as quantum  phase transitions [9] and learning topological invariants  [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The physics of nanoelectronic devices and quantum  information applications relies heavily on an accurate  and efficient description of many-body states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Tradition-  ally, the many-body systems have been approached by  mean-field theories like Hartree-Fock and density func-  tional theory (DFT), the latter being mostly employed  in the context of atomistic calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  However, for  applications that require q-bit level descriptions beyond  mean-field approaches, computationally more demanding  ∗ These authors contributed equally to this work  † Corresponding author:  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Nemnes (nemnes@solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='fizica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='unibuc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='ro)  methods such as the exact diagonalization (ED) method  [11–13] are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Many-electron states have been  previously analyzed in quantum dot (QD) systems with  top gate arrays [14], where the exponential increase in  the number of gate voltage configurations leads to a pro-  hibitively large computation effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Efficiently solving  a large number of many-body Hamiltonian diagonaliza-  tions is typically required in the design of nanoelectronic  devices and this is a suitable task for ML approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Visualization has always been essential for the under-  standing and interpretation of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In the context of  condensed matter, one idea is the use of graphs as means  to encode the information about atomic and molecular  structures [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Along with the development of ad-  vanced deep learning methods, it also became possible to  create algorithms that gain insights into raw representa-  tions such as pixels of an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' For this particular do-  main, convolutional neural networks (CNNs) have proved  to be decisive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In material physics, CNNs have been em-  ployed for a variety of applications, from the prediction of  the ionic conductivity of a ceramic material from image  quality maps [18] to the prediction of the space groups  and the crystallographic dimensionality of thin film ma-  terials from XRD spectral inputs [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A lot of work is  now invested in explaining how CNNs make accurate pre-  dictions and the common problem is that usually that  this type of network makes predictions based on sepa-  rate parts of a real object, while in condensed matter  and material science one is more interested in statistical  distributions [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' For example, after convolutional neu-  ral networks were used to predict material elastic prop-  erties from a synthetic high contrast material dataset,  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='02122v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='mes-hall]  5 Jan 2023  2  the authors argued that CNNs actually learn physically  relevant information about the structure and that the  convolutional filters are connected to the n-point spatial  correlation theory [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Autoencoders, which have CNNs embedded in their ar-  chitecture, were used to learn low-dimensional represen-  tations of the data from a material database and subse-  quently incorporate it in a data-driven solver to improve  efficiency [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' One step further from conventional con-  volutional networks, Generative Adversarial Networks  (GANs) are an emerging deep learning technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Sev-  eral research articles have focused on using GANs for mi-  crostructure synthesis [23, 24] and materials design, by  capturing the characteristics of complex materials and  learning the mapping between latent variables and the  structure [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Another remarkable network architecture that is suit-  able for image processing is the conditional adversarial  network (cGAN), in which both the discriminator and  generator are given additional information and, from this  point of view, are trained in a conditional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Isola et  al proposed an algorithm for image translation known as  pix2pix which employs a ”U-Net” based architecture for  the generator and a convolutional ”Patch-GAN” classi-  fier for the discriminator [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The advantage of this type  of algorithm is that it learns a loss function that adapts  to the data and can be applied to a wide range of image  processing related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' This type of model is a valuable  tool in image dehazing models, which aim to for improve  the quality of images and increase visibility, with applica-  tions video surveillance and obstacle avoidance systems  [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Pix2pix is also already employed in the field of  medical imaging, with the purpose of facilitating the col-  lection of larger data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' For example, Toda et al have  used a pix2pix based model (StylePix2pix) to generate  lesion images from tumor sketches, which proved to be an  effective approach for data augmentation [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Mehmood  et al have used the image translation algorithm for the  detection, colorization and classification of tumor images  [29], while Tahri et al used pix2pix to generate synthetic  CT images MRI radiotherapy planning [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In the field  of industry research, pix2pix was employed for the pur-  pose of generating new images with surface quality de-  fects, relevant in the production of metal workpieces [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Due to the popularity of the algorithm, there is also con-  siderable interest to increase the speed of training and  improve its efficiency [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  In this paper, we investigate cGANs implemented in  pix2pix method for predicting many-body charge densi-  ties in the ground state, for randomly generated quantum  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In the training process of the cGAN, the map-  ping is performed between the confinement potentials  and the densities corresponding to Coulomb interacting  systems, calculated by ED method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A similar mapping  is performed to yield the non-interacting densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  In  this way, the exact diagonalization is bypassed, which  is a considerable advantage as diagonalizing many-body  Hamiltonians becomes prohibitive when the number of  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  An interacting quantum system with random  A-B type domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The corresponding potential map in a  typical configuration, Vxy, obtained using the procedure de-  scribed in text, is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The system is defined on  a two-dimensional square region of area L × L, with vanish-  ing boundary conditions for the wavefunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Starting from  a step potential (yellow regions), Vs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='5 eV, a connected  set of quantum wells (black regions), V0 = 0 eV, defines the  confinement potential for electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  systems grows too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Using the pix2pix mapping, an  efficient and accurate prediction of the interacting den-  sities is achieved for new test systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In addition, we  provide a proof-of-concept for the inverse problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  generating a potential from an input density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In Section II, the  class of model systems is described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In the next section,  the numerical implementations of the ED and pix2pix  methods are detailed and some measures for quality as-  sessment of generated images are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The results  obtained for different types of pix2pix mappings, involv-  ing potentials, non-interacting and interacting densities,  are discussed in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The accuracies of predicted  densities are analyzed for several cGAN configurations  and optimal configurations are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Moreover, the  method is shown to produce accurate results for the in-  verse problem as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  MODEL SYSTEMS  The quantum systems consist of N electrons confined  in randomly generated potentials Vxy, defined on a two-  dimensional square region, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The  potentials Vxy correspond to connected groups of cir-  cular quantum wells (QWs) with different radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' These  potential configurations resemble systems of interacting  Nint3  QDs such as two-dimensional self-assembly functional-  ized graphene QDs [33], randomly distributed QDs for  memristive elements[34] or random geometric graphs of  QDs [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Monolayer graphene - hexagonal boron ni-  tride films can form arbitrary shaped A-B type domains  [36, 37], where A and B are conductive and insulating  domains, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Moreover, the choice of random  potential maps also ensures a thorough evaluation of the  pix2pix method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  In order to get a balanced distribution of QWs in  given potential map, the following scheme was consid-  ered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Starting with a flat potential step of height Vs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='5  eV, a number of Nqw = 25 flat QWs are placed inside  the square region of linear size L = 30 nm, having the  base potential V0 = 0 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The centers of the QWs are  randomly chosen, as well as their radius in the interval  L/16 < R < L/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  When a new QW is added, it is  allowed to partly overlap with the current QW, but no  more than 3/4 of its area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' If the new QW is disconnected  from the current QW, it is discarded and a new position  and radius are chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The process continues until all  Nqw are placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Finally, if more than 80% of the L2 area  is set by the QWs (V0 = 0 eV), the potential map is dis-  carded and the process is started from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In  this way, a connected ensemble of QWs is formed, with a  high degree of variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A total number of NV = 5000  potential instances are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  COMPUTATIONAL METHODS  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The exact diagonalization method  The non-interacting one-body Hamiltonian for an elec-  tron in a two-dimensional confinement potential V (r) is:  H0 = − ℏ2  2m∗  ∂2  ∂r2 + V (r) ,  (1)  where m∗ is the effective mass and r ≡ (x, y) is the posi-  tion vector in two-dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The N-particle Hamiltonian is written as a sum of  the single particle operators and the two-body operator,  which describes the Coulomb interaction:  H =  N  �  i=1  Hi + 1  2  �  i  �  j  Vij,  (2)  where  Hi = H0 (ri) ,  Vij = VC (ri, rj) =  e2  4πϵ0ϵr  1  |ri − rj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  (3)  The Hamiltonian in the second quantization becomes:  H =  �  a  ϵac†  aca + 1  2  �  abcd  Vabcdc†  ac†  bcdcc,  (4)  where ϵa are the energies of the single-particle states  and the Vabcd coefficients corresponding to the Coulomb  interaction are calculated based on the orbital compo-  nents of the single-particle states, {φa,σz}:  Vabcd =  �  dr  �  dr′ �  σz,σ′z  φ∗  a,σz(r)φ∗  b,σ′z (r′)  ×  e2  4πϵ0ϵr  1  |r − r′|φc,σz(r)φd,σ′z (r′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  (5)  Solving the time independent Schr¨oedinger equation  HΨn = EnΨn,  (6)  one obtains the eigenvalues En and eigenvectors Ψn ≡  Ψn (r1, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' , rN, sN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Then, the particle density in the  ground state is:  n0(r) = N  �  s1  · ·  �  sN  �  dr2 · · ·  ×  �  drN|Ψ0 (r, s1, r2, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' , rN, sN) |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  (7)  The numerical implementation of the ED method is  described in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Choosing an appropriate  single-particle basis, which fulfills the boundary condi-  tions, we first solve the one-particle problem for a given  two-dimensional potential, using a basis size N 2  b = 322,  on a grid Nx × Ny = 64 × 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Next, using the single-  particle eigenfunctions, {Φi(r)}, a two-particle basis of  Slater determinants is assembled in the occupation num-  ber representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  By diagonalizing the two-particle  Hamiltonian one obtains the ground state particle den-  sity:  n0(r) =  �  k  |C0k|2  N  �  p=1  �  |φip(k),↑(r)|2 + |φip(k),↓(r)|2�  ,  (8)  where C0k is the expansion coefficient corresponding to  the k-th Slater determinant and φip(k),↑(r), φip(k),↓(r)  are the orbital components of the single-particle states  Φip(k)(r), with spin up and spin down, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Pix2pix cGAN for image-to-image translation  The method developed by Isola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' [26] makes use  of a cGAN for general-purpose image-to-image transla-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Like in other cGAN approaches, the generator-  discriminator architecture of pix2pix is set to optimize a  global goal, namely that the generated output is made in-  distinguishable from the reference (ground truth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' How-  ever, the main differences compared to other cGAN type  4  approaches consists in the use of a U-Net architecture  for the generator and a PatchGAN for the discriminator,  which is more sensitive to local details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  We employ this approach to create three mappings of  type α �→ β, as follows: (i) Vxy �→ ˜n0, (ii) Vxy �→ ˜nint,  (iii) n0 �→ ˜nint, where Vxy = V (x, y) is the confinement  potential, n0(x, y) and ˜n0(x, y) are the calculated and  generated non-interacting particle densities, respectively,  and ˜nint(x, y) is the generated interacting particle den-  sity, which shall be compared to the calculated interact-  ing particle density, nint(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Following the standard pix2pix approach [26], the  generator G performs a mapping from an input quantity-  image α ∼ Vxy or n0 to an output quantity-image β ∼ ˜n0  or ˜nint, except, of course, the trivial n0 �→ ˜n0 mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Typically, in the image generation process a random noise  vector is added, denoted by γ, so that we may describe  the generator mapping as G : {α, γ} �→ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The gen-  erator is trained to produce better and better images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  On the other hand, the discriminator, D, is adversarially  trained to classify the input it receives as real or fake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  As the generator typically learns to ignore the noise in-  troduced by the random vector γ, it is implemented in  form of dropout in some layers of the generator, since its  overall influence is rather small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The loss function of the cGAN is expressed using the  binary cross entropy as [26]:  LcGAN(G, D) = Eα,β[log D(α, β)]  + Eα,γ[log(1 − D(α, G(α, γ))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  (9)  The objective G∗ is found as G tries to minimize LcGAN  and D tries to maximize it and, in addition, an LL1 loss,  representing the difference between generated and refer-  ence images, is included:  G∗ = arg min  G  max  D  LcGAN(G, D) + λLL1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  (10)  The parameter λ = 100 sets the relative importance be-  tween LcGAN and LL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Error and accuracy measures for generated  densities  In many applications of image-to-image translation like  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' maps �→ aerial photographs or the opposite, a per-  ceptual validation is often employed [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Assessing the  quality of the generated images or comparing them with  target images is generally not an easy task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  A quantitative approach often employed is the struc-  tural similarity index measure (SSIM), which com-  bines luminance, contrast and structure components [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  However, for the current aim of mapping the charge den-  sities in quantum systems, the stochasticity of the model  is limited and a strict comparison based on L1, L2 and  L∞ norms also becomes a suitable assessment with a  transparent interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The L1 norm reflects the  amount of displaced charge in generated vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  reference  systems, L2 is related to the root mean squared error  and L∞ corresponds to a local maximum error in the  evaluation of the charge densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  In order to evaluate the difference between the gener-  ated and reference grid-based quantities, denoted by β  and βref, we consider the L1, L2 and L∞ norms as pos-  sible measures, the first two being scaled by the number  of grid points (pixels):  L1 =  1  Nx × Ny  ∥β − βref∥1  (11)  L2 =  1  �  Nx × Ny  ∥β − βref∥2  (12)  L∞ = ∥β − βref∥∞  (13)  On the other hand, SSIM can provide further assessment  on the structural differences between the generated and  target densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In addition, we calculate a mean SSIM  (MSSIM) employing a uniformly weighted 8 × 8 square  window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In the subsequent analysis, for the calculation  of SSIM and MSSIM we use the typical parameters sug-  gested in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The prediction accuracy in an ensemble of Nsys gener-  ated and reference pairs, {(βi, βref,i)}, can be described  by the R2 coefficient of determination, calculated from  the residual sum of squares, SSres, and the total sums of  squares, SStot :  R2 = 1 − SSres  SStot  ,  (14)  with  SSres =  Nsys  �  i=1  ∥βref,i − βi∥2  2,  (15)  SStot =  Nsys  �  i=1  ∥βref,i − ¯βref,i∥2  2,  (16)  where ¯βref =  1  Nsys  �Nsys  i=1 βref,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In the vector space of the  grid-based quantities {βi}, we define βi ±βj as pixel-wise  addition and subtraction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The reference cGAN architecture  The architecture of the cGAN is specified by a num-  ber of parameters corresponding to the generator and  discriminator networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  We shall first assume a typi-  cal configuration, called reference configuration, which  is then modified for further optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The cGAN  translates grid-quantities set on Nx×Ny = 64×64 pixels,  which are potentials and charge densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The generator  has an encoder-decoder configuration with 6 downsam-  pling convolutional layers and 6 upsampling deconvolu-  tional layers, all with strides SG = 2, which keeps the  size of the output equal to the size of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The dis-  criminator receives two pairs of images, (input image,  5  nint  n0  Vxy  n0 �→ ˜nint  Vxy �→ ˜nint  Vxy �→ ˜n0  Reference  Predicted  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Non-interacting and interacting densities generated  by the pix2pix cGAN, for a two-particle system confined in  a random potential Vxy: (i) Vxy �→ ˜n0, (ii) Vxy �→ ˜nint, (iii)  n0 �→ ˜nint mappings, indicated by red, green and blue arrows,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The left column shows the images of the calcu-  lated grid-based quantities: Vxy, n0, nint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In the right column  the generated images are depicted: ˜n0 and ˜nint, the latter be-  ing determined from either Vxy or n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The solid lines indicate  the actual mapping, while the dashed lines indicate an associ-  ation between the calculated (reference) data and generated  (predicted) images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  reference image) and (input image, generated image),  which should be classified as real and fake, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Its architecture includes 5 convolutional layers with the  strides-sequence SD = (2, 2, 2, 1, 1), which reduces the  input to an output of 6×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' This corresponds to a convo-  luted response of patch classification in real or fake, the  patch size being dependent on the discriminator’s archi-  tecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The resulting patch size is Npt×Npt = 70×70 is  larger than the size of the image, in which case the cGAN  is referred to as ImageGAN [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Decreasing the number  of layers in the discriminator, the patch size decreases in  the sequence Npt = 34, 16, 7, 4, 1, where the limiting case  with Npt = 1 is termed PixelGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The kernel size for  both G and D is κ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  RESULTS AND DISCUSSION  The quantum systems considered here consist of Np =  2 particles confined in randomly generated potentials  {Vxy} following the scheme described in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Using  the reference cGAN configuration we perform the three  mappings, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 2 for a typical instance: (i)  Vxy �→ ˜n0, (ii) Vxy �→ ˜nint, (iii) n0 �→ ˜nint, where the  ’∼’ symbol denotes generated quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The potentials  {Vxy} are readily available as input data, while the non-  interacting densities, {n0}, can be determined by one-  particle calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The interacting densities, {nint},  are determined using the ED method, using the non-  interacting many-particle states obtained in the previ-  ous step, which are used to set-up the two-particle basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The first two mappings produce densities directly from  the input potentials and, in particular, the second one,  Vxy �→ ˜nint, is of highest importance, as it yields the in-  teracting density without any diagonalization procedure  after the model is trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The third mapping starts from  the non-interacting density, rather than the confinement  potential, and it is performed for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Evaluated by visual inspection, all three mappings de-  picted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 2 reproduce quite well the key features  of the reference (calculated) densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In a typical non-  interacting calculation, the ground state charge density,  n0, is mostly localized in the quantum well region where  the confinement is weaker, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' the wider part of the quan-  tum well, so that the kinetic energy is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In this  case, the two electrons with opposite spins occupy the  same space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' However, when the Coulomb interaction is  considered, the charge density in the ground state, nint, is  more delocalized, being distributed in the quantum well  of arbitrarily shape, also in regions with stronger confine-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Qualitatively, the distribution of nint is set by the  tradeoff between the larger kinetic energy in stronger con-  finement regions and the Coulomb interaction between  the particles occupying the same space in a region with  weaker confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Figure 2 shows that the cGAN is  able to learn the non-trivial features, so that the target  quantities are reproduced with a high degree of accu-  racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Additional examples are indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A1 and  A2 in the Supplementary Information (SI, AppendixA),  for mappings from potentials and non-interacting densi-  ties, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The training process of the cGAN was performed using  Ntrain = 4800 image pairs, while a number of Nval = 100  and Ntest = 100 distinct samples, were used for valida-  tion and test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' For the relatively large training set, the  averages of the potentials, non-interacting and interact-  ing densities indicate a balanced distributions, as shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A3 of the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' These averages are later used to  calculate the R2 coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  During the training process, we monitor the loss func-  tions of the generator and discriminator, which are de-  picted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 3 for a typical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In contrast to the usual  deep learning architectures, where the loss functions are  specified, in cGANs the discriminator loss is learned from  the input data, which usually brings large fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Therefore, instead of seeking the minima, the model be-  comes suitably trained when the loss functions are stabi-  lized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' This also poses a problem for the train-stopping-  criterion, which is often optimized by visually checking a  sequence of steps at the end of the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  6  0  10000  20000  30000  40000  0  1  2  3  4  Loss  Discriminator loss  0  10000  20000  30000  40000  0  1  2  3  4  5  6  7  8  Generator GAN loss  0  10000  20000  30000  40000  Steps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='7  Loss  Generator L1 loss  0  10000  20000  30000  40000  Steps  0  10  20  30  40  50  60  70  Generator total loss  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Generator and discriminator loss functions vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' the  number of steps in a typical training run, for the mapping  Vxy �→ ˜nint (thin lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A smoothing is applied to all four  data sets using the Savitzky-Golay filter with 3rd degree poly-  nomial and a window of 40 points, to better illustrate the  trends (thick lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In spite of the relatively large fluctua-  tions, typical for cGAN architectures, the loss functions tend  to stabilize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  0  10000  20000  30000  40000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='0  SSIM  0  20000  0  10000  20000  30000  40000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='6  L  1  0  10000  20000  30000  40000  Steps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='0  MSSIM  0  20000  Counts  0  10000  20000  30000  40000  Steps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='7  L  2  0  20000  40000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='5  L  ∞  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The evolution of SSIM and MSSIM during training  for the mapping Vxy �→ ˜nint, for a group of 30 test instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Histograms of all SSIM and MSSIM values collected during  training are depicted on the right hand sides of each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' It  is worth noting that the norms L1, L2 and, in part, L∞ are  closely correlated with SSIM and MSSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  In order to assess the quality of the generated den-  sities in the Vxy �→ ˜nint mapping, we calculate SSIM  and MSSIM for a group of 30 samples from the test set  and monitor their individual evolution as the model is  trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The data shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 4 indicates that high  values (up to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='9995) for both SSIM and MSSIM can  be obtained, when the generated density becomes very  similar to the target density, while at beginning of the  0  5000  10000  15000  20000  25000  30000  35000  40000  Steps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='74  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='84  R  2  Layers: 0+0,  Kernel: 4,  Patch Size: 1×1  Layers: 0+1,  Kernel: 4,  Patch Size: 4×4  Layers: 0+2,  Kernel: 4,  Patch Size: 7×7  Layers: 1+2,  Kernel: 4,  Patch Size: 16×16  Layers: 2+2,  Kernel: 4,  Patch Size: 34×34  Layers: 3+2,  Kernel: 4,  Patch Size: 70×70  Layers: 3+2,  Kernel: 3,  Patch Size: 47×47  Layers: 3+2,  Kernel: 2,  Patch Size: 24×24  Layers: 3+2,  Kernel: 1,  Patch Size: 1×1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Prediction accuracies obtained with different cGAN  architectures, based on tuning several key parameters, num-  ber of convolutional layers in the discriminator network and  the kernel size, resulting in the different patch sizes, Npt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' De-  pending on Npt values with respect of the image size, three  groups of networks can be identified: ImageGAN (Npt > 64),  PatchGAN (1 < Npt < 64) and PixelGAN (Npt = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  training these values are below ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='3, when the first gen-  erated densities resemble the input potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A number  of outliers are evidenced for which this procedure would  produce somewhat worse results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' These instances are de-  scribed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A4 of the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' It is important to note that  SSIM and MSSIM are in close correlation with the er-  ror measures based on L1, L2 and L∞, which are also  represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 4 for the same instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The overall accuracy of generated grid-based quantities  on a set of examples is evaluated by the R2 coefficient of  determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The evolution of R2 for the test set vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  time step is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 5 for several cGAN architec-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' We focus on the discriminator’s architecture and  vary the number of convolutional layers and the kernel  size, which determines the patch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The PixelGANs  (Npt = 1) perform better compared to an ImageGAN in  the standard configuration, with 3+2 convolutional layers  and a kernel k = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' However, overall, there are relatively  small differences between all these configurations, with  R2 values in the interval 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='78 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Although, in contrast to standard (dense or convolu-  tional) artificial neural networks, the utility of validation  in GANs is questionable, we observe a systematic corre-  lation between the training and a separate validation set,  as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A5(a) in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' This is particularly  useful as one difficulty observed in the training of the  cGANs consists in the sharp variations of the loss func-  tions with the time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The correlation between the  training and validation sets enables us to optimize the  training interval (Nsteps), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' it provides a stopping cri-  terion so that the model produces accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Then,  the model is frozen and new densities are generated for  the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Decreasing the number of input images the  R2 parameter is reduced, as one can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A5(b),  while the relatively high values reflect the overall resem-  7  n(r)  int  Vxy  ˜Vxy  nint  ED  ED  pix2pix  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The inverse problem: generating potentials from  interacting charge densities, according to the mapping nint �→  ˜Vxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Choosing an input potential, we calculate the interact-  ing density by ED, which becomes the input image for the  pix2pix approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The resulting potential, ˜Vxy, is tested by  computing its corresponding density, n(r)  int, which is very sim-  ilar to the initial density nint, calculated from Vxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  blance between the potential and the associated density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  We also investigated the effect of random jitter by re-  sizing the images to Nresize × Nresize and then randomly  cropping back to the original size, 64×64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' This procedure  was employed in a number of image translation problems  discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' [26], like Map ↔ aerial photograph, day  → night images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In other cases, like black/white → color  images no jittering was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A systematic investiga-  tion with respect to Nresize taking values from 64 to 80  in steps of 2 pixels shows that, for the nint �→ ˜Vxy map-  ping, no-resize (Nresize = 64) leads to the best results,  R2 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='9, and it decreases for larger Nresize values, as  it can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A6 in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' This is further con-  firmed by L1, L2 and L∞ norms, where the first two are  well correlated, while, as expected, there are larger fluc-  tuations for the L∞ norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The quality of the generated  images is also consistent with this trend, as the charge  distribution becomes less diffuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The inverse problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  mapping an input den-  sity to a generated potential, is highly important from  both fundamental and technological perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' How-  ever, not every proposed ground state density can be  obtained from a potential, which is known as the V-  representability problem[39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Therefore, the inverse map-  ping nint �→ ˜Vxy is here performed starting from com-  puted densities, rather than arbitrary ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  This pro-  vides a proof-of-concept for a solution to the inverse prob-  lem based on pix2pix approach, if the target potential  exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' As shown by Kohn in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' [40] small enough devi-  ations from a V-representable density is still in the same  class, leading to a slightly different potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  A typical nint �→ ˜Vxy, starting from an ED-computed  density is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  We use the same pair  (Vxy, nint), but this time nint serves as input and the  generated image contains the potential ˜Vxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Then, we  recalculate the density corresponding to the generated  potential, ˜Vxy, which is denoted by n(r)  int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Comparing ˜Vxy  with Vxy and n(r)  int with nint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  generated vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  input  quantities, one observes a large degree of similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' To  further support this, we plotted additional instances in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A2(b) in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' There are still some small differences  visible in the generated potentials compared to the orig-  inal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In most cases, these differences occur for the  regions with high confinement that are isolated from the  main quantum well [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' as it is found in the instances 5  and 6 from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A2(b) in the SI], which contain a small  amount of localized charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Consequently, as these QW  regions are removed in the pix2pix-generated potential  by the cGAN model, the recalculated charge, n(r)  int, will  not differ much from the input density, nint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Note that  even the small islands present in some of the generated  potentials are well represented compared to the originals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Then, as expected, the largest deviations occur at the  boundaries, in particular at the edges of the square re-  gion, where the wavefunction vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Although, in general, the ML methods are not very  transparent with respect to their inner workings it is in-  teresting to observe the evolution of generated images  representing densities and potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Figure 7 shows the  sequential improvement of the generated images starting  from the input images, as the model is improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In the  first row, the initial assumption for the density resem-  bles the potential, with larger values outside the region  corresponding to the quantum well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' This is reversed in  less than 10 steps and the charge is spread rather evenly  inside the quantum well region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Starting with 200-300  steps, the density begins to localize inside the quantum  well, while continuously changing its shape towards the  target density, with two localized maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' For the in-  verse problem, the evolution is shown in the second row  of snapshots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' This time, the input is the inter-  acting charge density and the first generated potential  resembles it closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' However, in less than 10 steps, two  quantum wells are individualized, then extending and  merging in the first 100 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Subsequently, the shape of  the generated potential becomes gradually closer to the  target potential, which is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The capac-  ity of the method to reproduce the desired quantities is  further confirmed by the SSIM values calculated for the  pairs generated - reference, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A7 in the  SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Overall, the pix2pix approach provides an accurate  and efficient alternative to predict the ground state den-  sity from the input potential or, conversely, to gener-  ate a potential from a given density, known to be V-  representable, once the cGAN is trained on a distinct set  of calculated examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Further investigations on excited  states, as well as on quantum systems with larger num-  bers of particles can be pursued in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  8  Density  Step = 0  Step = 10  Step = 30  Step = 40  Step = 100  Step = 200  Step = 320  Step = 820  Step = 1600  Step = 4540  Potential  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Evolution of the generated grid-based quantities ˜nint (first row) and ˜Vxy (second row), according to the mappings  Vxy �→ ˜nint and nint �→ ˜Vxy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' In the two mappings, the initially generated images resemble the input potential  and input density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Then, the images are gradually transformed, becoming more and more similar to the target density and  potential, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  CONCLUSIONS  We introduced an image-to-image translation approach  based on the pix2pix method to predict bi-particle  charge densities from the confinement potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The  quantum systems are defined on two-dimensional square  region with randomly generated potentials and the cor-  responding ground state densities are determined by ex-  act diagonalization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A large number of pair im-  ages is generated, corresponding to the confinement po-  tentials and calculated interacting densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Using the  cGANs implemented in pix2pix we perform three types  of mappings: potential to non-interacting density, poten-  tial to interacting density and non-interacting to interact-  ing density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Although all three mappings result in accu-  rate predictions, the focus is on generating an interacting  density from a given potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Several cGAN architec-  tures have been considered, by varying the number of  convolutional layers and kernel size in the discriminator  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' This analysis shows that a PixelGAN is most  accurate, although other configurations yield comparable  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The possibility to perform an inverse mapping, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  starting from a density and generating a potential, is out-  lined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Here, we considered as input a calculated density,  which ensures the V-representability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The generated po-  tential is then tested and confirmed by calculating the  ground state density associated with it and comparing  this density with the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The cGAN based approach provides an efficient so-  lution for predicting non-interacting and interacting  ground state densities when a large set of systems from  a given class is required to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Interestingly, the  inverse problem can also be approached using this tech-  nique, which is important for the design of nanoelectronic  devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The pix2pix method is shown to be accurate  for describing interacting quantum systems and appears  to be further well suited for a range of condensed matter  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This work was supported by a grant of the Roma-  nian Ministry of Research, Innovation and Digitalization,  CNCS - UEFISCDI, project number PN-III-P4-ID-PCE-  2020-1142, within PNCDI III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
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+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
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+page_content='  Chiang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
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+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Shen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
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+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
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+page_content='  [34] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Yan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Pei, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Zhao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Zhou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Wang,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Qin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Xiao,  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Zhao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Ren, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Zhou,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
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+page_content=' Sheikh, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Simoncelli, IEEE  Transactions on Image Processing 13, 600 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
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+page_content=' Levy, Proceedings of the National Academy of Sci-  ences 76, 6062 (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
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+page_content=' Kohn, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 51, 1596 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  10  Appendix A: Supplementary Information  Potential  Non-Interacting  Predicted  Error  Potential  Interacting  Predicted  Error  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Mapping from potentials to (left) non-interacting and (right) interacting densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The error maps correspond to  differences between the target and predicted distributions (in absolute value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  11  Non-Interacting  Interacting  Predicted  Error  Interacting  Potential  Predicted  Error  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Mapping from charge densities: (left) non-interacting to interacting densities, n0 �→ ˜nint and (right) the inverse  problem, nint �→ ˜Vxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The error maps correspond to differences between the target and predicted distributions (in absolute  value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  M12  ¯  Vxy  ¯n0  ¯nint  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Averages of confinement potential ( ¯Vxy), non-interacting density (¯n0) and interacting density (¯nint) calculated using  the training set (Ntrain = 4800).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' These average maps are used in the calculation of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' All three images indicate the balanced  distribution of potential shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The average non-interacting density is more concentrated in the center of square compared to  the interacting density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Potential  Interacting  Predicted  Error  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Three examples of outliers, which exhibit the largest deviations from the reference, as identified by the SSIM analysis  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  13  0  5000  10000  15000  20000  25000  30000  35000  40000  Steps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='90  R  2  (b)  1 Images  3 Images  4 Images  5 Images  7 Images  9 Images  10 Images  20 Images  30 Images  50 Images  100 Images  400 Images  800 Images  1800 Images  4800 Images  0  5000  10000  15000  20000  25000  30000  35000  40000  Steps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='74  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='86  R  2  (a)  Train  T  est  Validation  Real  1  3  4  5  7  9  10  20  30  50  100  400  800  1800  4800  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  Accuracies measured by R2 during training: (a) R2 for training, validation and test sets, for the mapping Vxy �→ ˜nint,  with the standard cGAN configuration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' (b) R2 values for the test set, while varying the number of training examples, Ntrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  The improvement of the final generated image for different sizes of the train sets is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' For Ntrain < 100, the generated  density merely resembles the potential (input image), while for Ntrain > 800 two individualized maxima can be observed, while  further fine-tuning occurs for larger Ntrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  0  5000  10000  15000  20000  25000  30000  35000  40000  Steps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
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+page_content='  Analysis of the random jitter by applying resizing to Nresize × Nresize and then randomly cropping the images to the  initial size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' (a) The generated densities shows that the best results are obtained for no-resize (Resize = 64 px).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' (b) The R2  coefficients and (c) the L1, L2, L∞ norms show consistently that the accuracy is reduced by increasing the resize parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  14  0  1000  2000  3000  4000  5000  Potential index  0,4  0,5  0,6  0,7  0,8  0,9  1  SSIM [nint]  (a)  0  1000  2000  3000  4000  5000  Potential index  0,3  0,4  0,5  0,6  0,7  0,8  0,9  1  SSIM [Vxy]  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' A7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='  SSIM values for pairs of densities and potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' One pair consists of the reference instance (index i0 = 3236)  from the test set, described in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' 2 and 6, and one other instance in the set of 5000 instances: (a) (nint,i0, nint,i) and (b)  (Vxy,i0, Vxy,i), depicted by black dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' For i = i0 we have SSIM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content=' The red dots indicate the comparisons between reference  and generated quantities, for (a) densities (nint,i0, ˜nint,i0), SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='993 and (b) potentials (Vxy,i0, ˜Vxy,i0), SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
+page_content='957, showing  that the generated density (˜nint,i0) and potential ( ˜Vxy,i0) have higher similarity with their references compared to any other  instance in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNA0T4oBgHgl3EQfLv9Z/content/2301.02122v1.pdf'}
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+Text sampling strategies for predicting missing bibliographic links
+F. V. Krasnova, I. S. Smaznevicha*, E. N. Baskakovaa
+a NAUMEN R&D, 49A, Tatishcheva st., Yekaterinburg, 620028, Russian Federation
+*ismaznevich@naumen.ru
+Krasnov Fedor Vladimirovich – Ph. D. in Engineering (Doctor of Technical Sciences). Expert at
+the Department of Semantic Systems, NAUMEN R&D. Research interests: intelligent text
+analysis. Author of over 70 scientific publications. 
+ORCID: 0000-0002-9881-7371 
+e-mail: fkrasnov@naumen.ru
+Smaznevich Irina Sergeevna – Business analyst at the Department of Semantic Systems,
+NAUMEN R&D. Graduate of the Faculty of Computational Mathematics and Cybernetics,
+Lomonosov Moscow State University. Research interests: use of intelligent algorithms in
+applied information systems. The number of scientific publications – 8.
+ORCID: 0000-0002-5996-4635 
+e-mail: ismaznevich@naumen.ru
+Baskakova Elena Nikolaevna – Leading system analyst at the Department of Semantic Systems,
+NAUMEN  R&D.  Graduate  of  the  Faculty  of  Informatics  and  Control  Systems,  Bauman
+Moscow State Technical University. Research interests: use of intelligent algorithms in applied
+information systems. The number of scientific publications – 5.
+ORCID: 0000-0002-7071-8961
+e-mail: enbaskakova@naumen.ru
+Word count – 3199
+
+The presence of a citation
+Thesentencecontains
+There is no citation marker
+marker in the sentence
+a citation marker
+in the sentence
+does not matter
+Positive sample
+Negative sample
+S;Bag of Words S;
+Estimators S;
+Documento
+Sentence
+OH
+Sampling strategies
+BoW1
+Ensemblevoting Si
+Estimator
+Sentence
+Si-n
+P1(S)
+Sentence
+BoW2
+P2(S)
+-P(S)→
+S;
+Si-n
+Estimator2
+Sentence
+P3(S)
+Si+m
+888
+1+m
+Sentence
+BoW3
+Estimator3
+Documentn
+Sentence
+SentenceStrategy#0
+Strategy #2
+Strategy #3
+0.016
+301.5 ± 154.0
+105.5 ± 39.2
+157.0 ± 49.5
+0.006
+273.9 ± 170.8
+0.0200
+103.0 ± 57.2
+0.014
+155.1 ± 76.1
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+0.005
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+E00'0
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+0.0050 -
+0.004
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+0.0025
+0.002
++000°0
+0.0000
+0.000 +
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500
+Strategy #5
+Strategy #7
+Strategy #9
+0.010 
+260.6 ± 74.9
+362.4 ± 92.6
+465.4 ± 113.2
+259.7 ± 105.7
+0.007
+364.4 ± 127.5
+0.006
+469.7 ± 149.2
+800°0
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+0.980
+0.975
+0.970
+weighted
+0.965
+a
+0.960
+-scor
+0.955
+0.950
+0.945
+Hard Voting
+Soft Voting
++
+0.940
+4
+5
+6
+7
+8
+9
+10
+NumberofestimatorsText sampling strategies for predicting missing bibliographic links
+The paper proposes various strategies for sampling text data when performing
+automatic  sentence  classification  for  the  purpose  of  detecting  missing
+bibliographic links. We construct samples based on sentences as semantic units
+of the text and add their immediate context which consists of several neighboring
+sentences. We examine a number of sampling strategies that differ in context size
+and position.
+The experiment is carried out on the collection of STEM scientific papers.
+Including the context of sentences into samples improves the result of their
+classification. We automatically determine the optimal sampling strategy for a
+given text collection by implementing an ensemble voting when classifying the
+same data sampled in different ways. Sampling strategy taking into account the
+sentence context with hard voting procedure leads to the classification accuracy
+of 98% (F1-score). This method of detecting missing bibliographic links can be
+used in recommendation engines of applied intelligent information systems.
+Keywords: text sampling, sampling strategy, citation analysis, bibliographic link
+prediction, sentence classification.
+1. Introduction
+Scientific research is impossible without correlating the results obtained with the work
+of other scientists. Other works should be mentioned by inserting bibliographic links in
+the article. Experts in scientometrics rationalize the need to establish such links between
+studies and formulate various citation theories.
+The normative theory of citation, which draws on the principles of scientific
+ethics formulated by Merton (1973), assumes that references in scientific papers are
+made in order to indicate the works that are the basis for research or topically related,
+describe the research methods used and are necessary to discuss the results. According
+to the reflexive theory, links between scientific works indicate the state of science and
+help to create its formalized representation, e.g. maps of science (Akoev et al. 2014).
+
+The presence of a citation
+Thesentencecontains
+There is no citation marker
+marker in the sentence
+a citation marker
+in the sentence
+does not matter
+Positive sample
+Negative sample
+S;Bag of Words S;
+Estimators S;
+Documento
+Sentence
+OH
+Sampling strategies
+BoW1
+Ensemblevoting Si
+Estimator
+Sentence
+Si-n
+P1(S)
+Sentence
+BoW2
+P2(S)
+-P(S)→
+S;
+Si-n
+Estimator2
+Sentence
+P3(S)
+Si+m
+888
+1+m
+Sentence
+BoW3
+Estimator3
+Documentn
+Sentence
+SentenceStrategy#0
+Strategy #2
+Strategy #3
+0.016
+301.5 ± 154.0
+105.5 ± 39.2
+157.0 ± 49.5
+0.006
+273.9 ± 170.8
+0.0200
+103.0 ± 57.2
+0.014
+155.1 ± 76.1
+0.0175
+0.005
+0.012
+0.0150
+0.004
+0.010 
+0.0125
+0.008
+E00'0
+0.0100
+0.0075
+0.006
+0.002
+0.0050 -
+0.004
+0.001
+0.0025
+0.002
++000°0
+0.0000
+0.000 +
+100
+200
+300
+400
+500
+0
+100
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+300
+400
+500
+100
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+300
+400
+500
+Strategy #5
+Strategy #7
+Strategy #9
+0.010 
+260.6 ± 74.9
+362.4 ± 92.6
+465.4 ± 113.2
+259.7 ± 105.7
+0.007
+364.4 ± 127.5
+0.006
+469.7 ± 149.2
+800°0
+0.006
+0.005
+0.005
+0.006
+0.004
+0.004
+E00'0
+0.004
+E000
+0.002
+0.002 
+0.002
+0.001 
+0.001
+000°0
+0.000 +
+0.000 +
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+0
+100
+200
+400
+500DependenceofFl-score(weighted)fromnumberofestimators
+0.980
+0.975
+0.970
+weighted
+0.965
+a
+0.960
+-scor
+0.955
+0.950
+0.945
+Hard Voting
+Soft Voting
++
+0.940
+4
+5
+6
+7
+8
+9
+10
+NumberofestimatorsThus, the beneficiary of scientific citation correctness is the entire scientific
+community, both researchers who create articles on their results and administrators who
+monitor achievements in various scientific fields. Mentioning relevant and remarkable
+results  of other scientists  is  one of the basic requirements  in the construction  of
+scientific texts, in particular from the point of view of the editors of scientific journals.
+These requirements are noted in academic writing guidelines (Emerson et al. 2005;
+Gray et al. 2008; Pears and Shields 2019) and are confirmed in practice, for example, by
+the results of studies of publication activity in top-rated international journals (Arsyad et
+al. 2020).
+Authors of scientific papers choose the sources for citation and positions for the
+links by themselves and at present, this process is not automated. In this work, we
+investigate the possibility of creating a recommendation algorithm that allows one to
+find missing bibliographic references in a scientific article, that is, to identify those text
+fragments where it is necessary to mention another research work. For this purpose, we
+estimate  the  probability  of  link  presence  in  fragments  of  the  text  using a semi-
+supervised machine learning approach. The formal statement of the problem under
+consideration is the following: it is required to automatically find in the text of a
+scientific article those fragments (sentences) where the link is absent, but necessary,
+using a set of labeled fragments with and without links as training data.
+The task of classifying text fragments in relation to the presence of the links in
+them is methodologically similar to the task of Sentiment Analysis, in which texts are
+automatically classified, mainly as positive and negative, according to their emotional
+characteristics.  In  addition  to  dividing  fragments  into  positive  and  negative,  the
+sentiment analysis approach is used to distinguish other classes, including citation
+significance detection (Aljuaid et al. 2021; Prester et al. 2021; Varanasi et al. 2021;
+
+The presence of a citation
+Thesentencecontains
+There is no citation marker
+marker in the sentence
+a citation marker
+in the sentence
+does not matter
+Positive sample
+Negative sample
+S;Bag of Words S;
+Estimators S;
+Documento
+Sentence
+OH
+Sampling strategies
+BoW1
+Ensemblevoting Si
+Estimator
+Sentence
+Si-n
+P1(S)
+Sentence
+BoW2
+P2(S)
+-P(S)→
+S;
+Si-n
+Estimator2
+Sentence
+P3(S)
+Si+m
+888
+1+m
+Sentence
+BoW3
+Estimator3
+Documentn
+Sentence
+SentenceStrategy#0
+Strategy #2
+Strategy #3
+0.016
+301.5 ± 154.0
+105.5 ± 39.2
+157.0 ± 49.5
+0.006
+273.9 ± 170.8
+0.0200
+103.0 ± 57.2
+0.014
+155.1 ± 76.1
+0.0175
+0.005
+0.012
+0.0150
+0.004
+0.010 
+0.0125
+0.008
+E00'0
+0.0100
+0.0075
+0.006
+0.002
+0.0050 -
+0.004
+0.001
+0.0025
+0.002
++000°0
+0.0000
+0.000 +
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500
+Strategy #5
+Strategy #7
+Strategy #9
+0.010 
+260.6 ± 74.9
+362.4 ± 92.6
+465.4 ± 113.2
+259.7 ± 105.7
+0.007
+364.4 ± 127.5
+0.006
+469.7 ± 149.2
+800°0
+0.006
+0.005
+0.005
+0.006
+0.004
+0.004
+E00'0
+0.004
+E000
+0.002
+0.002 
+0.002
+0.001 
+0.001
+000°0
+0.000 +
+0.000 +
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+0
+100
+200
+400
+500DependenceofFl-score(weighted)fromnumberofestimators
+0.980
+0.975
+0.970
+weighted
+0.965
+a
+0.960
+-scor
+0.955
+0.950
+0.945
+Hard Voting
+Soft Voting
++
+0.940
+4
+5
+6
+7
+8
+9
+10
+NumberofestimatorsFärber and Ashwath 2019). The problem of identifying missing or unnecessary links in
+the text resembles sentiment analysis and the sought-for sentiment here is the author’s
+need to confirm the formulated statement.
+Another  close  line  of  research  is  Named  Entity  Recognition  (NER)  using
+prediction by classifiers. A similar problem is considered in (Fu et al. 2021), where
+NER problem is solved in the Span Prediction approach. NER can be performed in two
+stages:  identifying  fragments  with  a  high  probability  of  containing  entities,  and
+determination of the exact positions of these entities (Ziyadi et al. 2020; Li 2021). Some
+methods of NER also take into account the context of entities, both local and global, or
+external (Wang et al. 2021).
+The task of sentence classification accounting for their nearest context has been
+discussed in a number of studies. Fiok et al. (2020) used contextualized embeddings
+created by language models, which high quality comes at the price of speed. Glazkova
+(2020) studied topical classification and showed that models taking context as input
+performed better than context-free models. In those works, context size is determined
+once based on some bias and may not be optimal for a certain text corpus. 
+The method  introduced  in  this  work also can  be considered  as  kind  of a
+resampling technique. Until now resampling has been used mainly for the purpose of
+balancing the class distribution in training datasets in order to improve the accuracy of
+class prediction, which is negatively affected by imbalanced data. Resampling methods
+are  classified  into  three  types,  namely  undersampling,  oversampling  and  hybrid
+techniques. Undersampling eliminates some data of a majority class (see, e.g., Maya
+and Jayasudha 2017; Akkasi et al. 2017), while oversampling either replicates existing
+instances of a minority class or creates new ones (Luo et al. 2019; Li et al. 2018;
+
+The presence of a citation
+Thesentencecontains
+There is no citation marker
+marker in the sentence
+a citation marker
+in the sentence
+does not matter
+Positive sample
+Negative sample
+S;Bag of Words S;
+Estimators S;
+Documento
+Sentence
+OH
+Sampling strategies
+BoW1
+Ensemblevoting Si
+Estimator
+Sentence
+Si-n
+P1(S)
+Sentence
+BoW2
+P2(S)
+-P(S)→
+S;
+Si-n
+Estimator2
+Sentence
+P3(S)
+Si+m
+888
+1+m
+Sentence
+BoW3
+Estimator3
+Documentn
+Sentence
+SentenceStrategy#0
+Strategy #2
+Strategy #3
+0.016
+301.5 ± 154.0
+105.5 ± 39.2
+157.0 ± 49.5
+0.006
+273.9 ± 170.8
+0.0200
+103.0 ± 57.2
+0.014
+155.1 ± 76.1
+0.0175
+0.005
+0.012
+0.0150
+0.004
+0.010 
+0.0125
+0.008
+E00'0
+0.0100
+0.0075
+0.006
+0.002
+0.0050 -
+0.004
+0.001
+0.0025
+0.002
++000°0
+0.0000
+0.000 +
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500
+Strategy #5
+Strategy #7
+Strategy #9
+0.010 
+260.6 ± 74.9
+362.4 ± 92.6
+465.4 ± 113.2
+259.7 ± 105.7
+0.007
+364.4 ± 127.5
+0.006
+469.7 ± 149.2
+800°0
+0.006
+0.005
+0.005
+0.006
+0.004
+0.004
+E00'0
+0.004
+E000
+0.002
+0.002 
+0.002
+0.001 
+0.001
+000°0
+0.000 +
+0.000 +
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+0
+100
+200
+400
+500DependenceofFl-score(weighted)fromnumberofestimators
+0.980
+0.975
+0.970
+weighted
+0.965
+a
+0.960
+-scor
+0.955
+0.950
+0.945
+Hard Voting
+Soft Voting
++
+0.940
+4
+5
+6
+7
+8
+9
+10
+NumberofestimatorsChawla et al. 2002), and hybrid resampling techniques aim to combine benefits of both
+(Taha et al. 2021).
+Local and global contexts are taken into account by modern neural network
+architectures for text analysis. Since text is a unidirectional list of terms, context is
+usually  understood  as  some  neighboring  words  before  or  after  the  term  under
+consideration  (Gallant  1991;  Huang  2012).  In  convolutional  neural  networks, an
+increase in the size of context leads to a significant increase in the dimension of tensors
+and, consequently, in the number of parameters of a model, which requires an increase
+in the size of collections. In deep learning models known as transformers, context is
+explored by means of an attention mechanism, and the local context is combined with
+the broader context (BERT [Devlin et al. 2018], GPT-3 [Brown et al. 2020]).
+It is important that all of the above algorithms do not take into account the
+natural structural units of texts (i.e. sentences and paragraphs) since these algorithms are
+adjusted to a certain size of the context, which is a fixed number of words, while the
+size of sentences and paragraphs varies.
+2. Methods
+The task of determining missing links is formalized as finding text fragments where the
+link is absent, but necessary, or, conversely, is present, but not needed.
+We solve the problem using automatic binary classification with two classes
+namely positive and negative. For each fragment of a scientific article, our algorithm
+determines the probability of a bibliographic link in it. A collection of text documents is
+given such that each document consists of fragments. A fragment is a sequence of
+words (terms) of different lengths. Fragments can overlap each other and vary in size.
+Each fragment is a sample and is labeled as one of the two possible classes with class
+labels: positive or negative. The class label corresponds to whether or not the given
+
+The presence of a citation
+Thesentencecontains
+There is no citation marker
+marker in the sentence
+a citation marker
+in the sentence
+does not matter
+Positive sample
+Negative sample
+S;Bag of Words S;
+Estimators S;
+Documento
+Sentence
+OH
+Sampling strategies
+BoW1
+Ensemblevoting Si
+Estimator
+Sentence
+Si-n
+P1(S)
+Sentence
+BoW2
+P2(S)
+-P(S)→
+S;
+Si-n
+Estimator2
+Sentence
+P3(S)
+Si+m
+888
+1+m
+Sentence
+BoW3
+Estimator3
+Documentn
+Sentence
+SentenceStrategy#0
+Strategy #2
+Strategy #3
+0.016
+301.5 ± 154.0
+105.5 ± 39.2
+157.0 ± 49.5
+0.006
+273.9 ± 170.8
+0.0200
+103.0 ± 57.2
+0.014
+155.1 ± 76.1
+0.0175
+0.005
+0.012
+0.0150
+0.004
+0.010 
+0.0125
+0.008
+E00'0
+0.0100
+0.0075
+0.006
+0.002
+0.0050 -
+0.004
+0.001
+0.0025
+0.002
++000°0
+0.0000
+0.000 +
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500
+Strategy #5
+Strategy #7
+Strategy #9
+0.010 
+260.6 ± 74.9
+362.4 ± 92.6
+465.4 ± 113.2
+259.7 ± 105.7
+0.007
+364.4 ± 127.5
+0.006
+469.7 ± 149.2
+800°0
+0.006
+0.005
+0.005
+0.006
+0.004
+0.004
+E00'0
+0.004
+E000
+0.002
+0.002 
+0.002
+0.001 
+0.001
+000°0
+0.000 +
+0.000 +
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+0
+100
+200
+400
+500DependenceofFl-score(weighted)fromnumberofestimators
+0.980
+0.975
+0.970
+weighted
+0.965
+a
+0.960
+-scor
+0.955
+0.950
+0.945
+Hard Voting
+Soft Voting
++
+0.940
+4
+5
+6
+7
+8
+9
+10
+Numberofestimatorsfragment contains a bibliographic link. The task of this study is to find a strategy for the
+construction samples of fragments, which gives the highest accuracy in determining the
+labels of the class by a certain classifier.
+The hypothesis of our study is the following: text sampling strategies that take
+into account the context increase the accuracy of sentence classification used to predict
+missing bibliographic links in scientific articles.
+We suggest that a positive sample consists of a bibliographic link surrounded by
+its  context  from  the  original  text,  and  a  negative  sample  is  a fragment  with  no
+bibliographic link in it. In order to avoid duplication of samples, we consider a sentence
+with two or more links only once. The context of the link is limited to the sentence
+containing it, or the context is extended and it includes neighboring sentences as well.
+The best option is when the boundaries of a link context coincide with the
+boundaries of the complete author’s statement to which this link belongs. In this case, a
+semantic unit of text can be either one or several sentences, which makes it difficult to
+set  the  size  of  the  context.  Nevertheless,  to  approach  the  specified  goal in  the
+proposed algorithm, as a context we consider a fragment which size is determined by
+the number of sentences, and not words, unlike neural network algorithms. Thus, in our
+algorithm context is formed on the basis of natural structural units of text.
+The feature space is constructed automatically based on vocabulary statistics
+within the Bag-of-Words model (BoW). The vocabulary of the model includes words
+and  all  the  original  punctuation  marks  and  typographical  symbols.  As  additional
+features, we consider named entities.
+Algorithm
+The algorithm consists of the following stages.
+
+The presence of a citation
+Thesentencecontains
+There is no citation marker
+marker in the sentence
+a citation marker
+in the sentence
+does not matter
+Positive sample
+Negative sample
+S;Bag of Words S;
+Estimators S;
+Documento
+Sentence
+OH
+Sampling strategies
+BoW1
+Ensemblevoting Si
+Estimator
+Sentence
+Si-n
+P1(S)
+Sentence
+BoW2
+P2(S)
+-P(S)→
+S;
+Si-n
+Estimator2
+Sentence
+P3(S)
+Si+m
+888
+1+m
+Sentence
+BoW3
+Estimator3
+Documentn
+Sentence
+SentenceStrategy#0
+Strategy #2
+Strategy #3
+0.016
+301.5 ± 154.0
+105.5 ± 39.2
+157.0 ± 49.5
+0.006
+273.9 ± 170.8
+0.0200
+103.0 ± 57.2
+0.014
+155.1 ± 76.1
+0.0175
+0.005
+0.012
+0.0150
+0.004
+0.010 
+0.0125
+0.008
+E00'0
+0.0100
+0.0075
+0.006
+0.002
+0.0050 -
+0.004
+0.001
+0.0025
+0.002
++000°0
+0.0000
+0.000 +
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500
+Strategy #5
+Strategy #7
+Strategy #9
+0.010 
+260.6 ± 74.9
+362.4 ± 92.6
+465.4 ± 113.2
+259.7 ± 105.7
+0.007
+364.4 ± 127.5
+0.006
+469.7 ± 149.2
+800°0
+0.006
+0.005
+0.005
+0.006
+0.004
+0.004
+E00'0
+0.004
+E000
+0.002
+0.002 
+0.002
+0.001 
+0.001
+000°0
+0.000 +
+0.000 +
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+0
+100
+200
+400
+500DependenceofFl-score(weighted)fromnumberofestimators
+0.980
+0.975
+0.970
+weighted
+0.965
+a
+0.960
+-scor
+0.955
+0.950
+0.945
+Hard Voting
+Soft Voting
++
+0.940
+4
+5
+6
+7
+8
+9
+10
+Numberofestimators1. Text preprocessing
+
+Text cleaning: removal of service characters (tabulation, line feed, etc.), words
+(journal titles, ISBN, etc.), and sections (funding, reference);
+
+Tokenization: 
+o
+splitting text into sentences;
+o
+terms normalization.
+2. Data labeling
+
+For each document (journal article), beginning and end marks are added;
+
+For each sentence:
+o
+if the  sentence  contains  a bibliographic  link (citation  marker),  it  is
+labeled as belonging to the class "With links";
+o
+if there is no citation marker in the sentence, it gets "Without links" class
+label. 
+o
+After labeling sentences, citation markers are removed.
+3. Named Entities processing 
+
+Detection of named entities in the text;
+
+Replacement of named entities with special marks.
+4. Construction of samples
+Samples are constructed in different ways depending on the class (positive or negative):
+
+To form a positive sample, we take a sentence “With links” and add n previous
+sentences and  m subsequent sentences, all sentences are taken in the original
+order.
+
+The presence of a citation
+Thesentencecontains
+There is no citation marker
+marker in the sentence
+a citation marker
+in the sentence
+does not matter
+Positive sample
+Negative sample
+S;Bag of Words S;
+Estimators S;
+Documento
+Sentence
+OH
+Sampling strategies
+BoW1
+Ensemblevoting Si
+Estimator
+Sentence
+Si-n
+P1(S)
+Sentence
+BoW2
+P2(S)
+-P(S)→
+S;
+Si-n
+Estimator2
+Sentence
+P3(S)
+Si+m
+888
+1+m
+Sentence
+BoW3
+Estimator3
+Documentn
+Sentence
+SentenceStrategy#0
+Strategy #2
+Strategy #3
+0.016
+301.5 ± 154.0
+105.5 ± 39.2
+157.0 ± 49.5
+0.006
+273.9 ± 170.8
+0.0200
+103.0 ± 57.2
+0.014
+155.1 ± 76.1
+0.0175
+0.005
+0.012
+0.0150
+0.004
+0.010 
+0.0125
+0.008
+E00'0
+0.0100
+0.0075
+0.006
+0.002
+0.0050 -
+0.004
+0.001
+0.0025
+0.002
++000°0
+0.0000
+0.000 +
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500
+Strategy #5
+Strategy #7
+Strategy #9
+0.010 
+260.6 ± 74.9
+362.4 ± 92.6
+465.4 ± 113.2
+259.7 ± 105.7
+0.007
+364.4 ± 127.5
+0.006
+469.7 ± 149.2
+800°0
+0.006
+0.005
+0.005
+0.006
+0.004
+0.004
+E00'0
+0.004
+E000
+0.002
+0.002 
+0.002
+0.001 
+0.001
+000°0
+0.000 +
+0.000 +
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+0
+100
+200
+400
+500DependenceofFl-score(weighted)fromnumberofestimators
+0.980
+0.975
+0.970
+weighted
+0.965
+a
+0.960
+-scor
+0.955
+0.950
+0.945
+Hard Voting
+Soft Voting
++
+0.940
+4
+5
+6
+7
+8
+9
+10
+Numberofestimators
+A negative sample is constructed of k sentences "Without links" in a row in the
+original text (adjacent sentences), where k = n+1+m.
+The visualization of the contents of the samples is shown in Figure 1.
+Figure 1. Construction of samples.
+5. Classification of samples
+
+Balancing  the  class  distribution  in  the  training  set  is  done  by  random
+undersampling.
+
+For each sample, a vector model is built using count vectorizer as the fastest and
+most computationally effective text representation. 
+
+Vectorized set of samples is processed using a classifier.
+6. Optimal sampling strategy determination
+Further, an ensemble method is used to automatically determine the optimal sampling
+strategy. We give the same data sampled in different ways to the estimators of the same
+type and implement a voting procedure.
+The  flowchart  of  the  whole  algorithm  is  shown  in  Figure  2.  Each  BoWj
+corresponds to one sampling strategy, and for each sampling strategy, we run its own
+estimator.  All  the  estimators  implement  the  same  classification  method  but  take
+different types of samples as input data.
+
+The presence of a citation
+Thesentencecontains
+There is no citation marker
+marker in the sentence
+a citation marker
+in the sentence
+does not matter
+Positive sample
+Negative sample
+S;Bag of Words S;
+Estimators S;
+Documento
+Sentence
+OH
+Sampling strategies
+BoW1
+Ensemblevoting Si
+Estimator
+Sentence
+Si-n
+P1(S)
+Sentence
+BoW2
+P2(S)
+-P(S)→
+S;
+Si-n
+Estimator2
+Sentence
+P3(S)
+Si+m
+888
+1+m
+Sentence
+BoW3
+Estimator3
+Documentn
+Sentence
+SentenceStrategy#0
+Strategy #2
+Strategy #3
+0.016
+301.5 ± 154.0
+105.5 ± 39.2
+157.0 ± 49.5
+0.006
+273.9 ± 170.8
+0.0200
+103.0 ± 57.2
+0.014
+155.1 ± 76.1
+0.0175
+0.005
+0.012
+0.0150
+0.004
+0.010 
+0.0125
+0.008
+E00'0
+0.0100
+0.0075
+0.006
+0.002
+0.0050 -
+0.004
+0.001
+0.0025
+0.002
++000°0
+0.0000
+0.000 +
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500
+Strategy #5
+Strategy #7
+Strategy #9
+0.010 
+260.6 ± 74.9
+362.4 ± 92.6
+465.4 ± 113.2
+259.7 ± 105.7
+0.007
+364.4 ± 127.5
+0.006
+469.7 ± 149.2
+800°0
+0.006
+0.005
+0.005
+0.006
+0.004
+0.004
+E00'0
+0.004
+E000
+0.002
+0.002 
+0.002
+0.001 
+0.001
+000°0
+0.000 +
+0.000 +
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+0
+100
+200
+400
+500DependenceofFl-score(weighted)fromnumberofestimators
+0.980
+0.975
+0.970
+weighted
+0.965
+a
+0.960
+-scor
+0.955
+0.950
+0.945
+Hard Voting
+Soft Voting
++
+0.940
+4
+5
+6
+7
+8
+9
+10
+NumberofestimatorsFigure 2. The algorithm flowchart for classifying sentences into the classes "With links"
+and "Without links" using various sampling strategies (for n=m=1).
+3. Experiment
+To test the hypothesis experimentally, we took the dataset of STEM journal articles,
+collected from scientific repository arXiv.org by Cohan et al. (2018). Documents of this
+dataset contain only texts, while figures and tables are removed. Math formulas and
+citation  markers  are  replaced  with  special  tokens  @xmath<number> and  @xcite.
+Documents contain only the sections up to the conclusion section and all sections after
+the conclusion are removed.
+The size of the dataset is the following: the number of documents – 215K, the
+average document length – 4938 words, the average summary length – 220 words.    
+The files are in jsonlines format where each line is a json object corresponding
+to one scientific article. Each line contains an abstract, sections and a body of the article,
+and all these texts are sentence tokenized. 
+In our experiment we consider sentences that are more than 30 words long. With
+this restriction we got the set of 458774 sentences in total. 
+
+The presence of a citation
+Thesentencecontains
+There is no citation marker
+marker in the sentence
+a citation marker
+in the sentence
+does not matter
+Positive sample
+Negative sample
+S;Bag of Words S;
+Estimators S;
+Documento
+Sentence
+OH
+Sampling strategies
+BoW1
+Ensemblevoting Si
+Estimator
+Sentence
+Si-n
+P1(S)
+Sentence
+BoW2
+P2(S)
+-P(S)→
+S;
+Si-n
+Estimator2
+Sentence
+P3(S)
+Si+m
+888
+1+m
+Sentence
+BoW3
+Estimator3
+Documentn
+Sentence
+SentenceStrategy#0
+Strategy #2
+Strategy #3
+0.016
+301.5 ± 154.0
+105.5 ± 39.2
+157.0 ± 49.5
+0.006
+273.9 ± 170.8
+0.0200
+103.0 ± 57.2
+0.014
+155.1 ± 76.1
+0.0175
+0.005
+0.012
+0.0150
+0.004
+0.010 
+0.0125
+0.008
+E00'0
+0.0100
+0.0075
+0.006
+0.002
+0.0050 -
+0.004
+0.001
+0.0025
+0.002
++000°0
+0.0000
+0.000 +
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500
+Strategy #5
+Strategy #7
+Strategy #9
+0.010 
+260.6 ± 74.9
+362.4 ± 92.6
+465.4 ± 113.2
+259.7 ± 105.7
+0.007
+364.4 ± 127.5
+0.006
+469.7 ± 149.2
+800°0
+0.006
+0.005
+0.005
+0.006
+0.004
+0.004
+E00'0
+0.004
+E000
+0.002
+0.002 
+0.002
+0.001 
+0.001
+000°0
+0.000 +
+0.000 +
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+0
+100
+200
+400
+500DependenceofFl-score(weighted)fromnumberofestimators
+0.980
+0.975
+0.970
+weighted
+0.965
+a
+0.960
+-scor
+0.955
+0.950
+0.945
+Hard Voting
+Soft Voting
++
+0.940
+4
+5
+6
+7
+8
+9
+10
+NumberofestimatorsSentences containing citation markers @xcite are assigned to the positive class
+("With links"), and after that citation markers are removed. Sentences without citation
+markers  are labeled as  negative  ("Without  links"). The  ratio  of  classes  is:  24%  –
+positive sentences, 76% – negative sentences. This is assumed as sampling strategy #0,
+and the classification result on data sampled that way is considered as a baseline:
+classification accuracy with sampling strategy #0 measured by F1-score is 0.7866.
+After establishing the baseline, we test various strategies of data sampling in
+order to improve the classification accuracy. The main idea of sampling is to take into
+account  some  context  of  the  sentences  with  a  link.  Different  strategies
+of sampling assume various directions, positions, and sizes of the context, determined
+by a number of surrounding sentences. In different sampling strategies, each sentence [i]
+is included in different types (variants) of samples.
+In the experiment we test 10 strategies with the following parameters: n: [0, 1, 2
+3 4 5], m: [0, 1, 2, 3, 4], k: [1, 3]. All the sample types corresponding to the chosen
+sampling strategies are presented in Table 1.
+Table 1. Sampling strategies tested in the experiment.
+#
+Sampling strategy
+(sample
+ 
+constructing
+algorithm)
+Number of sentences per sample
+Positive
+Negative
+0
+Sentence [i]
+2640 
+12352
+1
+Sentences [i: i +2]
+2640 
+10639
+2
+Sentences [i -1: i+1]
+2640 
+10639
+3
+Sentences [i -1: i+2]
+2640 
+9376
+4
+Sentences [i -2: i+2]
+2640 
+8384
+5
+Sentences [i -3: i+2]
+2640 
+7574
+
+The presence of a citation
+Thesentencecontains
+There is no citation marker
+marker in the sentence
+a citation marker
+in the sentence
+does not matter
+Positive sample
+Negative sample
+S;Bag of Words S;
+Estimators S;
+Documento
+Sentence
+OH
+Sampling strategies
+BoW1
+Ensemblevoting Si
+Estimator
+Sentence
+Si-n
+P1(S)
+Sentence
+BoW2
+P2(S)
+-P(S)→
+S;
+Si-n
+Estimator2
+Sentence
+P3(S)
+Si+m
+888
+1+m
+Sentence
+BoW3
+Estimator3
+Documentn
+Sentence
+SentenceStrategy#0
+Strategy #2
+Strategy #3
+0.016
+301.5 ± 154.0
+105.5 ± 39.2
+157.0 ± 49.5
+0.006
+273.9 ± 170.8
+0.0200
+103.0 ± 57.2
+0.014
+155.1 ± 76.1
+0.0175
+0.005
+0.012
+0.0150
+0.004
+0.010 
+0.0125
+0.008
+E00'0
+0.0100
+0.0075
+0.006
+0.002
+0.0050 -
+0.004
+0.001
+0.0025
+0.002
++000°0
+0.0000
+0.000 +
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500
+Strategy #5
+Strategy #7
+Strategy #9
+0.010 
+260.6 ± 74.9
+362.4 ± 92.6
+465.4 ± 113.2
+259.7 ± 105.7
+0.007
+364.4 ± 127.5
+0.006
+469.7 ± 149.2
+800°0
+0.006
+0.005
+0.005
+0.006
+0.004
+0.004
+E00'0
+0.004
+E000
+0.002
+0.002 
+0.002
+0.001 
+0.001
+000°0
+0.000 +
+0.000 +
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+0
+100
+200
+400
+500DependenceofFl-score(weighted)fromnumberofestimators
+0.980
+0.975
+0.970
+weighted
+0.965
+a
+0.960
+-scor
+0.955
+0.950
+0.945
+Hard Voting
+Soft Voting
++
+0.940
+4
+5
+6
+7
+8
+9
+10
+Numberofestimators6
+Sentences [i -3: i+3]
+2640 
+6880
+7
+Sentences [i -4: i+3]
+2640 
+6287
+8
+Sentences [i -4: i+4]
+2640 
+5776
+9
+Sentences [i -5: i+4]
+2640 
+5326
+The distribution of the length (number of words) in positive and negative
+samples of different types is shown in Figure 3.
+Figure 3. The distribution of the length as a number of words in positive and negative
+samples of different types (‘salmon’ color refers to the positive class, ‘olive’ refers to
+the negative one).
+After equalizing classes by random undersampling the data is divided into
+training and test sets with the proportion parameter test_size=0.33.
+Vector representation is build using CountVectorizer method of the Scikit-learn
+library. The vocabulary includes unigrams and bigrams and is reduced by frequency
+with the parameters min_df=3, max_df=0.7
+
+The presence of a citation
+Thesentencecontains
+There is no citation marker
+marker in the sentence
+a citation marker
+in the sentence
+does not matter
+Positive sample
+Negative sample
+S;Bag of Words S;
+Estimators S;
+Documento
+Sentence
+OH
+Sampling strategies
+BoW1
+Ensemblevoting Si
+Estimator
+Sentence
+Si-n
+P1(S)
+Sentence
+BoW2
+P2(S)
+-P(S)→
+S;
+Si-n
+Estimator2
+Sentence
+P3(S)
+Si+m
+888
+1+m
+Sentence
+BoW3
+Estimator3
+Documentn
+Sentence
+SentenceStrategy#0
+Strategy #2
+Strategy #3
+0.016
+301.5 ± 154.0
+105.5 ± 39.2
+157.0 ± 49.5
+0.006
+273.9 ± 170.8
+0.0200
+103.0 ± 57.2
+0.014
+155.1 ± 76.1
+0.0175
+0.005
+0.012
+0.0150
+0.004
+0.010 
+0.0125
+0.008
+E00'0
+0.0100
+0.0075
+0.006
+0.002
+0.0050 -
+0.004
+0.001
+0.0025
+0.002
++000°0
+0.0000
+0.000 +
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500
+Strategy #5
+Strategy #7
+Strategy #9
+0.010 
+260.6 ± 74.9
+362.4 ± 92.6
+465.4 ± 113.2
+259.7 ± 105.7
+0.007
+364.4 ± 127.5
+0.006
+469.7 ± 149.2
+800°0
+0.006
+0.005
+0.005
+0.006
+0.004
+0.004
+E00'0
+0.004
+E000
+0.002
+0.002 
+0.002
+0.001 
+0.001
+000°0
+0.000 +
+0.000 +
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+0
+100
+200
+400
+500DependenceofFl-score(weighted)fromnumberofestimators
+0.980
+0.975
+0.970
+weighted
+0.965
+a
+0.960
+-scor
+0.955
+0.950
+0.945
+Hard Voting
+Soft Voting
++
+0.940
+4
+5
+6
+7
+8
+9
+10
+NumberofestimatorsFor  classification  we  use  a  neural  network  multilayer  perceptron  (the
+MLPClassifier  method  of  the  Scikit-learn  library). Classification  performance
+depending on the sampling strategy used is presented in Table 2. 
+Table 2. The result of sentence classification by the MLPClassifier depending on the
+sampling strategy (weighed average).
+Sampling strateg
+y
+F1
+Precision
+Recall
+0
+0.7866
+0.7866
+0.7866 
+1
+0.8882
+0.8881
+0.8881
+2
+0.8884
+0.8881
+0.8881
+3
+0.9214
+0.9214 
+0.9214
+4
+0.9444
+0.9443
+0.9443
+5
+0.9410
+0.9409 
+0.9409
+6
+ 0.9601
+0.9598
+0.9598
+7
+0.9640
+0.9639
+0.9639
+8
+0.9593
+0.9593
+0.9593    
+9
+ 0.9581 
+0.9581 
+0.9581
+It can be seen from the table that the best result is achieved using the sampling
+strategy  #7.  A  further  increase  in  the  number  of  sentences  per  sample  does  not
+significantly improve the accuracy, since it tends to the asymptote.
+For each sentence we compare the result of classification obtained with all the
+sampling strategies and further improve it by voting procedure, both soft and hard. For
+soft voting, we set the threshold value 0.5 for the mean predicted probability. For hard
+voting, we summarize all the predicted probabilities and compare the sum with the
+
+The presence of a citation
+Thesentencecontains
+There is no citation marker
+marker in the sentence
+a citation marker
+in the sentence
+does not matter
+Positive sample
+Negative sample
+S;Bag of Words S;
+Estimators S;
+Documento
+Sentence
+OH
+Sampling strategies
+BoW1
+Ensemblevoting Si
+Estimator
+Sentence
+Si-n
+P1(S)
+Sentence
+BoW2
+P2(S)
+-P(S)→
+S;
+Si-n
+Estimator2
+Sentence
+P3(S)
+Si+m
+888
+1+m
+Sentence
+BoW3
+Estimator3
+Documentn
+Sentence
+SentenceStrategy#0
+Strategy #2
+Strategy #3
+0.016
+301.5 ± 154.0
+105.5 ± 39.2
+157.0 ± 49.5
+0.006
+273.9 ± 170.8
+0.0200
+103.0 ± 57.2
+0.014
+155.1 ± 76.1
+0.0175
+0.005
+0.012
+0.0150
+0.004
+0.010 
+0.0125
+0.008
+E00'0
+0.0100
+0.0075
+0.006
+0.002
+0.0050 -
+0.004
+0.001
+0.0025
+0.002
++000°0
+0.0000
+0.000 +
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500
+Strategy #5
+Strategy #7
+Strategy #9
+0.010 
+260.6 ± 74.9
+362.4 ± 92.6
+465.4 ± 113.2
+259.7 ± 105.7
+0.007
+364.4 ± 127.5
+0.006
+469.7 ± 149.2
+800°0
+0.006
+0.005
+0.005
+0.006
+0.004
+0.004
+E00'0
+0.004
+E000
+0.002
+0.002 
+0.002
+0.001 
+0.001
+000°0
+0.000 +
+0.000 +
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+0
+100
+200
+400
+500DependenceofFl-score(weighted)fromnumberofestimators
+0.980
+0.975
+0.970
+weighted
+0.965
+a
+0.960
+-scor
+0.955
+0.950
+0.945
+Hard Voting
+Soft Voting
++
+0.940
+4
+5
+6
+7
+8
+9
+10
+Numberofestimatorsthreshold value 3. We test different numbers of estimators. To form combinations of
+sampling strategies we start from the group of estimators #7, #8, and #9 and then add
+estimators one by one in reverse order. The voting results depending on the number of
+estimators considered are shown in Figure 4.
+Figure 4. The result of classification with hard and soft voting depending on the number
+of estimators included.  
+The graph demonstrates that the voting procedure further improves classification
+performance and increases F1-score by 1,5%. With all the combinations of estimators
+tested the result is consistently high, but the best result (>98%) is achieved with hard
+voting of 7,8 or 10 estimators classifying long samples. 
+4. Result
+The  formulated  research  hypothesis  has  been  confirmed  experimentally.  We  have
+
+The presence of a citation
+Thesentencecontains
+There is no citation marker
+marker in the sentence
+a citation marker
+in the sentence
+does not matter
+Positive sample
+Negative sample
+S;Bag of Words S;
+Estimators S;
+Documento
+Sentence
+OH
+Sampling strategies
+BoW1
+Ensemblevoting Si
+Estimator
+Sentence
+Si-n
+P1(S)
+Sentence
+BoW2
+P2(S)
+-P(S)→
+S;
+Si-n
+Estimator2
+Sentence
+P3(S)
+Si+m
+888
+1+m
+Sentence
+BoW3
+Estimator3
+Documentn
+Sentence
+SentenceStrategy#0
+Strategy #2
+Strategy #3
+0.016
+301.5 ± 154.0
+105.5 ± 39.2
+157.0 ± 49.5
+0.006
+273.9 ± 170.8
+0.0200
+103.0 ± 57.2
+0.014
+155.1 ± 76.1
+0.0175
+0.005
+0.012
+0.0150
+0.004
+0.010 
+0.0125
+0.008
+E00'0
+0.0100
+0.0075
+0.006
+0.002
+0.0050 -
+0.004
+0.001
+0.0025
+0.002
++000°0
+0.0000
+0.000 +
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500
+Strategy #5
+Strategy #7
+Strategy #9
+0.010 
+260.6 ± 74.9
+362.4 ± 92.6
+465.4 ± 113.2
+259.7 ± 105.7
+0.007
+364.4 ± 127.5
+0.006
+469.7 ± 149.2
+800°0
+0.006
+0.005
+0.005
+0.006
+0.004
+0.004
+E00'0
+0.004
+E000
+0.002
+0.002 
+0.002
+0.001 
+0.001
+000°0
+0.000 +
+0.000 +
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+0
+100
+200
+400
+500DependenceofFl-score(weighted)fromnumberofestimators
+0.980
+0.975
+0.970
+weighted
+0.965
+a
+0.960
+-scor
+0.955
+0.950
+0.945
+Hard Voting
+Soft Voting
++
+0.940
+4
+5
+6
+7
+8
+9
+10
+Numberofestimatorsshown that the choice of the sampling strategy affects the result of text classification. 
+The  baseline  is  established  with  sampling  strategy  #0.  In  this  case,  the
+classification performance measured using the F1-score is only 79 %, which is not
+sufficient for practical use in industrial information systems. 
+The improvement is achieved due to the data sampling strategy which assumes
+automatic determination of the optimal sample type. That is provided by applying the
+voting procedure to the decisions made by different estimators. The proposed algorithm
+shows 98% accuracy (F1-score), which is comparable to the state-of-the-art results for
+NER using automatic classification and other text classification tasks. It is important
+that  the  proposed  algorithm  provides  high  accuracy  but  doesn’t  require  huge
+computational resources to be implemented.
+5. Conclusion
+The paper proposes a new method of determining the probability of a bibliographic link
+in fragments of a scientific article. The approach assumes sentence classification with
+ensemble voting, in which different data sampling strategies correspond to estimators
+implementing the same classification method. The problem statement made by the
+authors is close to well-studied areas NER and sentiment analysis but is new from
+the real application point of view.
+The main innovation of the proposed method is finding the link context that
+maximally affects the probability of detecting a missing bibliographic link in a sentence.
+In  the  proposed  algorithm,  the  best  size  and  position  of  context  are  determined
+automatically. The size is based on the boundaries of semantic units of the text and is
+measured by the number of sentences, not words, thus we utilize the fact that a sentence
+is a more semantically capacious (meaningful) unit than a word. Most existing text
+classification methods do not assume fragment context as significantly important, but
+
+The presence of a citation
+Thesentencecontains
+There is no citation marker
+marker in the sentence
+a citation marker
+in the sentence
+does not matter
+Positive sample
+Negative sample
+S;Bag of Words S;
+Estimators S;
+Documento
+Sentence
+OH
+Sampling strategies
+BoW1
+Ensemblevoting Si
+Estimator
+Sentence
+Si-n
+P1(S)
+Sentence
+BoW2
+P2(S)
+-P(S)→
+S;
+Si-n
+Estimator2
+Sentence
+P3(S)
+Si+m
+888
+1+m
+Sentence
+BoW3
+Estimator3
+Documentn
+Sentence
+SentenceStrategy#0
+Strategy #2
+Strategy #3
+0.016
+301.5 ± 154.0
+105.5 ± 39.2
+157.0 ± 49.5
+0.006
+273.9 ± 170.8
+0.0200
+103.0 ± 57.2
+0.014
+155.1 ± 76.1
+0.0175
+0.005
+0.012
+0.0150
+0.004
+0.010 
+0.0125
+0.008
+E00'0
+0.0100
+0.0075
+0.006
+0.002
+0.0050 -
+0.004
+0.001
+0.0025
+0.002
++000°0
+0.0000
+0.000 +
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500
+Strategy #5
+Strategy #7
+Strategy #9
+0.010 
+260.6 ± 74.9
+362.4 ± 92.6
+465.4 ± 113.2
+259.7 ± 105.7
+0.007
+364.4 ± 127.5
+0.006
+469.7 ± 149.2
+800°0
+0.006
+0.005
+0.005
+0.006
+0.004
+0.004
+E00'0
+0.004
+E000
+0.002
+0.002 
+0.002
+0.001 
+0.001
+000°0
+0.000 +
+0.000 +
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+0
+100
+200
+400
+500DependenceofFl-score(weighted)fromnumberofestimators
+0.980
+0.975
+0.970
+weighted
+0.965
+a
+0.960
+-scor
+0.955
+0.950
+0.945
+Hard Voting
+Soft Voting
++
+0.940
+4
+5
+6
+7
+8
+9
+10
+Numberofestimatorsthis  study  shows  the  critical  importance  of  taking  it  into  account.  The
+considerable impact  of  the  context  on  the  classification  performance  demonstrates
+that semantics related to a bibliographic link can be localized in fragments of different
+lengths.
+The accuracy of the proposed algorithm reaches 98% (F1-score). It is important
+to note the high computational efficiency of the described method in comparison with
+convolutional artificial neural networks. This advantage is achieved due to the bigger
+size of samples. The investigated approach to text analysis expands the principle of the
+attention mechanism aimed at training a language model to understand the impact of
+global and local context. Automatic determination of the context boundaries correlates
+with the idea of automatic selection of significant features in artificial neural networks.
+The  proposed  method  can  be  used  in  recommendation  engines  of  applied
+intelligent information systems, including assistance for constructing documents and
+composing texts with probable links to other documents, or help in checking the
+document correctness. Such functions are useful in many fields e.g. science, law, or
+journalism, where documents contain statements that should be confirmed by references
+to legal acts or other sources.
+In accordance with the company's policy, we do not publish the source code.
+References
+1. Akkasi, A., E. Varoğlu, and N. Dimililer. 2018. “Balanced undersampling: a novel 
+sentence-based undersampling method to improve recognition of named entities 
+in chemical and biomedical text.” Applied Intelligence 48(8): 1965–1978. doi: 
+https://doi.org/10.1007/s10489-017-0920-5
+2. Akoev, M., V. Markusova, O. Moskaleva, and V. Pislyakov. 
+2014. Rukovodstvo po Naukometrii: 
+Indikatori Razvitiia Naukii Tehnologii [Handbook for Scientometrics: Indicators
+
+The presence of a citation
+Thesentencecontains
+There is no citation marker
+marker in the sentence
+a citation marker
+in the sentence
+does not matter
+Positive sample
+Negative sample
+S;Bag of Words S;
+Estimators S;
+Documento
+Sentence
+OH
+Sampling strategies
+BoW1
+Ensemblevoting Si
+Estimator
+Sentence
+Si-n
+P1(S)
+Sentence
+BoW2
+P2(S)
+-P(S)→
+S;
+Si-n
+Estimator2
+Sentence
+P3(S)
+Si+m
+888
+1+m
+Sentence
+BoW3
+Estimator3
+Documentn
+Sentence
+SentenceStrategy#0
+Strategy #2
+Strategy #3
+0.016
+301.5 ± 154.0
+105.5 ± 39.2
+157.0 ± 49.5
+0.006
+273.9 ± 170.8
+0.0200
+103.0 ± 57.2
+0.014
+155.1 ± 76.1
+0.0175
+0.005
+0.012
+0.0150
+0.004
+0.010 
+0.0125
+0.008
+E00'0
+0.0100
+0.0075
+0.006
+0.002
+0.0050 -
+0.004
+0.001
+0.0025
+0.002
++000°0
+0.0000
+0.000 +
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500
+Strategy #5
+Strategy #7
+Strategy #9
+0.010 
+260.6 ± 74.9
+362.4 ± 92.6
+465.4 ± 113.2
+259.7 ± 105.7
+0.007
+364.4 ± 127.5
+0.006
+469.7 ± 149.2
+800°0
+0.006
+0.005
+0.005
+0.006
+0.004
+0.004
+E00'0
+0.004
+E000
+0.002
+0.002 
+0.002
+0.001 
+0.001
+000°0
+0.000 +
+0.000 +
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+0
+100
+200
+400
+500DependenceofFl-score(weighted)fromnumberofestimators
+0.980
+0.975
+0.970
+weighted
+0.965
+a
+0.960
+-scor
+0.955
+0.950
+0.945
+Hard Voting
+Soft Voting
++
+0.940
+4
+5
+6
+7
+8
+9
+10
+Numberofestimatorsof science and technology development] (in Russian). Yekaterinburg: Ural 
+University Publishing House. doi: https://doi.org/10.19181/smtp.2019.1.1.5
+3. Aljuaid, H., R. Iftikhar, S. Ahmad, M. Asif, and M. T. Afzal. 2021. “Important 
+citation identification using sentiment analysis of in-text citations.” Telematics 
+and Informatics 56: 101492. doi: https://doi.org/10.1016/j.tele.2020.101492
+4. Arsyad, S., S. Ramadhan, and I. Maisarah, 2020. “The rhetorical problems 
+experienced by Indonesian lecturers in social sciences and humanities in writing 
+research articles for international journals.” The Asian Journal of Applied 
+Linguistics 7(1): 116–129.
+5. Brown, T. B., B. Mann, N. Ryder, M. Subbiah, J. Kaplan, P. Dhariwal, A. 
+Neelakantan, et al. 2020. “Language models are few-shot learners”. arXiv 
+preprint arXiv:2005.14165.
+6. Chawla, N. V., K. W. Bowyer, L. O. Hall, and W. P. Kegelmeyer. 2002. “SMOTE: 
+synthetic minority over-sampling technique.” Journal of artificial intelligence 
+research 16: 321–357. doi: https://doi.org/10.1613/jair.953
+7. Cohan, A., F. Dernoncourt, D. S. Kim, T. Bui, S. Kim, W. Chang, and N. 
+Goharian. 2018. “A discourse-aware attention model for abstractive 
+summarization of long documents.” arXiv preprint arXiv:1804.05685
+8. Devlin, J., M. W. Chang, K. Lee, and K. Toutanova. 2018 “Bert: Pre-training of deep 
+bidirectional transformers for language understanding.” arXiv preprint 
+arXiv:1810.04805.
+9. Emerson, L., M. T. Rees, and B. MacKay. 2005. “Scaffolding academic integrity: 
+Creating a learning context for teaching referencing skills.” Journal of university
+teaching & learning practice 2 (3): 3. doi: https://doi.org/10.14453/jutlp.v2i3.3
+10. ExplainaBoard — Named Entity Recognition, 
+http://explainaboard.nlpedia.ai/leaderboard/task-ner/, last accessed 2021/12/13.
+11. Färber, M., and S. Ashwath. 2019. "Determining how citations are used in citation 
+contexts." International Conference on Theory and Practice of Digital Libraries.
+Springer, Cham. doi: https://doi.org/10.1007/978-3-030-30760-8_38
+12. Fiok, K., et al. 2020. "Comparing the quality and speed of sentence classification 
+with modern language models." Applied Sciences 10.10: 3386. doi: 
+https://doi.org/10.3390/app10103386
+13. Fu, J., X. Huang, and P. Liu. 2021. “SpanNer: Named Entity Re-/Recognition as 
+Span Prediction.” arXiv preprint arXiv:2106.00641.
+
+The presence of a citation
+Thesentencecontains
+There is no citation marker
+marker in the sentence
+a citation marker
+in the sentence
+does not matter
+Positive sample
+Negative sample
+S;Bag of Words S;
+Estimators S;
+Documento
+Sentence
+OH
+Sampling strategies
+BoW1
+Ensemblevoting Si
+Estimator
+Sentence
+Si-n
+P1(S)
+Sentence
+BoW2
+P2(S)
+-P(S)→
+S;
+Si-n
+Estimator2
+Sentence
+P3(S)
+Si+m
+888
+1+m
+Sentence
+BoW3
+Estimator3
+Documentn
+Sentence
+SentenceStrategy#0
+Strategy #2
+Strategy #3
+0.016
+301.5 ± 154.0
+105.5 ± 39.2
+157.0 ± 49.5
+0.006
+273.9 ± 170.8
+0.0200
+103.0 ± 57.2
+0.014
+155.1 ± 76.1
+0.0175
+0.005
+0.012
+0.0150
+0.004
+0.010 
+0.0125
+0.008
+E00'0
+0.0100
+0.0075
+0.006
+0.002
+0.0050 -
+0.004
+0.001
+0.0025
+0.002
++000°0
+0.0000
+0.000 +
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500
+Strategy #5
+Strategy #7
+Strategy #9
+0.010 
+260.6 ± 74.9
+362.4 ± 92.6
+465.4 ± 113.2
+259.7 ± 105.7
+0.007
+364.4 ± 127.5
+0.006
+469.7 ± 149.2
+800°0
+0.006
+0.005
+0.005
+0.006
+0.004
+0.004
+E00'0
+0.004
+E000
+0.002
+0.002 
+0.002
+0.001 
+0.001
+000°0
+0.000 +
+0.000 +
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+0
+100
+200
+400
+500DependenceofFl-score(weighted)fromnumberofestimators
+0.980
+0.975
+0.970
+weighted
+0.965
+a
+0.960
+-scor
+0.955
+0.950
+0.945
+Hard Voting
+Soft Voting
++
+0.940
+4
+5
+6
+7
+8
+9
+10
+Numberofestimators14. Gallant, S. I. 1991. “A practical approach for representing context and for 
+performing word sense disambiguation using neural networks.” Neural 
+Computation 3(3): 293–309. doi: https://doi.org/10.1162/neco.1991.3.3.293
+15. Glazkova, A. V. 2020. “Topical Classification of Text Fragments Accounting for 
+Their Nearest Context.” Automation and Remote Control 81: 2262–2276. doi: 
+https://doi.org/10.1134/S0005117920120097
+16. Gray, K., C. Thompson, R. Clerehan, J. Sheard, and M. Hamilton. 2008. “Web 2.0 
+authorship: Issues of referencing and citation for academic integrity.” The 
+Internet and Higher Education 11(2): 112–118. doi: 
+https://doi.org/10.1016/j.iheduc.2008.03.001
+17. Huang, E. H., R. Socher, C. D. Manning, and A. Y. Ng. 2012. “Improving word 
+representations via global context and multiple word prototypes.” Proceedings 
+of the 50th Annual Meeting of the Association for Computational Linguistics 1: 
+Long Papers: 873–882.
+18. John, M., and J. S. Jayasudha. 2017. “Enhancing Performance of Deep Learning 
+Based Text Summarizer”. Int. J. Appl. Eng. Res. 12 (24): 15986–15993.
+19. Li, B. 2021. “Named entity recognition in the style of object detection”. arXiv 
+preprint arXiv:2101.11122.
+20. Li, Y., H. Guo, Q. Zhang, M. Gu, and J. Yang. 2018. “Imbalanced text sentiment 
+classification using universal and domain-specific knowledge.” Knowledge-
+Based Systems 160: 1–15. doi: https://doi.org/10.1016/j.knosys.2018.06.019
+21. Luo, Y., H. Feng, X. Weng, K. Huang, and H. Zheng. 2019. “A novel oversampling 
+method based on SeqGAN for imbalanced text classification.” 2019 IEEE 
+International Conference on Big Data (Big Data): 2891–2894. IEEE. doi: 
+10.1109/BigData47090.2019.9006138
+22. Merton, R. K. 1973. The sociology of science: Theoretical and empirical 
+investigations. University of Chicago press.
+23. Pears, R., and G. J. Shields. 2019. Cite them right: the essential referencing guide. 
+Macmillan International Higher Education.
+24. Prester, J., G. Wagner, G. Schryen, and N. R. Hassan. 2021. “Classifying the 
+ideational impact of information systems review articles: A content-enriched 
+deep learning approach.” Decision Support Systems 140: 113432. doi: 
+https://doi.org/10.1016/j.dss.2020.113432
+
+The presence of a citation
+Thesentencecontains
+There is no citation marker
+marker in the sentence
+a citation marker
+in the sentence
+does not matter
+Positive sample
+Negative sample
+S;Bag of Words S;
+Estimators S;
+Documento
+Sentence
+OH
+Sampling strategies
+BoW1
+Ensemblevoting Si
+Estimator
+Sentence
+Si-n
+P1(S)
+Sentence
+BoW2
+P2(S)
+-P(S)→
+S;
+Si-n
+Estimator2
+Sentence
+P3(S)
+Si+m
+888
+1+m
+Sentence
+BoW3
+Estimator3
+Documentn
+Sentence
+SentenceStrategy#0
+Strategy #2
+Strategy #3
+0.016
+301.5 ± 154.0
+105.5 ± 39.2
+157.0 ± 49.5
+0.006
+273.9 ± 170.8
+0.0200
+103.0 ± 57.2
+0.014
+155.1 ± 76.1
+0.0175
+0.005
+0.012
+0.0150
+0.004
+0.010 
+0.0125
+0.008
+E00'0
+0.0100
+0.0075
+0.006
+0.002
+0.0050 -
+0.004
+0.001
+0.0025
+0.002
++000°0
+0.0000
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+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500
+Strategy #5
+Strategy #7
+Strategy #9
+0.010 
+260.6 ± 74.9
+362.4 ± 92.6
+465.4 ± 113.2
+259.7 ± 105.7
+0.007
+364.4 ± 127.5
+0.006
+469.7 ± 149.2
+800°0
+0.006
+0.005
+0.005
+0.006
+0.004
+0.004
+E00'0
+0.004
+E000
+0.002
+0.002 
+0.002
+0.001 
+0.001
+000°0
+0.000 +
+0.000 +
+0
+100
+200
+300
+400
+500
+0
+100
+200
+300
+400
+500
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+100
+200
+400
+500DependenceofFl-score(weighted)fromnumberofestimators
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+0.965
+a
+0.960
+-scor
+0.955
+0.950
+0.945
+Hard Voting
+Soft Voting
++
+0.940
+4
+5
+6
+7
+8
+9
+10
+Numberofestimators25. Taha, A. Y., Tiun, S., Abd Rahman, A. H., & Sabah, A. 2021. “Multilabel Over-
+sampling and Under-sampling with Class Alignment for Imbalanced Multilabel 
+Text Classification.” Journal of Information and Communication Technology 20
+(3): 423–456. doi: https://doi.org/10.32890/jict2021.20.3.6
+26. Varanasi, K. K., T. Ghosal, P. Tiwary, and M. Singh. 2021. “Iitp-cuni@ 3c: 
+Supervised approaches for citation classification (task a) and citation 
+significance detection (task b).” Proceedings of the Second Workshop on 
+Scholarly Document Processing: 140-145.
+27. Wang, X., Y. Jiang, N. Bach, T. Wang, Z. Huang, F. Huang, and K. Tu. 2021. 
+“Improving Named Entity Recognition by External Context Retrieving and 
+Cooperative Learning.” arXiv preprint arXiv:2105.03654.
+28. Ziyadi, M., Y. Sun, A. Goswami, J. Huang, and W. Chen. 2020. “Example-based 
+named entity recognition.” arXiv preprint arXiv:2008.10570.
+
+The presence of a citation
+Thesentencecontains
+There is no citation marker
+marker in the sentence
+a citation marker
+in the sentence
+does not matter
+Positive sample
+Negative sample
+S;Bag of Words S;
+Estimators S;
+Documento
+Sentence
+OH
+Sampling strategies
+BoW1
+Ensemblevoting Si
+Estimator
+Sentence
+Si-n
+P1(S)
+Sentence
+BoW2
+P2(S)
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+Estimator2
+Sentence
+P3(S)
+Si+m
+888
+1+m
+Sentence
+BoW3
+Estimator3
+Documentn
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+SentenceStrategy#0
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+Strategy #3
+0.016
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+0.0175
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+E00'0
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++000°0
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+100
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+0.007
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+0.006
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+800°0
+0.006
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+0.004
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+E00'0
+0.004
+E000
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+000°0
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+0.945
+Hard Voting
+Soft Voting
++
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+4
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+7
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+10
+Numberofestimators
\ No newline at end of file
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+page_content=' Author of over 70 scientific publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content=' Graduate of the Faculty of Computational Mathematics and Cybernetics,  Lomonosov Moscow State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Research interests: use of intelligent algorithms in  applied information systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content=' We automatically determine the optimal sampling strategy for a  given text collection by implementing an ensemble voting when classifying the  same data sampled in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='  Keywords: text sampling, sampling strategy, citation analysis, bibliographic link  prediction, sentence classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Introduction  Scientific research is impossible without correlating the results obtained with the work  of other scientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Other works should be mentioned by inserting bibliographic links in  the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Experts in scientometrics rationalize the need to establish such links between  studies and formulate various citation theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The normative theory of citation, which draws on the principles of scientific  ethics formulated by Merton (1973), assumes that references in scientific papers are  made in order to indicate the works that are the basis for research or topically related,  describe the research methods used and are necessary to discuss the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The presence of a citation  Thesentencecontains  There is no citation marker  marker in the sentence  a citation marker  in the sentence  does not matter  Positive sample  Negative sample  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='  Documento  Sentence  OH  Sampling strategies  BoW1  Ensemblevoting Si  Estimator  Sentence  Si-n  P1(S)  Sentence  BoW2  P2(S)  P(S)→  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='940  4  5  6  7  8  9  10  NumberofestimatorsThus, the beneficiary of scientific citation correctness is the entire scientific  community, both researchers who create articles on their results and administrators who  monitor achievements in various scientific fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Mentioning relevant and remarkable  results  of other scientists  is  one of the basic requirements  in the construction  of  scientific texts, in particular from the point of view of the editors of scientific journals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  These requirements are noted in academic writing guidelines (Emerson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Gray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Pears and Shields 2019) and are confirmed in practice, for example, by  the results of studies of publication activity in top-rated international journals (Arsyad et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Authors of scientific papers choose the sources for citation and positions for the  links by themselves and at present, this process is not automated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' In this work, we  investigate the possibility of creating a recommendation algorithm that allows one to  find missing bibliographic references in a scientific article, that is, to identify those text  fragments where it is necessary to mention another research work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' For this purpose, we  estimate  the  probability  of  link  presence  in  fragments  of  the  text  using a semi-  supervised machine learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' The formal statement of the problem under  consideration is the following: it is required to automatically find in the text of a  scientific article those fragments (sentences) where the link is absent, but necessary,  using a set of labeled fragments with and without links as training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The task of classifying text fragments in relation to the presence of the links in  them is methodologically similar to the task of Sentiment Analysis, in which texts are  automatically classified, mainly as positive and negative, according to their emotional  characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' In  addition  to  dividing  fragments  into  positive  and  negative,  the  sentiment analysis approach is used to distinguish other classes, including citation  significance detection (Aljuaid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Prester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Varanasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The presence of a citation  Thesentencecontains  There is no citation marker  marker in the sentence  a citation marker  in the sentence  does not matter  Positive sample  Negative sample  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='Bag of Words S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Estimators S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Documento  Sentence  OH  Sampling strategies  BoW1  Ensemblevoting Si  Estimator  Sentence  Si-n  P1(S)  Sentence  BoW2  P2(S)  P(S)→  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content=' NER can be performed in two  stages:  identifying  fragments  with  a  high  probability  of  containing  entities,  and  determination of the exact positions of these entities (Ziyadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Li 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Some  methods of NER also take into account the context of entities, both local and global, or  external (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The task of sentence classification accounting for their nearest context has been  discussed in a number of studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Fiok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' (2020) used contextualized embeddings  created by language models, which high quality comes at the price of speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Glazkova  (2020) studied topical classification and showed that models taking context as input  performed better than context-free models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' In those works, context size is determined  once based on some bias and may not be optimal for a certain text corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The method  introduced  in  this  work also can  be considered  as  kind  of a  resampling technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Until now resampling has been used mainly for the purpose of  balancing the class distribution in training datasets in order to improve the accuracy of  class prediction, which is negatively affected by imbalanced data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Resampling methods  are  classified  into  three  types,  namely  undersampling,  oversampling  and  hybrid  techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content=', Maya  and Jayasudha 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Akkasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' 2017), while oversampling either replicates existing  instances of a minority class or creates new ones (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The presence of a citation  Thesentencecontains  There is no citation marker  marker in the sentence  a citation marker  in the sentence  does not matter  Positive sample  Negative sample  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='  Estimators S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Documento  Sentence  OH  Sampling strategies  BoW1  Ensemblevoting Si  Estimator  Sentence  Si-n  P1(S)  Sentence  BoW2  P2(S)  P(S)→  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='945  Hard Voting  Soft Voting  +  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='940  4  5  6  7  8  9  10  Numberofestimatorsfragment contains a bibliographic link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' The task of this study is to find a strategy for the  construction samples of fragments, which gives the highest accuracy in determining the  labels of the class by a certain classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The hypothesis of our study is the following: text sampling strategies that take  into account the context increase the accuracy of sentence classification used to predict  missing bibliographic links in scientific articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  We suggest that a positive sample consists of a bibliographic link surrounded by  its  context  from  the  original  text,  and  a  negative  sample  is  a fragment  with  no  bibliographic link in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' In order to avoid duplication of samples, we consider a sentence  with two or more links only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' The context of the link is limited to the sentence  containing it, or the context is extended and it includes neighboring sentences as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The best option is when the boundaries of a link context coincide with the  boundaries of the complete author’s statement to which this link belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' In this case, a  semantic unit of text can be either one or several sentences, which makes it difficult to  set  the  size  of  the  context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Nevertheless,  to  approach  the  specified  goal in  the  proposed algorithm, as a context we consider a fragment which size is determined by  the number of sentences, and not words, unlike neural network algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Thus, in our  algorithm context is formed on the basis of natural structural units of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The feature space is constructed automatically based on vocabulary statistics  within the Bag-of-Words model (BoW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' The vocabulary of the model includes words  and  all  the  original  punctuation  marks  and  typographical  symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' As  additional  features, we consider named entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Algorithm  The algorithm consists of the following stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The presence of a citation  Thesentencecontains  There is no citation marker  marker in the sentence  a citation marker  in the sentence  does not matter  Positive sample  Negative sample  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='Bag of Words S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Estimators S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Documento  Sentence  OH  Sampling strategies  BoW1  Ensemblevoting Si  Estimator  Sentence  Si-n  P1(S)  Sentence  BoW2  P2(S)  P(S)→  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Si-n  Estimator2  Sentence  P3(S)  Si+m  888  1+m  Sentence  BoW3  Estimator3  Documentn  Sentence  SentenceStrategy#0  Strategy #2  Strategy #3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='945  Hard Voting  Soft Voting  +  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='940  4  5  6  7  8  9  10  Numberofestimators1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Text preprocessing Text cleaning: removal of service characters (tabulation, line feed, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' ), words  (journal titles, ISBN, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' ), and sections (funding, reference);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Tokenization:  o  splitting text into sentences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  o  terms normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Data labeling For each document (journal article), beginning and end marks are added;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' For each sentence:  o  if the  sentence  contains  a bibliographic  link (citation  marker),  it  is  labeled as belonging to the class "With links";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  o  if there is no citation marker in the sentence, it gets "Without links" class  label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  o  After labeling sentences, citation markers are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Named Entities processing Detection of named entities in the text;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Replacement of named entities with special marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Construction of samples  Samples are constructed in different ways depending on the class (positive or negative): To form a positive sample, we take a sentence “With links” and add n previous  sentences and  m subsequent sentences, all sentences are taken in the original  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The presence of a citation  Thesentencecontains  There is no citation marker  marker in the sentence  a citation marker  in the sentence  does not matter  Positive sample  Negative sample  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='  Estimators S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='945  Hard Voting  Soft Voting  +  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='940  4  5  6  7  8  9  10  Numberofestimators\uf0b7  A negative sample is constructed of k sentences "Without links" in a row in the  original text (adjacent sentences), where k = n+1+m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The visualization of the contents of the samples is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Construction of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Classification of samples Balancing  the  class  distribution  in  the  training  set  is  done  by  random  undersampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' For each sample, a vector model is built using count vectorizer as the fastest and  most computationally effective text representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Vectorized set of samples is processed using a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Optimal sampling strategy determination  Further, an ensemble method is used to automatically determine the optimal sampling  strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' We give the same data sampled in different ways to the estimators of the same  type and implement a voting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The  flowchart  of  the  whole  algorithm  is  shown  in  Figure  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Each  BoWj  corresponds to one sampling strategy, and for each sampling strategy, we run its own  estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' All  the  estimators  implement  the  same  classification  method  but  take  different types of samples as input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The presence of a citation  Thesentencecontains  There is no citation marker  marker in the sentence  a citation marker  in the sentence  does not matter  Positive sample  Negative sample  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='Bag of Words S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Estimators S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Documento  Sentence  OH  Sampling strategies  BoW1  Ensemblevoting Si  Estimator  Sentence  Si-n  P1(S)  Sentence  BoW2  P2(S)  P(S)→  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Si-n  Estimator2  Sentence  P3(S)  Si+m  888  1+m  Sentence  BoW3  Estimator3  Documentn  Sentence  SentenceStrategy#0  Strategy #2  Strategy #3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='5 ± 154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='0  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='5 ± 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='2  157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='0 ± 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='1 ± 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='940  4  5  6  7  8  9  10  NumberofestimatorsFigure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' The algorithm flowchart for classifying sentences into the classes "With links"  and "Without links" using various sampling strategies (for n=m=1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Experiment  To test the hypothesis experimentally, we took the dataset of STEM journal articles,  collected from scientific repository arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='org by Cohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Documents of this  dataset contain only texts, while figures and tables are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Math formulas and  citation  markers  are  replaced  with  special  tokens  @xmath<number> and  @xcite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Documents contain only the sections up to the conclusion section and all sections after  the conclusion are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The size of the dataset is the following: the number of documents – 215K, the  average document length – 4938 words, the average summary length – 220 words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The files are in jsonlines format where each line is a json object corresponding  to one scientific article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Each line contains an abstract, sections and a body of the article,  and all these texts are sentence tokenized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  In our experiment we consider sentences that are more than 30 words long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' With  this restriction we got the set of 458774 sentences in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The presence of a citation  Thesentencecontains  There is no citation marker  marker in the sentence  a citation marker  in the sentence  does not matter  Positive sample  Negative sample  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='  Estimators S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Documento  Sentence  OH  Sampling strategies  BoW1  Ensemblevoting Si  Estimator  Sentence  Si-n  P1(S)  Sentence  BoW2  P2(S)  P(S)→  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='940  4  5  6  7  8  9  10  NumberofestimatorsSentences containing citation markers @xcite are assigned to the positive class  ("With links"), and after that citation markers are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content=' The  ratio  of  classes  is:  24%  –  positive sentences, 76% – negative sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' This is assumed as sampling strategy #0,  and the classification result on data sampled that way is considered as a baseline:  classification accuracy with sampling strategy #0 measured by F1-score is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='  After establishing the baseline, we test various strategies of data sampling in  order to improve the classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' The main idea of sampling is to take into  account  some  context  of  the  sentences  with  a  link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Different  strategies  of sampling assume various directions, positions, and sizes of the context, determined  by a number of surrounding sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' In different sampling strategies, each sentence [i]  is included in different types (variants) of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  In the experiment we test 10 strategies with the following parameters: n: [0, 1, 2  3 4 5], m: [0, 1, 2, 3, 4], k: [1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' All the sample types corresponding to the chosen  sampling strategies are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Sampling strategies tested in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='Sampling strategy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='(sample  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='constructing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='algorithm)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='Number of sentences per sample  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='Positive  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='Negative  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='Sentence [i]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='2640  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='12352  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='Sentences [i: i +2]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='Sentences [i -1: i+1]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='Sentences [i -1: i+2]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='Sentences [i -3: i+2]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='The presence of a citation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='Thesentencecontains  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='  Documento  Sentence  OH  Sampling strategies  BoW1  Ensemblevoting Si  Estimator  Sentence  Si-n  P1(S)  Sentence  BoW2  P2(S)  P(S)→  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Si-n  Estimator2  Sentence  P3(S)  Si+m  888  1+m  Sentence  BoW3  Estimator3  Documentn  Sentence  SentenceStrategy#0  Strategy #2  Strategy #3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='9  362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='2  259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='2  800°0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content=' The distribution of the length as a number of words in positive and negative  samples of different types (‘salmon’ color refers to the positive class, ‘olive’ refers to  the negative one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='  Vector representation is build using CountVectorizer method of the Scikit-learn  library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='2  157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='9581  It can be seen from the table that the best result is achieved using the sampling  strategy  #7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' A  further  increase  in  the  number  of  sentences  per  sample  does  not  significantly improve the accuracy, since it tends to the asymptote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  For each sentence we compare the result of classification obtained with all the  sampling strategies and further improve it by voting procedure, both soft and hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' For  soft voting, we set the threshold value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='5 for the mean predicted probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' For hard  voting, we summarize all the predicted probabilities and compare the sum with the  The presence of a citation  Thesentencecontains  There is no citation marker  marker in the sentence  a citation marker  in the sentence  does not matter  Positive sample  Negative sample  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='  Estimators S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Documento  Sentence  OH  Sampling strategies  BoW1  Ensemblevoting Si  Estimator  Sentence  Si-n  P1(S)  Sentence  BoW2  P2(S)  P(S)→  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content=' We test different numbers of estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' To form combinations of  sampling strategies we start from the group of estimators #7, #8, and #9 and then add  estimators one by one in reverse order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' The voting results depending on the number of  estimators considered are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' The result of classification with hard and soft voting depending on the number  of estimators included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The graph demonstrates that the voting procedure further improves classification  performance and increases F1-score by 1,5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' With all the combinations of estimators  tested the result is consistently high, but the best result (>98%) is achieved with hard  voting of 7,8 or 10 estimators classifying long samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Result  The  formulated  research  hypothesis  has  been  confirmed  experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' We  have  The presence of a citation  Thesentencecontains  There is no citation marker  marker in the sentence  a citation marker  in the sentence  does not matter  Positive sample  Negative sample  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='Bag of Words S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Estimators S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Documento  Sentence  OH  Sampling strategies  BoW1  Ensemblevoting Si  Estimator  Sentence  Si-n  P1(S)  Sentence  BoW2  P2(S)  P(S)→  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Si-n  Estimator2  Sentence  P3(S)  Si+m  888  1+m  Sentence  BoW3  Estimator3  Documentn  Sentence  SentenceStrategy#0  Strategy #2  Strategy #3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='2  157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='945  Hard Voting  Soft Voting  +  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='940  4  5  6  7  8  9  10  Numberofestimatorsshown that the choice of the sampling strategy affects the result of text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The  baseline  is  established  with  sampling  strategy  #0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' In  this  case,  the  classification performance measured using the F1-score is only 79 %, which is not  sufficient for practical use in industrial information systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The improvement is achieved due to the data sampling strategy which assumes  automatic determination of the optimal sample type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' That is provided by applying the  voting procedure to the decisions made by different estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' The proposed algorithm  shows 98% accuracy (F1-score), which is comparable to the state-of-the-art results for  NER using automatic classification and other text classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' It is important  that  the  proposed  algorithm  provides  high  accuracy  but  doesn’t  require  huge  computational resources to be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Conclusion  The paper proposes a new method of determining the probability of a bibliographic link  in fragments of a scientific article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' The approach assumes sentence classification with  ensemble voting, in which different data sampling strategies correspond to estimators  implementing the same classification method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' The problem statement made by the  authors is close to well-studied areas NER and sentiment analysis but is new from  the real application point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The main innovation of the proposed method is finding the link context that  maximally affects the probability of detecting a missing bibliographic link in a sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  In  the  proposed  algorithm,  the  best  size  and  position  of  context  are  determined  automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' The size is based on the boundaries of semantic units of the text and is  measured by the number of sentences, not words, thus we utilize the fact that a sentence  is a more semantically capacious (meaningful) unit than a word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Most existing text  classification methods do not assume fragment context as significantly important, but  The presence of a citation  Thesentencecontains  There is no citation marker  marker in the sentence  a citation marker  in the sentence  does not matter  Positive sample  Negative sample  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='Bag of Words S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Estimators S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Documento  Sentence  OH  Sampling strategies  BoW1  Ensemblevoting Si  Estimator  Sentence  Si-n  P1(S)  Sentence  BoW2  P2(S)  P(S)→  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Si-n  Estimator2  Sentence  P3(S)  Si+m  888  1+m  Sentence  BoW3  Estimator3  Documentn  Sentence  SentenceStrategy#0  Strategy #2  Strategy #3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='5 ± 154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='0  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='5 ± 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='  The accuracy of the proposed algorithm reaches 98% (F1-score).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' It is important  to note the high computational efficiency of the described method in comparison with  convolutional artificial neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' This advantage is achieved due to the bigger  size of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' The investigated approach to text analysis expands the principle of the  attention mechanism aimed at training a language model to understand the impact of  global and local context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Automatic determination of the context boundaries correlates  with the idea of automatic selection of significant features in artificial neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  The  proposed  method  can  be  used  in  recommendation  engines  of  applied  intelligent information systems, including assistance for constructing documents and  composing texts with probable links to other documents, or help in checking the  document correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Such functions are useful in many fields e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content="  In accordance with the company's policy, we do not publish the source code." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Akkasi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Varoğlu, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Dimililer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' “Balanced undersampling: a novel  sentence-based undersampling method to improve recognition of named entities  in chemical and biomedical text.” Applied Intelligence 48(8): 1965–1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' doi:  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='1007/s10489-017-0920-5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Akoev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=', V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Markusova, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Moskaleva, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Pislyakov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content=' Rukovodstvo po Naukometrii:  Indikatori Razvitiia Naukii Tehnologii [Handbook for Scientometrics: Indicators  The presence of a citation  Thesentencecontains  There is no citation marker  marker in the sentence  a citation marker  in the sentence  does not matter  Positive sample  Negative sample  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='Bag of Words S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Estimators S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Documento  Sentence  OH  Sampling strategies  BoW1  Ensemblevoting Si  Estimator  Sentence  Si-n  P1(S)  Sentence  BoW2  P2(S)  P(S)→  S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='  Si-n  Estimator2  Sentence  P3(S)  Si+m  888  1+m  Sentence  BoW3  Estimator3  Documentn  Sentence  SentenceStrategy#0  Strategy #2  Strategy #3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='5 ± 154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='0  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='2  157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='0 ± 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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+page_content='1 ± 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQftP2f/content/2301.01673v1.pdf'}
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